A SPATIO-TEMPORAL MODEL FOR WHITE MATTER
TRACTOGRAPHY IN DIFFUSION TENSOR IMAGING

By

Juna Goo

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Statistics — Doctor of Philosophy

2020

ABSTRACT

A SPATIO-TEMPORAL MODEL FOR WHITE MATTER TRACTOGRAPHY IN

DIFFUSION TENSOR IMAGING

By

Juna Goo

This dissertation focuses on the theoretical and applied aspects of a spatio-temporal

modeling for the reconstruction of in-vivo fiber tracts in white matter when a single brain is

scanned with magnetic resonance imaging (MRI) on several occasions. The objective of this

research is twofold: one is how to estimate the spatial trajectory of a nerve fiber bundle at a

given time point in the presence of measurement noise and the other is how to incorporate

a progressive deterioration of brain connectivity into a hypothesis test.

This dissertation leverages the spatio-temporal behavior of water diffusion in a region

of the brain where the estimation of fiber trajectories is made from smoothing the time-

varying diffusion tensor field via the Nadaraya-Watson type kernel regression estimator to

its eigenvector field. The estimated fiber pathway takes the form of confidence ellipsoids

given the estimates of mean and covariance functions.

Furthermore, this dissertation proposes a hypothesis test in which the null hypothesis

states that true fiber trajectories remain the same over a certain time interval. This null

hypothesis indicates no substantial pathological changes of fiber pathways in that region of

the brain during the observed time period. The proposed test statistic is shown to follow

the limiting chi-square distribution under the null hypothesis. The power of the test is

illustrated via Monte Carlo simulations. Lastly, this dissertation demonstrates the test can

also be applied to a real longitudinal DTI study of a single brain repeatedly measured across

time.

This dissertation is dedicated to my beloved parents and sister.

iii

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor, Dr. Lyudmila Sakhanenko, for

her detailed guidance and constructive feedback on this dissertation. I would like to thank

the members of my dissertation committee, Drs. Yimin Xiao, Chih-Li Sung, and David Zhu

for their time. In particular, I appreciate Dr. David Zhu for providing real DTI data sets

and sharing his expertise with me.

I also thank my dear friends who have encouraged me throughout the past years in

graduate school, and last but not least, I thank my family for all the support and love.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Estimation and Hypothesis Test . . . . . . . . . . . . . . . . . . .
2.1 True parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Nadaraya-Watson type kernel estimator . . . . . . . . . . . . . . . . . . . . .
2.3 Uniform consistency of the estimators . . . . . . . . . . . . . . . . . . . . . .
2.4 Weak convergence of the sequence of stochastic processes . . . . . . . . . . .
2.5 Hypothesis test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 Pseudo Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 Theorem 2.4.1.
3.2 Theorem 2.5.1.

Chapter 4

Simulation and Application to Real Data . . . . . . . . . . . . .
4.1 Artificial data: semicircular trajectory over time . . . . . . . . . . . . . . . .
4.2 Real longitudinal DTI data
. . . . . . . . . . . . . . . . . . . . . . . . . . .

5
5
7
12
19
28

36
36
40

47
47
55

Chapter 5 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . .

63

Chapter 6 Proofs of Theorem 2.4.1., Theorem 2.4.2., and Theorem 2.4.3. 66
66
72
76
84
89
93

6.1 Asymptotic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Mean function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Covariance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Convergence of finite-dimensional distributions . . . . . . . . . . . . . . . . .
6.5 Asymptotic equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

v

LIST OF TABLES

Table 4.1: Monte Carlo simulation-based power analysis when HA1

is true . . . . . .

51

Table 4.2: Monte Carlo simulation-based power analysis when HA2

is true . . . . . .

52

Table 4.3: Monte Carlo simulation-based power analysis when HA3

is true . . . . . .

54

Table 4.4: Result of test statistics in both ROIs

. . . . . . . . . . . . . . . . . . . .

57

vi

LIST OF FIGURES

Figure 1.1: A geometric representation of the 3 × 3 diffusion tensor D in DTI with
. . . . . . . . . .

positive eigenvalues λ1, λ2 and λ3 in descending order

Figure 4.1: A solid blue line indicates the estimated trajectory given the 5th time
point under H0, whereas blue dotted lines represent the pointwise 95%
confidence ellipsoids along the estimated trajectory. . . . . . . . . . . . .

Figure 4.2: At each c value the 5th estimated trajectory and its 95% confidence el-
lipsoids under HA1
, both colored in red, are overlaid with those of the
reference value (c = 0.5) under H0, colored in blue. All 3D figures are
projected onto the xy-plane. . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.3: At each r value the 5th estimated trajectory and its 95% confidence el-
, both colored in red, are overlaid with those of the
lipsoids under HA2
reference value (r = 0.5) under H0, colored in blue. All 3D figures are
projected onto the xy-plane. . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.4: The estimated trajectory colored in red is projected onto the xy-plane
over the observed period from July 2014 to December 2018 in ascending
order in the anterior part of the CC.
. . . . . . . . . . . . . . . . . . . .

2

49

50

53

58

Figure 4.5: Anterior part of the CC scanned in December 2014 . . . . . . . . . . . .

59

Figure 4.6: The estimated trajectory colored in red is projected onto the xy-plane
over the observed period from July 2014 to December 2018 in ascending
order in the posterior part of the CC. . . . . . . . . . . . . . . . . . . . .

60

Figure 4.7: Posterior part of the CC scanned in December 2018 . . . . . . . . . . . .

61

Figure 4.8: Left isthmus of the cingulate cortex scanned in July 2014 . . . . . . . . .

62

vii

LIST OF ALGORITHMS

Algorithm 3.1: Fiber Trajectory Estimation . . . . . . . . . . . . . . . . . . . . . . .

Algorithm 3.2: Pre-step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Algorithm 3.3: Mean Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Algorithm 3.4: Noise Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Algorithm 3.5: Covariance Function . . . . . . . . . . . . . . . . . . . . . . . . . . .

Algorithm 3.6: Fiber Trajectory with Confidence Ellipsoids . . . . . . . . . . . . . .

Algorithm 3.7: Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Algorithm 3.8: Nested Variance Function and Nested Covariance Function . . . . . .

Algorithm 3.9: Covariance Function . . . . . . . . . . . . . . . . . . . . . . . . . . .

Algorithm 3.10: Hypothesis Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

37

38

38

39

40

43

44

45

46

viii

KEY TO SYMBOLS

(cid:62) : tensor transpose
Id : d × d identity matrix
0d : d × 1 vector with all elements being 0
1d : d × 1 vector with all elements being 1
|u| : L2 norm of vector u = [x1 x2 . . . xd t](cid:62)
(cid:107)f (u)(cid:107)G : the supremum of f over the compact set G, i.e., supu∈G|f (u)|
I{a≤u≤b} : indicator function, i.e., 1 if a ≤ u ≤ b; otherwise 0
C([a, b], Rd) : the space of all Rd-valued continuous functions on the interval [a, b]
Ck(G, Rd) : the space of all Rd-valued continuous and k times differentiable functions on G
Xn ⇒ X : Xn converges in distribution to X

ix

Chapter 1

Introduction

In the mid 1980s and early 1990s, diffusion tensor imaging (DTI) emerged from conventional

magnetic resonance imaging (MRI) with the concept of microscopic Gaussian diffusion of

water molecules within brain tissue. DTI focuses on the movement of water molecules within

brain tissue (in particular, white matter containing mainly nerve fibers) that diffuses with

different rates depending on the angle between orientation of fiber tracts and magnetic field

gradient directions (spatial variation in a magnetic field gradient). Since DTI generates

a collection of images obtained along at least six magnetic field gradient directions, DTI
characterizes the diffusion of water molecules within each voxel as a 3× 3 symmetric positive

definite diffusion tensor D in the x,y, and z directions such as



D =

 .

Dxx Dxy Dxz

Dyx Dyy Dyz

Dzx Dzy Dzz

In the presence of a single fiber bundle (uniformly oriented) within a voxel, DTI identifies

the shape of the diffusion tensor D as an ellipsoid where the three orthogonal eigenvectors of

the diffusion tensor are used as principal axes of the ellipsoid where lengths are scaled by the

corresponding three positive eigenvalues as diffusivities in the direction of each eigenvector

as illustrated in Figure 1.1. For an isotropic diffusion such as Brownian motion (the random

1

motion of water molecules without any obstacles), the diffusion tensor is visualized as a

sphere since diffusion is the same in all directions, resulting in a single diffusion coefficient.

For details see the pioneering papers by Le Bihan et al. (1986) and Basser et al. (1994).

Figure 1.1: A geometric representation of the 3 × 3 diffusion tensor D in DTI with positive
eigenvalues λ1, λ2 and λ3 in descending order

As a means of quantifying microscopic anisotropy of the diffusion tensor, there are scalar

DTI measures such as the fractional anisotropy (FA; the normalized standard deviation of

three eigenvalues), the mean diffusivity (MD; the average of three eigenvalues), the axial

diffusivity (AD; the largest eigenvalue), and the radial diffusivity (RD; the average of two

smaller eigenvalues perpendicular to the dominant eigenvector) as proposed by Basser (1995)

and Pierpaoli and Basser (1996). These scalar measures are often used for comparisons in a

region of interest (ROI) and also for visualization of the degree of anisotropic diffusivities in

brain images.

Furthermore, DTI is used to reconstruct in-vivo fiber tracts in white matter, which is

referred to as white matter fiber tractography. Over the past 20 years, many studies using

DTI have been conducted to develop the virtual map of fiber pathways in white matter. Such

DTI-based fiber tractography has been a popular technique for clinicians and researchers to

access the structure and connectivity of the brain. See relevant papers for white matter fiber

2

tractography by Mori and van Zijl (2002), Wakana et al. (2004), Assaf and Basser (2005),

and Behrens et al. (2007).

However, DTI has also been criticized for its low spatial resolution associated with its

inherent high noise level and its inability to distinguish complex fiber configurations such as

crossing, branching or kissing fibers. Since DTI is based on the assumption of a unimodal

anisotropic Gaussian distribution, DTI is not a proper technique when multiple fiber bundles

are present within a voxel. In the late 1990s and early 2000s, an advanced MRI technique

called high angular resolution diffusion imaging (HARDI) was introduced to overcome the

limitations of DTI. Compared to six non-collinear gradient directions used in DTI, HARDI

imposes diffusion-sensitizing gradients in a large number of gradient directions. HARDI

enables us to model multimodal diffusion, which reflects heterogenous fiber orientations in

white matter, with high image resolution using a high-order tensor (DTI uses a second-order

tensor). However, HARDI has longer acquisition time and higher computational complexity

compared to DTI. Detailed surveys can be found in Assemlal et al. (2011) and Jones et al.

(2013).

In this dissertation, we focus on an early application of statistical perspective in DTI

by Koltchinskii et al. (2007). Koltchinskii et al. (2007) identified the problem of tracing

fiber trajectories that were measured with random errors. They sought a solution to the

Cauchy problem for the first-order ordinary differential equation (ODE) with an initial value
x0 = x(0) ∈ X

dx(s)

ds

= v(x(s)), s ≥ 0,

where v was defined as a vector field in a bounded open set X ⊂ Rd and was observed with

additive random noise. Their methodology used the Nadaraya-Watson kernel regression

3

estimate of the vector field and then plugged the value into the ODE to estimate true fiber

trajectory. Later Carmichael and Sakhanenko (2015, 2016) extended the scope of their

methodology to include a tensor field in DTI and HARDI, respectively.

This dissertation further extends the statistical theory of white matter fiber tractography

to the realm of a time-dependent tensor field when a brain is scanned repeatedly over the

years. We aim to establish statistical reasoning with theoretical proofs in DTI-based tractog-

raphy using both spatial and temporal information, and to apply theoretical results where

brain tissue is progressively degenerative and connectivity shrinks over time. To be specific,

we address an estimation of the fiber spatial trajectory at a given time point in the presence

of noisy measurements arising from biological processes within living tissue and measurement

errors during image acquisitions. We also propose a statistical framework to incorporate a

patient’s progressive loss of brain connectivity, such as that caused by Alzheimer’s disease,

into the model.

In Chapter 2, we propose an estimation procedure which is extended to a time-dependent

tensor field model in DTI. Our estimators are for the true fiber spatial trajectory at a fixed

time point and the rate of change in true fiber trajectory with respect to time. We investigate

the asymptotic behavior of these estimators via weak convergence of stochastic processes.

Based on the asymptotic properties of the estimators, we provide a hypothesis test involving

the null hypothesis that the true fiber pathways remain the same over time, assuming a single

oriented fiber bundle exists within a voxel. In Chapter 3, we provide pseudo algorithms for

main theorems. In Chapter 4, Monte Carlo simulations and a real longitudinal DTI study

with a single healthy brain are presented. In Chapter 5, the limitations of this dissertation

and the directions for future research are addressed. In Chapter 6, detailed proofs of main

theorems are provided.

4

Chapter 2

Estimation and Hypothesis Test

2.1 True parameters

Let u = [x1 x2 . . . xd t](cid:62), where x = [x1 x2 . . . xd](cid:62) ∈ X , X is a d dimensional compact
hyperrectangle in Rd and t ∈ [0, T ], T > 0. Let G = X × [0, T ] for simplicity. At a fixed
u ∈ G, suppose D(u) is a d× d symmetric and positive definite tensor. Due to symmetry, the
2 × 1 vector as follows:
upper triangular elements of D(u), u ∈ G can be written as a d(d+1)
(cid:21)(cid:62)

(cid:20)

D(u) =

D11(u) D12(u)

. . . D1d(u) D22(u) D23(u)

. . . D2d(u)

. . . Ddd(u)

.

In the application of DTI, u is a value on the hypothetical d + 1 dimensional grid given the

parameter time t and D(u) is a second-order diffusion tensor at the value u. In practice,
d = 3. We assume that D ∈ C2(G, Rd(d+1)/2), which is twice continuously differentiable
on G. Note that its continuous differentiability implies locally Lipschitz continuity, and it

implies uniformly Lipschitz continuity on any compact set. D(u) is further assumed to have
a simple maximal eigenvalue at each u ∈ G since multiple eigenvalues, in general, are non-

differentiable. Therefore, the maximal eigenvalue and the corresponding eigenvector are also
of the class C2(G, Rd), and hence they belong to Lipschitz functions of x ∈ X uniformly in
t ∈ [0, T ]. For each u ∈ G, we denote λ(D(u)) as the largest eigenvalue of D(u) and v(D(u))

is the corresponding eigenvector which is normalized to unit length.

5

Then, there exists a unique solution to the first-order ordinary differential equation

(ODE) with the parameter time t ∈ [0, T ] and the initial value such that

x(s, t) = v(D(x(s, t), t)), s ∈ [0, S], x(0, t) = x0,

∂
∂s

(2.1)

starting at a time-invariant fixed location x0 ∈ X . Equivalently, the integral equation form

of this solution is

(cid:90) s

0

x(s, t) = x0 +

v(D(x(ξ, t), t))dξ, s ∈ [0, S], t ∈ [0, T ],

and its partial derivative with respect to time t, i.e., ∂

∂tx(s, t), exists. References on ODEs

can be found in Coddington and Levinson (1955).

For s ∈ [0, S], t ∈ [0, T ],

x(s, t + ∆t) − x(s, t)

∂
∂t

x(s, t) = lim
∆t→0

= lim
∆t→0

(cid:90) s

∆t

v(D(x(ξ, t + ∆t), t + ∆t)) − v(D(x(ξ, t), t))

0

∆t

by the dominated convergence theorem

dξ

dξ

(cid:90) s
(cid:90) s
(cid:90) s

0

0

0
∂
∂D

=

=

=

+

lim
∆t→0
d
dt

(cid:110) ∂

∂D

v(D(x(ξ, t + ∆t), t + ∆t)) − v(D(x(ξ, t), t))

∆t

v(D(x(ξ, t), t))dξ

v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)

∂
∂t

x(ξ, t)

(cid:111)

v(D(x(ξ, t), t))

∂
∂t

D(x(ξ, t), t)

dξ.

6

More precisely, the true fiber trajectory is defined as a tangent to the dominant eigenvec-

tor of the diffusion tensor at each point s at the given parameter time t. The rest of Chapter

2 consists of the following parts: (i) we estimate both the true fiber trajectory x(s, t) and

its rate of change with respect to time t holding s constant, i.e., ∂

∂tx(s, t) in Section 2.2. (ii)

we investigate the asymptotic behavior of the corresponding estimators in Section 2.3 and

Section 2.4. In Section 2.5, (iii) we construct a statistical test for the null hypothesis: the

true fiber trajectories remain the same over time, that is, ∂

∂tx(s, t) = 0d. Furthermore, (iv)

we study possible alternatives that specify the rate of change in the true fiber trajectories

across time.

2.2 Nadaraya-Watson type kernel estimator

At u ∈ G, we first define the component of a N × M response tensor Y (u) where N is the

number of magnetic field gradient directions and M is the number of repetitions at each visit

for a MRI scan. For i = 1, 2, . . . , N and j = 1, 2, . . . , M , the component in the ith row and
jth column of the response tensor Y (u), u ∈ G is defined as follows:

(cid:18) A(u, bij)

(cid:19)

Yij(u) := log

A(u, 0)

,

where bij is the ith spatial direction at the jth repetition, A(u, bij) is the signal intensity (echo

amplitude) measured at bij and A(u, 0) is the signal intensity measured without magnetic

field gradient directions. Details can be found in Basser and Pierpaoli (1998).

Then we suggest the following fixed linear model to estimate the unknown diffusion tensor

D(u), u ∈ G:

7

(cid:123)(cid:122)
(cid:125)
Y (u) = BD(u)1(cid:62)

(cid:124)

fixed

(cid:124)

+ Σ1/2(u)Ξ
,

M

(cid:123)(cid:122)

noise

(cid:125)

(2.2)

2

tensor determined from the set of N magnetic

where B is a known full rank N × d(d+1)
field gradient directions {b1, b2, . . . , bN} applied during image acquisition, D is a d(d+1)
2 × 1
vector representing the true diffusion tensor, 1M is a M × 1 vector with all elements being
1, Σ is a N × N symmetric positive definite tensor and Ξ is a N × M random noise tensor
which does not depend on u ∈ G. We assume Ξ has zero mean tensor and finite moments.
Additionally, for all j = 1, . . . , M, 1 ≤ k, l ≤ N, E[ΞkjΞlj] = 1 for k = l, and 0 otherwise.

We denote Ui, i = 1, 2, ..., n as observations on the non-random grid of discrete locations

Xj’s and time points Tk’s for j = 1, 2, . . . , nx, k = 1, 2, . . . , nt, and n = nxnt. However,

for a sufficiently large number of n, Ui’s are assumed to be i.i.d. uniformly distributed in
G. We further assume the independence of Xj’s and Tk’s. Our rationale for the uniformly

distributed Tk’s is based on the inverse transform method that can convert time points Tk’s,

which are recorded as the calendar dates, into uniformly distributed random numbers in

[0, 1] if their cumulative distribution function (CDF) is known.

Given Ui for i = 1, 2, . . . , n, the response tensor Y is known, however, the diffusion tensor

D is not directly observable in DTI acquisitions. Since the well-known Stejskal and Tanner

equation (1965), ordinary linear least squares (OLS), weighted linear least squares (WLS),

and nonlinear least squares (NLS) have been used as common approaches to estimate the

underlying diffusion tensor. In this dissertation, we focus on the OLS estimates of D(Ui),

i = 1, 2, . . . , n and we denote them by (cid:101)D(Ui), i = 1, 2, . . . , n. Then (cid:101)D(Ui), i = 1, 2, . . . , n can

be divided into the diffusion tensor and the random noise tensor at Ui, i = 1, 2, . . . , n.

8

(cid:101)D(Ui) =

=

1
M
1
M

(B(cid:62)B)−1B(cid:62)Y (Ui)1M

(B(cid:62)B)−1B(cid:62)(cid:0)BD(Ui)1(cid:62)

M + Σ1/2(Ui)Ξi

(cid:1)1M

:= D(Ui) + Γ(Ui),

2 × 1 random
where Γ(Ui) := 1
noise tensors being uncorrelated to D(Ui). Note that E[Γ(Ui)] = 0, i = 1, 2, . . . , n due to

M (B(cid:62)B)−1B(cid:62)Σ1/2(Ui)Ξi1M , i = 1, 2, . . . , n are i.i.d. d(d+1)

E[Ξi] = 0, i = 1, 2, . . . , n.

It is of significance to link discrete estimates with continuous realization of fiber trajec-

tory by imposing smoothness. Thus, we adopt the Nadaraya-Watson type kernel estimator
(NWE) as a locally weighted average of the OLS estimates from n observations for any u ∈ G

such that

(cid:98)Dn(u) :=

(cid:101)D(Ui)K

(cid:16) u − Ui

(cid:17)

hn

,

n(cid:88)

i=1

1

nhd+1

n

(2.3)

where K is a measurable kernel function on Rd+1 satisfying common conditions (K1)-(K3)

as well as one of (K4) or (K5) bandwidth condition for a particular purpose of interest:

(K1) Standard assumptions including

(cid:90)

(cid:90)

K(u)du = 1,

Rd+1

uK(u)du = 0,

sup

u∈Rd+1

|K(u)| < ∞,

Rd+1

(cid:90)

Rd+1

|u(cid:62)u|K(u)du < ∞.

(K2) K is non-negative and its partial derivatives are continuous on its bounded support.
(K3) For the class of the functions K = {K((u−)/hn) : hn > 0, u ∈ Rd+1}, we assume the
uniform entropy condition on K as follows: for some C > 0 and v > 0, N (ε,K) ≤ Cε−v, 0 <
ε < 1, where N (ε,K, L2(Q)) for a probability measure Q is the smallest number of balls of

9

radius ε in L2(Q) needed to cover K. An example of such kernel function that satisfies from

(K1) to (K3) is a d + 1 dimensional Gaussian kernel with the zero mean function and the

identity covariance function, i.e.,

K(u) = (2π)−(d+1)/2 exp (−0.5u(cid:62)u), u ∈ Rd+1.

Throughout this dissertation, we consider the above standard Gaussian kernel as a main

example of the kernel K. The following specific bandwidth condition either (K4) or (K5) is

satisfied depending on the purpose of its use.

(K4) For the fiber estimation and the test on the null hypothesis that ∂

∂tx(s, t) = 0d, the

bandwidth hn satisfies the following regularity conditions:

hn → 0, nhn → ∞,

n

nhd+1

|log hn| → ∞, and

|log hn|
log log n

→ ∞ as n → ∞.

nhd+4

n → β1 > 0 as n → ∞, where β1 is a known fixed number.

(K5) For the estimation of fiber’s first partial derivatives, the bandwidth hn satisfies the

following regularity conditions:

hn → 0, nhn → ∞,

n

nhd+3

|log hn| → ∞, and

|log hn|
log log n

→ ∞ as n → ∞.

nhd+6

n → β2 > 0 as n → ∞, where β2 is a known fixed number.

From estimated diffusion tensor (cid:98)Dn(u), u ∈ G, we compute its largest eigenvalue λ((cid:98)Dn(u))
and its corresponding normalized eigenvector v((cid:98)Dn(u)). Finally, the true trajectory given
the parameter time t ∈ [0, T ] in (2.1) is estimated by a plug-in estimator (cid:98)Xn(s, t) such that

10

(cid:98)Xn(s, t) = v((cid:98)Dn((cid:98)Xn(s, t), t)), s ∈ [0, S], (cid:98)Xn(0, t) = x0,

∂
∂s

(2.4)

where x0 ∈ X . This is equivalent to
(cid:90) s

(cid:98)Xn(s, t) = x0 +

0

v((cid:98)Dn((cid:98)Xn(ξ, t), t))dξ, s ∈ [0, S], t ∈ [0, T ].

Consecutively, this procedure can be carried further to estimate the partial derivative of

(cid:98)Xn(s, t) with respect to t at a fixed s ∈ [0, S] by plugging in as follows:

(cid:98)Xn(s, t) =

∂
∂t

=

+

(cid:90) s
(cid:90) s

0

0
∂
∂D

d
dt

v((cid:98)Dn((cid:98)Xn(ξ, t), t))dξ
(cid:110) ∂
v((cid:98)Dn((cid:98)Xn(ξ, t), t))
v((cid:98)Dn((cid:98)Xn(ξ, t), t))

∂D

∂
∂t

(cid:98)Dn((cid:98)Xn(ξ, t), t)

∂
∂x

(cid:111)
(cid:98)Dn((cid:98)Xn(ξ, t), t)

(cid:98)Xn(ξ, t)

∂
∂t

dξ, s ∈ [0, S], t ∈ [0, T ].

Furthermore, when we assume a regular grid for Xi and spatial local continuity for Σ(u)
in x ∈ X , Σ(u) for any u ∈ G can be estimated in two steps. First, we use a local spatial

averaging procedure. For i = 1, 2, . . . , n and j = 1, 2, . . . , nx,

(cid:101)ΣΓ,n(Ui) :=

1

#N (x)

Xj∈N (x)

(cid:88)

(cid:0)Y (Ui) − B(cid:101)D(Ui)(cid:1)(cid:0)Y (Ui) − B(cid:101)D(Ui)(cid:1)(cid:62)

,

where N (x) is the set of all neighbors of a point x ∈ X . In case of d = 3, the cardinality of

the set N (x) is 26. Second, we use the NWE such that

(cid:98)Σn(u) :=

n(cid:88)

(cid:101)ΣΓ,n(Ui)K

(cid:16)u − Ui

(cid:17)

hn

.

(2.5)

1

nhd+1

n

i=1

11

2.3 Uniform consistency of the estimators

In the following four lemmas, we show the strong consistency of the estimators. These

lemmas are used to prove major theorems in Section 2.4 and Section 2.5. Without loss of
generality, we consider Gδ = [−δ, 1 + δ]d+1 for some δ > 0 assuming Ui, i = 1, 2, . . . , n are
i.i.d. uniformly distributed in [0, 1]d+1, i.e., X = [0, 1]d and T = 1. Throughout this paper,

c, c1, c2, . . . represent constants. We refer to Gin´e and Guillou (2002), Einmahl and Mason

(2005) and Blondin (2007) for conditions on the bandwidth in uniform consistency.

Lemma 2.3.1. Suppose hn → 0, nhn → ∞, nhd+1

n|log hn| → ∞, and

|log hn|
log log n → ∞ as n → ∞.

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)

(cid:12)(cid:12)(cid:12) → 0,

in probability as n → ∞.

Then we have

sup
u∈Gδ

Proof. Note that

sup
u∈Gδ

E[(cid:98)Dn(u)] =

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:90)
(cid:90)
(cid:90)

Rd+1

Rd+1

(cid:110)

=

1

=

Rd+1

(cid:12)(cid:12)(cid:12) + sup

u∈Gδ

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − E[(cid:98)Dn(u)]
(cid:12)(cid:12)(cid:12).

(cid:12)(cid:12)(cid:12) ≤ sup

u∈Gδ

(cid:12)(cid:12)(cid:12)E[(cid:98)Dn(u)] − D(u)
(cid:16)u − w

(cid:17)

dw

D(w)K

hd+1
n
D(u − hnψ)K(ψ)dψ, by letting ψ =

hn

u − w
hn

D(u) + D(u − hnψ) − D(u)

K(ψ)dψ

(cid:111)

by Taylor’s theorem in a sufficiently small neighborhood of D(u)

(cid:90)

(cid:110) − hn

= D(u) +

Rd+1

(cid:111)

ψ(cid:62) ∂2

∂u2 D(u)ψ + o(h2
n)

K(ψ)dψ,

∂
∂u

D(u)ψ +

h2
n
2

12

where

∂uD(u)

is a Jacobian matrix of D evaluated at u = u0, i.e.,

∂D11
∂x1
...

∂Ddd
∂x1

. . .

. . .

. . .

∂D11
∂xd

∂D11

∂t

...

∂Ddd
∂xd

∂Ddd

∂t

is the corresponding 3-dimensional hypermatrix

(cid:111)

d(d+1)

2 ×(d+1)

(cid:111)

d(d+1)

2 ×(d+1)

=



(cid:110) ∂

(cid:110) ∂2

(cid:12)(cid:12)(cid:12)u=u0
(cid:110) ∂

D(u)

∂u

(cid:12)(cid:12)(cid:12)u=u0

(cid:12)(cid:12)(cid:12)u=u0
(cid:111)

and

∂u2 D(u)

(d+1)× d(d+1)
(i.e., a third-order tensor) such that

2 ×(d+1)

(cid:110) ∂2

∂u2 D(u)

(cid:111)

(cid:12)(cid:12)(cid:12)u=u0

(d+1)× d(d+1)

2 ×(d+1)

=

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)u=u0

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)u=u0

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)u=u0


...

...

∂2D11
∂x1∂t

∂2D11
∂t2

∂2Ddd
∂x1∂t

∂2Ddd
∂2t

.

. . .

. . .

. . .

. . .

. . .

. . .

∂2D11
∂x1∂xd

∂2D11
∂t∂xd

∂2Ddd
∂x1∂xd

∂2Ddd
∂t∂xd

∂2D11
∂t∂x1

∂2D11
∂x2
1
...

∂2Ddd
∂x2
1
...


...
(cid:13)(cid:13)(cid:13)(cid:13)Gδ
(cid:12)(cid:12)(cid:12) = O(h2

∂2Ddd
∂t∂x1

(cid:90)

13

(cid:12)(cid:12)(cid:12)ψ(cid:62)ψ

(cid:12)(cid:12)(cid:12)K(ψ)dψ

(cid:16)

(cid:17)

1 + op(1)

.

Rd+1

Thus, we have

sup
u∈Gδ

∂u2 D(u)

n
2

(cid:13)(cid:13)(cid:13)(cid:13) ∂2

(cid:12)(cid:12)(cid:12) ≤ h2
(cid:12)(cid:12)(cid:12)E[(cid:98)Dn(u)] − D(u)
and(cid:82)
(cid:12)(cid:12)(cid:12)E[(cid:98)Dn(u)] − D(u)

∂u2 D(u)(cid:107)Gδ

sup
u∈Gδ

Provided that (cid:107) ∂2

Rd+1|ψ(cid:62)ψ|K(ψ)dψ are bounded,

n) as n → ∞.

For the almost sure uniform convergence rate for the Nadaraya-Watson kernel estimator,

see references such as Einmahl and Mason (2005):

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − E[(cid:98)Dn(u)]
(cid:12)(cid:12)(cid:12) = O

sup
u∈G

(cid:18)(cid:115)|log hn|

nhd+1

n

(cid:19)

as n → ∞.

Thus we have

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:12)(cid:12)(cid:12) = O

(cid:18)

h2
n +

sup
u∈G

(cid:115)|log hn|

(cid:19)

nhd+1

n

as n → ∞.

Then the proof is complete under the stated assumptions.

Lemma 2.3.2. Suppose hn → 0, nhn → ∞, nhd+3

n|log hn| → ∞, and

|log hn|
log log n → ∞ as n → ∞.

Then we have

sup
u∈Gδ

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂x

(cid:98)Dn(u) − ∂

∂x

in probability as n → ∞.

(cid:12)(cid:12)(cid:12)(cid:12) → 0 and sup

u∈Gδ

D(u)

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂t

(cid:98)Dn(u) − ∂

∂t

(cid:12)(cid:12)(cid:12)(cid:12) → 0

D(u)

Proof. The proof is given only for the strong consistency of the partial derivative with respect

to t since the proof for the partial derivative with respect to x can be obtained in the same

manner. As we prove Lemma 2.3.1., we begin with

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂t

(cid:98)Dn(u) − ∂

∂t

D(u)

sup
u∈Gδ

(cid:98)Dn(u)] − ∂

∂t

∂
∂t

(cid:12)(cid:12)(cid:12)(cid:12) + sup

u∈Gδ

D(u)

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂t

(cid:98)Dn(u) − E[

(cid:12)(cid:12)(cid:12)(cid:12).
(cid:98)Dn(u)]

∂
∂t

(cid:12)(cid:12)(cid:12)(cid:12)E[

u∈Gδ

(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup
n(cid:88)

(cid:98)Dn(u)] =

E[

∂
∂t

1

nhd+2

n

i=1

E[ ˜D(Ui)K

(cid:16)u − Ui

(cid:17)

hn

]

(1)
t

14

(cid:90)

1

hd+2
n

Rd+1

=

(cid:16) u − w

(cid:17)

hn

dw

D(w)K

(1)
t

by letting ψ = u−w
hn

=

1
hn

(cid:90)

Rd+1

D(u − hnψ)K

(1)
t

(ψ)dψ

by Taylor’s theorem again

(cid:90)

(cid:110)

=

1
hn

+ h2
n

Rd+1
∂2
∂x∂t

D(u) − hn
h2
n
2

D(u)ψx − hn
∂
∂x
∂2
∂t2 D(u)ψ2

∂
∂t

D(u)ψxψt +

(cid:111)

t + o(h2
n)

K

(1)
t

(ψ)dψ,

D(u)ψt +

ψ(cid:62)

x

h2
n
2

∂2
∂x2 D(u)ψx

where ψ = (ψx, ψt). By choosing kernel Lt(ψ) = −ψtK

(1)
t

(cid:12)(cid:12)(cid:12)(cid:12)E[

sup
u∈Gδ

(cid:98)Dn(u)] − ∂

∂t

∂
∂t

D(u)

(cid:12)(cid:12)(cid:12)(cid:12) = O(cid:0)hn

(ψ),

(cid:1) as n → ∞.

By Theorem 2.2 as in Blondin (2007), we have the following rate of strong uniform consistency

for the partial derivatives of the Nadaraya-Watson kernel estimator:

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂t

(cid:98)Dn(u) − E[

∂
∂t

(cid:12)(cid:12)(cid:12)(cid:12) = O
(cid:98)Dn(u)]

(cid:18)(cid:115)|log hn|

(cid:19)

nhd+3

n

sup
u∈Gδ

as n → ∞.

Thereafter we have

sup
u∈Gδ

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂t

(cid:98)Dn(u) − ∂

∂t

(cid:12)(cid:12)(cid:12)(cid:12) = O

D(u)

(cid:18)

hn +

(cid:115)|log hn|

(cid:19)

nhd+3

n

as n → ∞.

15

Lemma 2.3.3. Suppose hn → 0, nhn → ∞, nhd+1

n|log hn| → ∞, and

|log hn|
log log n → ∞ as n → ∞.

Then we have

(cid:12)(cid:12)(cid:12)(cid:98)Xn(s, t) − x(s, t)

(cid:12)(cid:12)(cid:12) → 0,

in probability as n → ∞.

sup

s∈[0,S],t∈[0,T ]

Proof. Note that

(cid:98)Xn(s, t) − x(s, t) =

=

+

(cid:90) s
(cid:90) s
(cid:90) s

0

0

0

(cid:110)
(cid:111)
v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − v(D(x(ξ, t), t))
(cid:110)
(cid:111)
v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − v(D((cid:98)Xn(ξ, t), t))
(cid:111)
(cid:110)
v(D((cid:98)Xn(ξ, t), t)) − v(D(x(ξ, t), t))

dξ.

dξ

dξ

Then

(cid:12)(cid:12)(cid:12)(cid:98)Xn(s, t) − x(s, t)

(cid:12)(cid:12)(cid:12) ≤ sLv sup

u∈Gδ

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)

(cid:12)(cid:12)(cid:12) + LvD

(cid:90) s

0

(cid:12)(cid:12)(cid:12)(cid:98)Xn(ξ, t) − x(ξ, t)
(cid:12)(cid:12)(cid:12)dξ,

where Lv > 0, LvD > 0 are Lipschitz constants. By Gronwall–Bellman inequality,

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)

(cid:12)(cid:12)(cid:12) exp(cid:0)sLvD

(cid:1)

≤ sLv sup
u∈Gδ

and hence

(cid:12)(cid:12)(cid:12)(cid:98)Xn(s, t) − x(s, t)

(cid:12)(cid:12)(cid:12) ≤ SLv sup

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)

(cid:12)(cid:12)(cid:12) exp(cid:0)SLvD

(cid:1)

u∈Gδ

sup

s∈[0,S],t∈[0,T ]

Due to the bounded exponent and Lemma 2.3.1., the proof is complete.

16

Lemma 2.3.4. Suppose that the assumptions for Lemma 2.3.1., Lemma 2.3.2., and Lemma

2.3.3. hold. Then we have

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂t

(cid:98)Xn(s, t) − ∂

∂t

(cid:12)(cid:12)(cid:12)(cid:12) → 0, in probability as n → ∞.

x(s, t)

sup

s∈[0,S],t∈[0,T ]

Proof. In this proof, we use the first derivatives of the normalized eigenvector associated with

the largest eigenvalue with respect to the component of the d(d+1)

2 × 1 diffusion tensor in a
neighborhood of D0 as in (Carmichael and Sakhanenko, 2016, p. 313). In a neighborhood of

D0, λ(D0) denotes the largest eigenvalue of D0 and the corresponding normalized eigenvector

is denoted as v(D0). Let δkl be 1 if k = l and 0 elsewhere. We also simply define Z(D0) :=
λ(D0)Id − D0. For 1 ≤ k, l, p ≤ d,

(cid:12)(cid:12)(cid:12)(cid:12)D=D0
(cid:12)(cid:12)(cid:12)(cid:12)D=D0

∂λ(D)
∂Dkl
∂vp(D)
∂Dkl

= (2 − δkl)vk(D0)vl(D0)

= (1 − δkl/2)(cid:2)Z+(D0)pkvl(D0) + Z+(D0)plvk(D0)(cid:3),

where A+ is the Moore-Penrose inverse of A. See Theorem 8.9 in (Magnus, 2019, p. 180).

Applying these DEs enables us to decompose the following terms:

v((cid:98)Dn(u)) − ∂

∂t

∂
∂t

v(D(u)) =

+

∂v((cid:98)Dn(u))
(cid:110)∂v((cid:98)Dn(u))

∂D(u)

∂D(u)

×(cid:110)∂(cid:98)Dn(u)

∂t

− ∂v(D(u))
∂D(u)

− ∂D(u)

(cid:111)
(cid:111) × ∂D(u)

∂t

.

∂t

Provided that (cid:107) ∂D(u)

∂t (cid:107)Gδ
v((cid:98)Dn(u)) − ∂

∂t

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂t

sup
u∈Gδ

and (cid:107) ∂v((cid:98)Dn(u))
∂D(u) (cid:107)Gδ
(cid:12)(cid:12)(cid:12)(cid:12) ≤ c1 sup

(cid:12)(cid:12)(cid:12)(cid:12) ∂

u∈Gδ

∂t

v(D(u))

are bounded, we have

(cid:98)Dn(u) − ∂

∂t

D(u)

(cid:12)(cid:12)(cid:12)(cid:12) + c2 sup

u∈Gδ

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:12)(cid:12)(cid:12)

17

and likewise, provided that (cid:107) ∂D(u)

∂x (cid:107)Gδ

is bounded,

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂x

sup
u∈Gδ

v((cid:98)Dn(u)) − ∂

∂x

(cid:12)(cid:12)(cid:12)(cid:12) ≤ c1 sup

u∈Gδ

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂x

(cid:98)Dn(u) − ∂

∂x

(cid:12)(cid:12)(cid:12)(cid:12) + c2 sup

u∈Gδ

D(u)

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:12)(cid:12)(cid:12).

v(D(u))

Then Lemma 2.3.1. and 2.3.2. complete the following properties:

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂x

sup
u∈Gδ

v((cid:98)Dn(u)) − ∂

∂x

v(D(u))

v((cid:98)Dn(u)) − ∂

∂t

(cid:12)(cid:12)(cid:12)(cid:12) = op(1).

v(D(u))

Returning now to the main theorem, we have

(cid:98)Xn(s, t) − ∂

∂t

∂
∂t

x(s, t) =

dt

dt

∂t

∂x

u∈Gδ

(cid:12)(cid:12)(cid:12)(cid:12) ∂

(cid:12)(cid:12)(cid:12)(cid:12) = op(1), and sup
(cid:110) d
v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − d
(cid:110) ∂
v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − ∂
(cid:111)
(cid:98)Xn(ξ, t) − ∂
(cid:110) ∂
v(D((cid:98)Xn(ξ, t), t))
(cid:110) ∂
v(D((cid:98)Xn(ξ, t), t)) − ∂
(cid:110) ∂
v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − ∂
(cid:110) ∂
v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − ∂
(cid:110) ∂
v(D((cid:98)Xn(ξ, t), t)) − ∂

∂
∂x

x(ξ, t)

∂x

∂x

∂x

dξ

∂t

∂t

∂t

∂t

∂t

∂t

∂x

∂t

0

0

=

(cid:90) s
(cid:90) s
×(cid:110) ∂
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s

+

+

+

+

0

0

0

0

+

0

(cid:98)Xn(ξ, t) − ∂

∂x

dξ

v(D(x(ξ, t), t))

(cid:111)
(cid:111)
v(D((cid:98)Xn(ξ, t), t))
(cid:111)
(cid:111) ∂
(cid:111) ∂
v(D((cid:98)Xn(ξ, t), t))
(cid:111)
v(D((cid:98)Xn(ξ, t), t))
(cid:111)

v(D(x(ξ, t), t))

x(ξ, t)

dξ

dξ

∂t

∂t

v(D(x(ξ, t), t))

dξ.

x(ξ, t)dξ

x(ξ, t)dξ

∂t

Therefore, we have

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂t

(cid:98)Xn(s, t) − ∂

∂t

x(s, t)

(cid:12)(cid:12)(cid:12)(cid:12) ≤(cid:110)
(cid:90) s

×

sup
u∈Gδ

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂t

0

(cid:12)(cid:12)(cid:12)(cid:12) ∂
v((cid:98)Dn(u)) − ∂
(cid:98)Xn(ξ, t) − ∂

∂x

∂x

x(ξ, t)

∂t

(cid:12)(cid:12)(cid:12)(cid:12)dξ

v(D(u))

(cid:111)

(cid:12)(cid:12)(cid:12)(cid:12) + c1

18

(cid:110)

(cid:12)(cid:12)(cid:12)dξ

(cid:111)(cid:90) s
(cid:12)(cid:12)(cid:12)(cid:12) ∂
v((cid:98)Dn(u)) − ∂
(cid:12)(cid:12)(cid:12)(cid:12) ∂
v((cid:98)Dn(u)) − ∂

(cid:12)(cid:12)(cid:12)(cid:98)Xn(ξ, t) − x(ξ, t)
(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12),

v(D(u))

v(D(u))

∂x

∂t

+

c2LvxD + LvtD

0

+ sc3 sup
u∈Gδ

∂x

+ s sup
u∈Gδ

∂t

where LvxD > 0 and LvtD > 0 are Lipschitz constants provided with bounded (cid:107) ∂
and sups∈[0,S],t∈[0,T ]| ∂

∂xv(D(u))(cid:107)Gδ
∂tx(s, t)|. By applying Gronwall-Bellman inequality together with pre-

vious lemmas and proven properties, the proof is complete.

2.4 Weak convergence of the sequence of stochastic

processes

In this section, we show the weak convergence of the sequence of stochastic processes via

the functional central limit theorem. This leads to our main results as in Theorem 2.4.1.,

Theorem 2.4.2., and Theorem 2.4.3. with the pointwise rate of convergence accordingly. In

Theorem 2.4.1., we address the asymptotic behavior of the deviation processes between the

estimated fiber trajectory and the true fiber trajectory. In Theorem 2.4.2., we also consider

the deviation processes between the partial derivative of estimated fiber trajectory with

respect to time and the one of true fiber trajectory. Furthermore, Theorem 2.4.3. delivers the

limiting behavior of difference between deviation processes resulting from Theorem 2.4.1. at

different time points. Throughout these theorems, DTI data corrupted by noise is taken into

account by the covariance function of the limiting Gaussian process. That is, we quantify

that the higher noise level in signal is associated with the larger variance of confidence

ellipsoids for the true fiber trajectory at the fixed time point. Detailed proofs of Theorem

19

2.4.1., Theorem 2.4.2., and Theorem 2.4.3. are given in Chapter 6.

We define the second derivatives of the normalized eigenvector associated with the largest

eigenvalue with respect to each component of the d(d+1)

2 × 1 diffusion tensor based on
218). Recall that δkl = 1 if k = l and δkl = 0 if k (cid:54)= l, and
(Magnus, 2019, p.
Z(D0) = λ(D0)Id − D0. Then for 1 ≤ k, l, p, r, w ≤ d, we have the following second

derivative of the maximal eigenvalue with respect to each component of D:



(cid:12)(cid:12)(cid:12)(cid:12)D0

=

∂2λ(D)

∂Drw∂Dkl

(2 − δkl)[Z+(D0)krvr(D0)vl(D0) + Z+(D0)lrvr(D0)vk(D0)],

when r = w.
(2 − δkl)[Z+(D0)krvw(D0)vl(D0) + Z+

ktvr(D0)vl(D0)

+Z+(D0)lrvw(D0)vk(D0) + Z+(D0)lwvr(D0)vk(D0)],

when r (cid:54)= w.

For the second derivatives of the corresponding normalized eigenvector with respect to

each component of D,

(cid:12)(cid:12)(cid:12)(cid:12)D0
(cid:12)(cid:12)(cid:12)(cid:12)D0
(cid:12)(cid:12)(cid:12)(cid:12)D0

∂2vp(D)
∂Drr∂Dkk
∂2vp(D)
∂Drw∂Dkk

∂2vp(D)
∂Drw∂Dkl

=

=

∂Z+(D0)pk

∂Drr

∂Z+(D0)pk

∂Drw

vk(D0) + Z+(D0)pkZ+(D0)krvr(D0)

vk(D0) + Z+(D0)pkZ+(D0)krvw(D0)

+ Z+(D0)pkZ+(D0)kwvr(D0)

=

+

∂Z+(D0)pk

∂Drw

vl(D0) + Z+(D0)pk[Z+(D0)lrvw(D0) + Z+(D0)lwvr(D0)]

∂Z+(D0)pl

∂Drw

vk(D0) + Z+(D0)pl[Z+(D0)krvw(D0) + Z+(D0)kwvr(D0)].

20

Note that

∂Z+(D0)pl

∂Drw

∂Z(D0)kq

∂Drw

= − d(cid:88)

d(cid:88)

d(cid:88)

Z+(D0)pk

k=1

m=1

∂Drw
= (2 − δrw)vr(D0)vw(D0)δkq − δ∗
kq,

q=1

∂Z(D0)kq

Z+(D0)qm(Z(D0)Z+(D0))−1

ml

where δ∗

kq = 1 if either k = r and q = w or k = w and q = r, while δ∗

kq = 0 otherwise.

In the following theorems, we denote G as a d×d tensor-valued Green’s function satisfying

∂
∂s

∂
∂D

∂
G(s, ξ, t) =
∂x
G(ξ, ξ, t) = Id, ξ ∈ [0, s], s ∈ [0, S],

v(D(x(s, t), t))

D(x(s, t), t)G(s, ξ, t),

given the parameter time t ∈ [0, T ]. Equivalently,

(cid:90) s

ξ

G(s, ξ, t) = Id +

∂
∂D

v(D(x(τ, t), t))

∂
∂x

D(x(τ, t), t)G(τ, ξ, t)dτ, ξ ∈ [0, s], s ∈ [0, S].

For a fixed parameter time t ∈ [0, T ], G is continuous in (s, ξ) satisfying a Lipschitz condition
with respect to s ∈ [0, S]. Green’s function is used to provide the unique solution to the first-

order nonhomogeneous differential equation with the boundary value given the parameter
time t ∈ [0, T ]. See Coddington and Levinson (1955) for the use of Green’s function in

Chapter 6.

Theorem 2.4.1. Suppose that hn → 0, nhn → ∞, nhd+1
n → ∞. Suppose also that nhd+4

|log hn|
log log n → ∞ as
n → β1 > 0 as n → ∞, where β1 is a known fixed number.

n|log hn| → ∞, and

Then the sequence of stochastic processes

21

(cid:113)

nhd
n

(cid:16)(cid:98)Xn(s, t) − x(s, t)
(cid:17)

, s ∈ [0, S], t ∈ [0, T ]

converges weakly in the space of Rd-valued continuous functions on [0, S] to the Gaussian
process GP1(s, t), s ∈ [0, S], t ∈ [0, T ] with the mean function

√

β1
2

µβ1

(s, t) =

(cid:90) s

0

G(s, ξ, t)

∂
∂D

v(D(x(ξ, t), t))

ψ(cid:62) ∂2

∂u2 D(x(ξ, t), t)ψK(ψ)dψdξ,

Rd+1

and the covariance function for all pairs of spatial points (s, s∗) ∈ [0, S] at the given time
point t ∈ [0, T ]

C1((s, t), (s∗, t)) =

Ψ(v(D(x(ξ, t), t)))G(s, ξ, t)

∂
∂D

v(D(x(ξ, t), t))

D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)

(cid:105)

v(D(x(ξ, t), t))

G(cid:62)(s∗, ξ, t)dξ

(cid:90) s∧s∗
×(cid:104)
×(cid:16) ∂

0

∂D

(cid:90)

(cid:17)(cid:62)

where Ψ(v) := (cid:82)

R

(cid:82)

Rd+1 K(ψ)K(ψ + (τ v, 0))dψdτ . The pointwise rate of convergence is

O(n−4/(d+4)) given the parameter time t ∈ [0, T ].

√
Remark 2.4.1. For the standard Gaussian kernel with d = 3, Ψ(v) = 1
8π

π

.

Proof. Let d = 3. For simplicity, we use the notation (τ v, 0) := τ w, where w = [v1 v2 v3 0](cid:62).

(cid:90)

(cid:90)
(cid:90)

Ψ(v) =

=

(cid:18)

K(ψ)K(ψ + τ w)dψdτ

R

R

R4
1
16π2 exp

− τ 2w(cid:62)w

4

(cid:19)(cid:90)

1
π2 exp

R4

since the integral of the Gaussian distribution N(cid:0) − 0.5τ w, 0.5I4

(cid:17)(cid:19)

dψdτ

τ w
2

ψ +

(cid:18)

−(cid:16)

ψ +

τ w
2

(cid:17)(cid:62)(cid:16)
(cid:1) is 1,

22

(cid:90)

(cid:18)

√
w(cid:62)w
√
2
π

=

√
π

− τ 2w(cid:62)w
since the integral of the Gaussian distribution N(cid:0)0,

√
1
w(cid:62)w

exp

8π

R

4

(cid:19)

dτ

(cid:1) is 1,

2

w(cid:62)w

=

√
1
8π
π

since eigenvectors are normalized, i.e,

√

w(cid:62)w = 1.

Corollary 2.4.1. The Gaussian process GP1(s, t), s ∈ [0, S], t ∈ [0, T ] in Theorem 2.4.1.
satisfies the following stochastic differential equation (SDE) with GP1(0, t) = 0d:

(cid:90) s

0

A((s, t), (ξ, t))dW(ξ, t),

GP1(s, t) = µβ1

(s, t) +

where

A((s, t), (ξ, t)) := Ψ1/2(v(D(x(ξ, t), t)))G(s, ξ, t)

×(cid:104)

(cid:105)1/2
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)

v(D(x(ξ, t), t))

∂
∂D

and W(s, t) is the Wiener process indexed by s ∈ [0, S] given the parameter time t ∈ [0, T ].

Theorem 2.4.2. Suppose that hn → 0, nhn → ∞, nhd+3
n → ∞. Suppose also that nhd+6

n|log hn| → ∞, and

|log hn|
log log n → ∞ as
n → β2 > 0 as n → ∞, where β2 is a known fixed number.
(cid:16) ∂

(cid:17)

(cid:98)Xn(s, t) − ∂

, s ∈ [0, S], t ∈ [0, T ]

x(s, t)

∂t

(cid:113)

nhd+2

n

∂t

Then the sequence of stochastic processes

converges weakly in the space of Rd-valued continuous functions on [0, S] to the Gaussian
process GP2(s, t), s ∈ [0, S], t ∈ [0, T ] with the mean function

23

√

β2
2

µβ2

(s, t) =

G(s, ξ, t)

∂2
∂D2 v(D(x(ξ, t), t))

D(x(ξ, t), t)dξ

G(s, ξ, t)

∂2
∂D2 v(D(x(ξ, t), t))

(cid:90)
(cid:90)

Rd+1

Rd+1

ψ(cid:62) ∂2

∂u2 D(x(ξ, t), t)ψK(ψ)dψ

ψ(cid:62) ∂2

∂u2 D(x(ξ, t), t)ψK(ψ)dψ

D(x(ξ, t), t)

∂
∂t

x(ξ, t)dξ

G(s, ξ, t)

∂2
∂D2 v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)

(cid:90) ξ

0

G(ξ, ζ, t)

(cid:90)

∂
∂D

(cid:90)

v(D(x(ζ, t), t))

Rd+1

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζdξ

ψ(cid:62) ∂2

G(s, ξ, t)

v(D(x(ξ, t), t))

∂2
∂x∂t

D(x(ξ, t), t)

G(ξ, ζ, t)

(cid:90) ξ

0

v(D(x(ζ, t), t))

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζdξ

Rd+1
∂2
∂D2 v(D(x(ξ, t), t))

ψ(cid:62) ∂2
(cid:90)

G(s, ξ, t)

∂
∂D

G(s, ξ, t)

∂
∂D

v(D(x(ξ, t), t))

(cid:90)

∂
D(x(ξ, t), t)
∂x
ψ(cid:62) ∂2
∂2
∂x2 D(x(ξ, t), t)
ψ(cid:62) ∂2

G(ξ, ζ, t)

v(D(x(ζ, t), t))

Rd+1

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζ

G(ξ, ζ, t)

∂
∂D

v(D(x(ζ, t), t))

Rd+1

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζ

∂
∂x

D(x(ξ, t), t)

∂
∂t

∂
∂t

x(ξ, t)dξ

x(ξ, t)dξ

0

(cid:90) s
(cid:90) s
(cid:90) s

0

0

(cid:90) s

0

(cid:90) s

0

(cid:90) s

0

× ∂
√
∂t

+

β2
2
× ∂
√
∂x

+

+

β2
2
× ∂
∂D
√
β2
2
× ∂
√
∂D
β2
2

(cid:90) ξ

+

×

+

×

√

0
β2
2

(cid:90) ξ

0

and the covariance function for all pairs of spatial points (s, s∗) ∈ [0, S] at the given time
point t ∈ [0, T ]

C2((s, t), (s∗, t)) =

0

∂
∂D

v(D(x(ξ, t), t))

Ψt(v(D(x(ξ, t), t)))G(s, ξ, t)

(cid:90) s∧s∗
×(cid:104)
(cid:105)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)
×(cid:16) ∂
(cid:90) s∧s∗
×(cid:104)

(cid:105)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)

Ψx(v(D(x(ξ, t), t)))G(s, ξ, t)

G(cid:62)(s∗, ξ, t)dξ

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

(cid:17)(cid:62)

∂
∂D

∂D

+

0

24

(cid:17)(cid:62)

(cid:17)(cid:62)

v(D(x(ξ, t), t))

G(cid:62)(s∗, ξ, t)dξ

0

Ψtx(v(D(x(ξ, t), t)))G(s, ξ, t)

(cid:105)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)

v(D(x(ξ, t), t))

∂
∂D

v(D(x(ξ, t), t))

G(cid:62)(s∗, ξ, t)dξ,

∂D

×(cid:16) ∂
(cid:90) s∧s∗
×(cid:104)
×(cid:16) ∂

+

∂D

where

Ψt(v) :=

Ψx(v) :=

Ψtx(v) :=

(cid:90)
(cid:90)
(cid:90)

(cid:90)
(cid:90)
(cid:90)

K

(1)
t

(ψ)K

(1)
t

(ψ + (τ v, 0))dψdτ

R

Rd+1

(cid:17)(cid:62)(cid:16)

(cid:16) ∂
(cid:17)(cid:62)(cid:16)

∂t

(cid:17)(cid:62)

(cid:17)(cid:62)

x(ξ, t)

x(ξ, t)

K

(1)
x (ψ + (τ v, 0))

dψdτ

x(ξ, t)

K

(1)
x (ψ + (τ v, 0))

dψdτ.

K

(1)
x (ψ)

K

(1)
t

(ψ)

∂
∂t

(cid:16) ∂

∂t

R

Rd+1

R

Rd+1

The pointwise rate of convergence is O(n−4/(d+6)) given the parameter time t ∈ [0, T ].

Remark 2.4.2. Under the Gaussian kernel smoothing to a tensor field, however, both

stochastic processes in Theorem 2.4.1. and Theorem 2.4.2.
fail to converge in the space
of Rd-valued continuous functions on [0, S]× [0, T ] and the corresponding candidate limiting

processes behave as white noise in time t.

∂t(cid:98)Xn(s, t) − ∂

∂

In Theorem 2.4.3., we study the integration of the sequence of stochastic processes

∂tx(s, t), s ∈ [0, S], t ∈ [0, T ] with a time-dependent weight function on any

fixed time interval in [0, T ]. In particular, we consider a positive Lebesgue measurable weight

function. The simplest choice of the weight function is to assign equal weights to each ele-

∂t(cid:98)Xn(s, t)− ∂

∂tx(s, t), s ∈ [0, S], t ∈ [0, T ]. Since

ment of the sequence of stochastic processes ∂

such an integral no longer depends on the parameter time t, we can eliminate the problem

where the limiting processes of interest behave like white noise in time t.

25

Theorem 2.4.3. Suppose that the assumptions for Theorem 2.4.1. hold. Let w be a positive
vector-valued weight function. For 0 < a < b ≤ T, we define

(cid:113)

(cid:90) b

a

(cid:18) ∂

∂t

w(cid:62)(t)

(cid:19)

(cid:98)Xn(s, t) − ∂

∂t

x(s, t)

Wn(s) :=

nhd
n

dt, s ∈ [0, S].

Then we establish the weak convergence of the stochastic process in the space of R-valued
continuous functions on [0, S] to the Gaussian process GP3(s), s ∈ [0, S] with the mean

G(s, ξ, t)

∂2
∂D2 v(D(x(ξ, t), t))

G(s, ξ, t)

∂2
∂D2 v(D(x(ξ, t), t))

(cid:90)
(cid:90)

Rd+1

Rd+1

ψ(cid:62) ∂2

∂u2 D(x(ξ, t), t)ψK(ψ)dψ

ψ(cid:62) ∂2

∂u2 D(x(ξ, t), t)ψK(ψ)dψ

function

µβ1

(s) =

√

β1
2

D(x(ξ, t), t)dξdt

a

(cid:90) b
(cid:90) b
(cid:90) b

a

w(cid:62)(t)

w(cid:62)(t)

0

(cid:90) s
(cid:90) s
(cid:90) s

0
∂
∂t

a

0

(cid:90) b

(cid:90) s

∂
∂D
w(cid:62)(t)

a

0

(cid:90) b

(cid:90) s

∂
∂D
w(cid:62)(t)

a

0

(cid:90) b

(cid:90) s

∂
∂D
w(cid:62)(t)

a

0

+

× ∂
√
∂t
β1
2
× ∂
√
∂x
β1
2

(cid:90) ξ

+

×

+

×

+

×

+

×

√
0
β1
2

(cid:90) ξ

√
0
β1
2

(cid:90) ξ

√
0
β1
2

(cid:90) ξ

0

x(ξ, t)dξdt

D(x(ξ, t), t)
w(cid:62)(t)

G(s, ξ, t)

∂2
∂D2 v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)

G(ξ, ζ, t)

v(D(x(ζ, t), t))

ψ(cid:62) ∂2

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζdξdt

Rd+1

G(s, ξ, t)

∂
∂D

∂2
∂x∂t

D(x(ξ, t), t)

v(D(x(ξ, t), t))
ψ(cid:62) ∂2

Rd+1

G(ξ, ζ, t)

v(D(x(ζ, t), t))

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζdξdt

G(s, ξ, t)

∂2
∂D2 v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)

∂
∂x

D(x(ξ, t), t)

G(ξ, ζ, t)

v(D(x(ζ, t), t))

ψ(cid:62) ∂2

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζ

Rd+1

G(s, ξ, t)

∂
∂D

∂2
∂x2 D(x(ξ, t), t)

v(D(x(ξ, t), t))
ψ(cid:62) ∂2

Rd+1

G(ξ, ζ, t)

∂
∂D

v(D(x(ζ, t), t))

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζ

∂
∂t

∂
∂t

x(ξ, t)dξdt

x(ξ, t)dξdt

(cid:90)

(cid:90)

(cid:90)

(cid:90)

26

and the covariance function for all pairs of points (s, s∗) ∈ [0, S]

C3(s, s∗) =

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

+

∂D

a

0

∂
∂D

(cid:17)(cid:62)

v(D(x(ξ, t), t))

(cid:90) s∧s∗

(cid:101)Ψt(v(D(x(ξ, t), t)))w(cid:62)(t)G(s, ξ, t)

G(cid:62)(s∗, ξ, t)w(t)dξdt
(cid:101)Ψx(v(D(x(ξ, t), t)))w(cid:62)(t)G(s, ξ, t)

(cid:90) b
×(cid:104)
(cid:105)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)
×(cid:16) ∂
(cid:90) s∧s∗
(cid:90) b
×(cid:104)
(cid:105)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)
×(cid:16) ∂
(cid:90) s∧s∗
(cid:90) b
×(cid:104)
×(cid:16) ∂

(cid:105)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)

G(cid:62)(s∗, ξ, t))w(t)dξdt
(cid:101)Ψtx(v(D(x(ξ, t), t)))w(cid:62)(t)G(s, ξ, t)

G(cid:62)(s∗, ξ, t))w(t)dξdt,

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

(cid:17)(cid:62)

(cid:17)(cid:62)

∂
∂D

∂
∂D

a

0

a

0

∂D

+

∂D

v(D(x(ξ, t), t))

where

(cid:90)
(cid:101)Ψt(v) :=
(cid:90)
(cid:101)Ψx(v) :=
×(cid:16) ∂
(cid:90)
(cid:101)Ψtx(v) :=

∂t

(cid:90)
(cid:90)
(cid:90)

R

R

(cid:90)
(cid:90)
(cid:90)

(cid:0)ψ + (τ v + γ

∂
∂t

x(ξ, t), γ)(cid:1)dψdτ dγ

K

(1)
t

(ψ)K

(1)
t

R

R

Rd+1

K

(1)
x (ψ)

Rd+1

(cid:17)(cid:62)(cid:16)

x(ξ, t)

K

(1)
x

∂
∂t

x(ξ, t)

(cid:0)ψ + (τ v + γ
(cid:16) ∂

x(ξ, t), γ)(cid:1)(cid:17)(cid:62)
(cid:17)(cid:62)(cid:16)

∂
∂t

R

R

Rd+1

K

(1)
t

(ψ)

x(ξ, t)

∂t

K

(1)
x

dψdτ dγ

(cid:0)ψ + (τ v + γ

∂
∂t

x(ξ, t), γ)(cid:1)(cid:17)(cid:62)

dψdτ dγ.

The pointwise rate of convergence is O(n−4/(d+4)).

Remark 2.4.3. An example of choosing a and b is a = t1 and b = tnt.

27

2.5 Hypothesis test

Based on Theorem 2.4.3., we further establish Theorem 2.5.1. to test the null hypothesis

regarding the zero rate of change in true fiber trajectory with respect to time. On the

contrary, Theorem 2.5.2. investigates possible alternatives to the null hypothesis in order to

address time-varying fiber trajectories.

Theorem 2.5.1. Suppose that the assumptions for Theorem 2.4.3. hold. Consider the
testing problem for 0 < a < b ≤ T,

H0 :

∂
∂t

x(s, t) = 0d versus HA :

x(s, t) (cid:54)= 0d, s ∈ [0, S], t ∈ [a, b].

∂
∂t

Under the null hypothesis, Theorem 2.4.3. gives the weak convergence in C([0, S], R) of the

function of stochastic processes

(cid:99)Wn,0(s) :=

(cid:113)

nhd
n

(cid:90) b

a

(cid:98)Xn(s, t)dt, s ∈ [0, S],

w(cid:62)(t)

∂
∂t

to the Gaussian process GP3,0(s), s ∈ [0, S] with the zero mean function and the covariance
function for all pairs of points (s, s∗) ∈ [0, S]

C3,0(s, s∗) =

(cid:90) s∧s∗

(cid:101)Ψt,0(v(D(x(ξ, t), t)))w(cid:62)(t)G(s, ξ, t)

(cid:105)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)

∂
∂D

0

v(D(x(ξ, t), t))

a

(cid:90) b
×(cid:104)
×(cid:16) ∂
(cid:82)
(cid:82)

R

R

where (cid:101)Ψt,0(v) :=(cid:82)

(cid:17)(cid:62)

v(D(x(ξ, t), t))

∂D

G(cid:62)(s∗, ξ, t)w(t)dξdt,

Rd+1 K

(1)
t

(ψ)K

(1)
t

(ψ + (τ v, γ))dψdτ dγ.

28

That is, for any finite index set of elements s1, s2, . . . , sm ∈ [0, S],


 .

C3,0(s1, s1)

. . . C3,0(s1, sm)

C3,0(s2, s1)

. . . C3,0(s2, sm)

...

. . .

...

C3,0(sm, s1)

. . . C3,0(sm, sm)





(cid:99)Wn,0(s1)
(cid:99)Wn,0(s2)
. . .(cid:99)Wn,0(sm)




0

0

0

. . .

 ,



⇒ N

as n → ∞. This is simplified as

(cid:99)Wn,0(·) ⇒ W0(·), as n → ∞,
(cid:21)(cid:62)

(cid:20)(cid:99)Wn,0(s1) (cid:99)Wn,0(s2)

where(cid:99)Wn,0(·) :=
is a m×1 random vector obtained
by stacking the sequence of stochastic processes in ascending order and W0(·) is a m × 1

. . . (cid:99)Wn,0(sm)

random vector from the multivariate normal distribution given its zero mean vector and the
covariance matrix C3,0(·,·).

Provided that the covariance matrix C3,0(·,·) is invertible, the Wald test of level α rejects

H0 if and only if, for 0 < a < b ≤ T,
(cid:104)
(cid:99)W(cid:62)
n,0(·)

C3,0(·,·)

(cid:105)−1(cid:99)Wn,0(·) > χ2

α,df =m,

where χ2

α,df =m is the upper-tail critical value of the limiting chi-square distribution with m

degrees of freedom.

Proof. Recall that

∂
∂t

x(s, t) =

(cid:90) s

(cid:110) ∂

0

∂D

v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)

∂
∂t

x(ξ, t)

29

+

∂
∂D

v(D(x(ξ, t), t))

∂
∂t

D(x(ξ, t), t)

(cid:111)

dξ, s ∈ [0, S], t ∈ [0, T ].

Since ∂

implies ∂

∂D v(D(x(ξ, t), t)) (cid:54)= 0 d×0.5d(d+1) for all ξ ∈ [0, S] given t ∈ [0, T ], the null hypothesis
∂tD(x(ξ, t), t) = 0 0.5d(d+1)×1 for all ξ ∈ [0, S] given t ∈ [0, T ]. Thereafter, we have
(cid:62), where
Σ0 is a m × m nonsingular matrix, we have (cid:99)Wn,0 ⇒ Σ0Z, where Z is a m × 1 standard
3,0(cid:99)Wn,0 is
normal random vector. Then the limiting distribution of the test statistic(cid:99)W(cid:62)
n,0C−1

the zero mean function. Provided that the covariance function satisfies C3,0 = Σ0Σ0

(Σ0Z)(cid:62)(Σ0Σ0

(cid:62))−1Σ0Z = Z(cid:62)(Σ0

−1Σ0)Z = Z(cid:62)Z ∼ χ2
Remark 2.5.1. For the standard Gaussian kernel with d = 3, (cid:101)Ψt,0(v) = 1

−1Σ0)(cid:62)(Σ0

df =m.

4π .

Proof. It is a special case of Remark 2.5.2. when c3 = 03.

Theorem 2.5.2. Suppose that the assumptions for Theorem 2.4.3. hold. Let cd be a nonzero
constant vector ∈ X . Consider the following alternative hypothesis for 0 < a < b ≤ T,

HA :

∂
∂t

x(s, t) = cd, s ∈ [0, S], t ∈ [a, b].

Under HA, Theorem 2.4.3. gives the weak convergence in C([0, S], R) of the function of

stochastic processes

(cid:99)Wn,A(s) :=

(cid:113)

nhd
n

(cid:90) b

a

(cid:18) ∂

∂t

w(cid:62)(t)

(cid:19)

(cid:98)Xn(s, t) − cd

dt, s ∈ [0, S],

to the Gaussian process GP3,A(s), s ∈ [0, S] with the zero mean function and the following

covariance function

30

C3(s, s∗) =

a

0

+

∂D

∂
∂D

(cid:17)(cid:62)

v(D(x(ξ, t), t))

(cid:90) s∧s∗

(cid:101)Ψt,A(v(D(x(ξ, t), t)))w(cid:62)(t)G(s, ξ, t)

G(cid:62)(s∗, ξ, t)w(t)dξdt
(cid:101)Ψx,A(v(D(x(ξ, t), t)))w(cid:62)(t)G(s, ξ, t)

(cid:90) b
×(cid:104)
(cid:105)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)
×(cid:16) ∂
(cid:90) s∧s∗
(cid:90) b
×(cid:104)
(cid:105)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)
×(cid:16) ∂
(cid:90) s∧s∗
(cid:90) b
×(cid:104)
×(cid:16) ∂

∂
(cid:105)
∂D
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)

(cid:101)Ψtx,A(v(D(x(ξ, t), t)))w(cid:62)(t)G(s, ξ, t)

G(cid:62)(s∗, ξ, t)w(t)dξdt,

G(cid:62)(s∗, ξ, t)w(t)dξdt

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

(cid:17)(cid:62)

(cid:17)(cid:62)

∂
∂D

∂D

+

a

0

a

0

∂D

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

where

(cid:101)Ψt,A(v) :=
(cid:101)Ψx,A(v) :=
(cid:101)Ψtx,A(v) :=

(cid:90)
(cid:90)
(cid:90)

(cid:90)
(cid:90)
(cid:90)

(cid:90)
(cid:90)
(cid:90)

R

R

Rd+1

R

R

Rd+1

R

R

Rd+1

(cid:0)ψ + (τ v + cdγ, γ)(cid:1)dψdτ dγ
(cid:0)ψ + (τ v + cdγ, γ)(cid:1)(cid:17)(cid:62)
(cid:16)
(cid:0)ψ + (τ v + cdγ, γ)(cid:1)(cid:17)(cid:62)

(1)
x

(1)
x

K

K

(1)
t

(ψ)K

(1)
t

d

(1)

x (ψ)cdc(cid:62)
(cid:16)
(ψ)c(cid:62)

(1)
t

d

K

K

K

dψdτ dγ

dψdτ dγ.

Proof. Suppose ∂
cd ∈ X . Since ∂

∂tx(s, t) = cd, s ∈ [0, S], t ∈ [0, T ] given the non-zero constant vector
∂tx(s2, t) = cd and ∂

0 =

=

=

∂
∂t

(cid:90) s2
(cid:90) s2

x(s2, t) − ∂
∂t
(cid:110) ∂

d
dt

0

∂D

s1

x(s1, t)

v(D(x(ξ, t), t))dξ −

d
dt

0

v(D(x(ξ, t), t))

∂tx(s1, t) = cd for 0 ≤ s1 < s2 ≤ S and t ∈ [0, T ],
(cid:90) s1

v(D(x(ξ, t), t))dξ

(cid:111)

D(x(ξ, t), t)

dξ.

∂
∂x

D(x(ξ, t), t)cd +

∂
∂D

v(D(x(ξ, t), t))

∂
∂t

31

This implies for ξ ∈ [0, S]

− ∂
∂D

v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)cd =

∂
∂D

v(D(x(ξ, t), t))

∂
∂t

D(x(ξ, t), t).

Provided that ∂

∂D v(D(x(ξ, t), t)) has linearly independent columns,

(cid:18) ∂
(cid:18) ∂

∂D

∂D

−

=

(cid:19)+ ∂
(cid:19)+ ∂

∂D

∂D

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)cd

∂
∂t

D(x(ξ, t), t),

since A+ is the Moore-Penrose inverse of A, we have

− ∂
∂x

D(x(ξ, t), t)cd =

∂
∂t

D(x(ξ, t), t).

Followed by ∂

∂tx(ξ, t) = cd and − ∂

∂xD(x(ξ, t), t)cd = ∂

∂tD(x(ξ, t), t) for ξ ∈ [0, S] and t ∈

[0, T ], we have the zero mean function.

Remark 2.5.2. For the standard Gaussian kernel with d = 3,

(cid:113)
(cid:113)

8π

8π

(i) (cid:101)Ψt,A(v) =
(ii) (cid:101)Ψx,A(v) =
(iii) (cid:101)Ψtx,A(v) =

1 + c(cid:62)

1 + c(cid:62)

1

1

1 +

3 c3 − (v(cid:62)c3)2

(cid:18)
(cid:18)
3 c3 − (v(cid:62)c3)2
(cid:113)
3 c3 − (v(cid:62)c3)2
c(cid:62)
1 + c(cid:62)
3 c3 − (v(cid:62)c3)2)

c(cid:62)
3 c3 +

8π(1 + c(cid:62)

(cid:19)

,

1

1 + c(cid:62)

3 c3 − (v(cid:62)c3)2
(v(cid:62)c3)2(1 − c(cid:62)

3 c3) + (c(cid:62)
3 c3 − (v(cid:62)c3)2

3 c3)2

1 + c(cid:62)

(cid:19)

,

3 c3 − (v(cid:62)c3)2

.

Proof. Since v(cid:62)v = 1 and v(cid:62)c3 = c(cid:62)

3 v = constant, we note that

(τ v + c3γ)(cid:62)(τ v + c3γ) = (τ + γv(cid:62)c3)2 + γ2c(cid:62)

3 c3 − γ2(v(cid:62)c3)2.

32

(i) (cid:101)Ψt,A(v) =

(cid:90)
(cid:90)

R

R

(cid:90)
(cid:90)
(cid:90)
(cid:16) − γ2

R

4

K

(1)
t

R4
1
16π2 exp

=

R
× exp

(cid:90)

×

√
1
π

R3

π

exp

(1 + c(cid:62)
(cid:18)

(ψ)K

(1)
t

(cid:0)ψ + (τ v + c3γ, γ)(cid:1)dψdτ dγ
(cid:18)
−(cid:16)

(cid:17)
(cid:16) − (τ + γv(cid:62)c3)2
(cid:17)(cid:90)
3 c3 − (v(cid:62)c3)2)
−(cid:16)

R
(τ v + c3γ)

(cid:17)(cid:62)(cid:16)

ψ2
t√
π

exp

ψx +

ψx +

4

2

ψt +

γ
2
(τ v + c3γ)

2

since the integral of the Gaussian distribution N(cid:0) − 0.5(τ v + c3γ), 0.5I3

(cid:17)2(cid:19)
(cid:17)(cid:19)
(cid:1) is 1,

dψxdψtdτ dγ

(cid:90)
(cid:90)

=

×

(cid:90)

R

R

R
ψ2
t√
π

(cid:16) − (τ + γv(cid:62)c3)2
−(cid:16)

(cid:17)2(cid:19)

ψt +

4

1
16π2 exp

(cid:18)

exp

(cid:17)

dψtdτ dγ

γ
2

(cid:16) − γ2

4

exp

(cid:17)

(1 + c(cid:62)

3 c3 − (v(cid:62)c3)2)

since E[ψ2

t ] = 2+γ2

4

,

(cid:113)

16π

(cid:90)

=

×

R

1
1 + c(cid:62)

(cid:113)
3 c3 − (v(cid:62)c3)2

(2 + γ2)

1 + c(cid:62)
3 c3 − (v(cid:62)c3)2
√
2
π

exp

(cid:90)

R

√
1
2

π

exp

(cid:17)

dτ

(cid:16) − (τ + γv(cid:62)c3)2
(cid:16) − γ2

(1 + c(cid:62)

4

4

3 c3 − (v(cid:62)c3)2)

(cid:17)

dγ

since E[γ2] =

1+c(cid:62)

2

3 c3−(v(cid:62)c3)2 ,
(cid:113)

1

1 + c(cid:62)
√
1
2

π

3 c3 − (v(cid:62)c3)2

(cid:18)
(cid:16) − (τ + γv(cid:62)c3)2
(cid:18)

1 +

(cid:17)

=

8π

(cid:90)

(cid:19)

1

1 + c(cid:62)

3 c3 − (v(cid:62)c3)2

R

×

exp

(cid:113)

(cid:19)
since the integral of the Gaussian distribution N(cid:0) − γv(cid:62)c3, 2(cid:1) is 1.

3 c3 − (v(cid:62)c3)2

3 c3 − (v(cid:62)c3)2

1 + c(cid:62)

1 + c(cid:62)

1 +

8π

dτ

=

1

4

1

33

(cid:90)
(cid:90)
(cid:90)
(cid:90)

(cid:90)
(cid:90)

(ii) (cid:101)Ψx,A(v) =

=

×

×

R

R

(cid:90)
(cid:90)

(cid:90)

3

K

(1)

(1)
x

(cid:16)
−(cid:16)
(cid:17)2(cid:19)

x (ψ)c3c(cid:62)
(cid:16) − (τ + γv(cid:62)c3)2
(cid:18)
−(cid:16)

ψx +

exp

ψt +

4

K

R

R4
1
16π2 exp
R
x c3c(cid:62)
ψ(cid:62)
(cid:18)
√
3 ψx
π
π

exp

(cid:0)ψ + (τ v + c3γ, γ)(cid:1)(cid:17)(cid:62)
(cid:17)

exp

(1 + c(cid:62)

(cid:16) − γ2
(cid:17)(cid:62)(cid:16)

4

(τ v + c3γ)

2

ψx +

dψtdτ dγ

R3
1√
π

R

γ
2

dψdτ dγ

(cid:17)
3 c3 − (v(cid:62)c3)2)
(cid:17)(cid:19)
(τ v + c3γ)

dψx

2

since the integral of the Gaussian distribution N (−0.5γ, 0.5) is 1,

(cid:90)

=

×

1
16π2 exp
R
ψ(cid:62)
x c3c(cid:62)
√
3 ψx
π
π

R

R3

(cid:17)

(cid:16) − (τ + γv(cid:62)c3)2
(cid:18)

4

−(cid:16)

ψx +

exp

(cid:16) − γ2
(cid:17)(cid:62)(cid:16)

4

exp

(1 + c(cid:62)

(cid:17)
3 c3 − (v(cid:62)c3)2)
(cid:17)(cid:19)
(τ v + c3γ)

(τ v + c3γ)

2

ψx +

2

dψxdτ dγ

Note that ψx is a normal random vector such that N (−0.5(τ v + c3γ), 0.5I3). Then

E[ψ(cid:62)

x c3c(cid:62)

3 ψx] = 0.5c(cid:62)

3 c3 +

τ 2(v(cid:62)c3)2 + 2τ γv(cid:62)c3c(cid:62)

3 c3 + γ2(c(cid:62)

3 c3)2

.

Plugging it to the previous step,

(cid:113)
(cid:113)
(cid:113)
(cid:113)
(cid:113)

8π

(cid:90)

R

16π

(cid:90)

R

16π

c(cid:62)
3 c3
1 + c(cid:62)
3 c3 − (v(cid:62)c3)2
1 + c(cid:62)
3 c3 − (v(cid:62)c3)2
√
2

π

1
1 + c(cid:62)
3 c3 − (v(cid:62)c3)2
1 + c(cid:62)
3 c3 − (v(cid:62)c3)2
√
2
π
(c(cid:62)
3 c3)2
1 + c(cid:62)
3 c3 − (v(cid:62)c3)2

(ii) =

×

+

×

+

4

(cid:16) − (τ + γv(cid:62)c3)2

4

(cid:17)

dτ

(cid:17)

(cid:90)

R

√
1
2

exp

π

(cid:16) − γ2

exp

(cid:90)

R

exp

(cid:90)

R

4

(1 + c(cid:62)

3 c3 − (v(cid:62)c3)2)
τ 2(v(cid:62)c3)2 + 2τ γv(cid:62)c3c(cid:62)
√
3 c3
2
π

exp

(cid:16) − γ2

4

√
1
2

π

exp

(1 + c(cid:62)

(cid:17)
3 c3 − (v(cid:62)c3)2)
(cid:17)

(cid:16) − (τ + γv(cid:62)c3)2

dτ

dγ

4

34

dγ

(cid:16) − (τ + γv(cid:62)c3)2

(cid:17)

dτ

4

(cid:90)

R

×

γ2(cid:113)

1 + c(cid:62)
3 c3 − (v(cid:62)c3)2
√
2

π

(cid:16) − γ2

4

(cid:17)
3 c3 − (v(cid:62)c3)2)

(1 + c(cid:62)

dγ

exp

2

1+c(cid:62)

3 c3−(v(cid:62)c3)2 ,

since E[τ 2] = 2 + γ2(v(cid:62)c3)2, E[τ ] = −γv(cid:62)c3 and E[γ] = 0 and E[γ2] =
(cid:19)

(v(cid:62)c3)2(1 − c(cid:62)

1

3 c3)2

(cid:18)
c(cid:62)
3 c3 +

(cid:113)

,

3 c3) + (c(cid:62)
3 c3 − (v(cid:62)c3)2

1 + c(cid:62)

=

8π

1 + c(cid:62)

(iii) (cid:101)Ψtx,A(v) =

=

×

×

R

R

K

(1)
t

(cid:90)

3 c3 − (v(cid:62)c3)2
(cid:90)
(cid:90)
(cid:90)
(cid:90)
(cid:90)
(cid:90)

R4
1
16π2 exp
R
c(cid:62)
√
3 ψx
π
π
ψt√
π

(cid:18)

exp

exp

R3

R

R

(cid:16)

(cid:0)ψ + (τ v + c3γ, γ)(cid:1)(cid:17)(cid:62)
(cid:16) − γ2
(cid:17)(cid:62)(cid:16)

4
(τ v + c3γ)

(cid:17)

exp

4

ψx +

3

K

(1)
x

(ψ)c(cid:62)
(cid:16) − (τ + γv(cid:62)c3)2
(cid:18)
−(cid:16)
−(cid:16)

(cid:17)2(cid:19)

ψx +

ψt +

2

γ
2

dψtdτ dγ

dψdτ dγ

(cid:17)
3 c3 − (v(cid:62)c3)2)

(1 + c(cid:62)

(cid:17)(cid:19)

(τ v + c3γ)

2

dψx

since E[ψx] = −0.5(τ v + c3γ) and E[ψt] = −0.5γ,
(cid:90)

(cid:113)
(cid:113)

16π

(cid:90)

=

×

R

1
1 + c(cid:62)
3 c3 − (v(cid:62)c3)2
1 + c(cid:62)
3 c3 − (v(cid:62)c3)2
√
2

π

R

exp

(cid:16) − γ2

4

τ γv(cid:62)c3 + γ2c(cid:62)
3 c3

√
2
π

exp

4

(cid:17)
3 c3 − (v(cid:62)c3)2)

dγ

(1 + c(cid:62)

(cid:16) − (τ + γv(cid:62)c3)2

(cid:17)

since E[τ ] = −γv(cid:62)c3 and E[γ2] =

2

1+c(cid:62)

3 c3−(v(cid:62)c3)2 ,

(cid:113)
3 c3 − (v(cid:62)c3)2
c(cid:62)
1 + c(cid:62)

=

8π(1 + c(cid:62)

3 c3 − (v(cid:62)c3)2)

3 c3 − (v(cid:62)c3)2

.

35

Chapter 3

Pseudo Algorithms

3.1 Theorem 2.4.1.

The main contribution of Chapter 3 is to provide pseudo algorithms which can be easily

converted into any computer programming language. Pseudocodes are simply written to

deliver what programming codes might look like. Resulting from Theorem 2.4.1., Algorithm

3.1 is a pseudocode to implement the plug-in estimator (cid:98)Xn(s, t) as in (2.4). All ODEs are

approximated via Euler’s method.

Input:
Fix t ∈ [0, T ] and x0 ∈ X .

Fix β1 > 0 such that nhd+4

n → β1 as n → ∞.

Let s0 = 0. Let δ > 0 be a size of each step such that sk+1 = sk + δ, k = 0, 1, . . .

Initialize (cid:98)Xn(s0, t) = x0 at t ∈ [0, T ].
while sk+1 ≤ S do

(cid:98)Xn(sk+1, t) ≈ (cid:98)Xn(sk, t) + δv((cid:98)Dn((cid:98)Xn(sk, t), t))

end

Algorithm 3.1: Fiber Trajectory Estimation

36

Input:
Let hn,1 → 0, nhd+2
while sk ≤ S do
(cid:18)

∂
∂x

− 1
2

i=1

hn,1

× exp

n(cid:88)

n(cid:88)

n,2 → ∞ as n → ∞.

(cid:18)((cid:98)Xn(sk, t), t) − Ui

n,1 → ∞, hn,2 → 0, and nhd+3
(cid:101)D(Ui)(cid:0)(cid:98)Xn(sk, t) − Xi
(cid:98)Dn((cid:98)Xn(sk, t), t) = −(2π)−(d+1)/2(nhd+2
n,1 )−1
(cid:19)(cid:19)
(cid:19)(cid:62)(cid:18)((cid:98)Xn(sk, t), t) − Ui
(cid:101)D(Ui)(cid:0)t − Ti
(cid:98)Dn((cid:98)Xn(sk, t), t) = −(2π)−(d+1)/2(nhd+2
n,1 )−1
(cid:19)(cid:19)
(cid:19)(cid:62)(cid:18)((cid:98)Xn(sk, t), t) − Ui
(cid:18)(cid:0)(cid:98)Xn(sk, t) − Xi
(cid:98)Dn((cid:98)Xn(sk, t), t) = (2π)−(d+1)/2(nhd+3
(cid:101)D(Ui)
n,2 )−1
(cid:19)(cid:19)
(cid:19)(cid:62)(cid:18)((cid:98)Xn(sk, t), t) − Ui
(cid:18)(cid:0)t − Ti
(cid:19)
(cid:1)2 − 1
∂t2(cid:98)Dn((cid:98)Xn(sk, t), t) = (2π)−(d+1)/2(nhd+3
(cid:101)D(Ui)
n,2 )−1
(cid:19)(cid:62)(cid:18)((cid:98)Xn(sk, t), t) − Ui
(cid:19)(cid:19)

(cid:18)((cid:98)Xn(sk, t), t) − Ui

(cid:18)((cid:98)Xn(sk, t), t) − Ui

(cid:18)((cid:98)Xn(sk, t), t) − Ui

n(cid:88)

i=1

n(cid:88)

i=1

× exp

(cid:18)

(cid:18)

(cid:18)

i=1

hn,1

× exp

− 1
2

− 1
2

∂
∂t

∂2
∂x2
j

∂2

hn,1

hn,1

(cid:1)

.

hn,2

hn,2

hn,2

hn,2

× exp

− 1
2

(cid:1)(cid:62)

(cid:19)
(cid:1)2
j − 1

, j = 1, . . . , d.

1 ≤ p, r, w ≤ d,
∂
∂D

Let Z((cid:98)Dn) = λ((cid:98)Dn)Id − (cid:98)Dn. Define δrw = 1 if r = w and 0 otherwise. Then for
vp((cid:98)Dn((cid:98)Xn(sk, t), t)) = (1 − δrw/2)(cid:2)Z+((cid:98)Dn((cid:98)Xn(sk, t), t))prvw((cid:98)Dn((cid:98)Xn(sk, t), t))
+ Z+((cid:98)Dn((cid:98)Xn(sk, t), t))pwvr((cid:98)Dn((cid:98)Xn(sk, t), t))(cid:3)
TrH((cid:98)Xn(sk, t), t) =

(cid:98)Dn((cid:98)Xn(sk, t), t) +

∂t2(cid:98)Dn((cid:98)Xn(sk, t), t)

d(cid:88)

∂2

∂2
∂x2
j

j=1

end

Algorithm 3.2: Pre-step Functions

37

Algorithm 3.3 is intended to implement the mean function of the limiting Gaussian
(s, t), s ∈ [0, S], t ∈ [0, T ].

process given any fixed time point as in Theorem 2.4.1., i.e., µβ1

(s0, t) = 0d at t ∈ [0, T ].

Input:

Initialize(cid:98)µβ1
while sk+1 ≤ S do
(sk+1, t) ≈(cid:98)µβ1

(cid:98)µβ1

δ

+

end

v((cid:98)Dn((cid:98)Xn(sk, t), t))

(cid:98)Dn((cid:98)Xn(sk, t), t)(cid:98)µβ1

∂
∂x

∂
∂D

(sk, t) + δ

√
β1
2

∂
∂D

v((cid:98)Dn((cid:98)Xn(sk, t), t))TrH((cid:98)Xn(sk, t), t)

(sk, t)

Algorithm 3.3: Mean Function

For the covariance function of the limiting Gaussian process in Theorem 2.4.1., we first

define Algorithm 3.4 as follows:

while sk ≤ S do
(cid:101)ΣΓ,n(Ui)
#N ((cid:98)Xn(sk, t))
(cid:98)Σn((cid:98)Xn(sk, t), t) = (2π)−(d+1)/2(nhd+1

Xj∈N ((cid:98)Xn(sk,t))

(cid:88)

=

1

× exp

end

(cid:0)Y (Ui) − B(Ui)(cid:101)D(Ui)(cid:1)(cid:0)Y (Ui) − B(Ui)(cid:101)D(Ui)(cid:1)(cid:62)

n(cid:88)
(cid:18)((cid:98)Xn(sk, t), t) − Ui

)−1

i=1

n

(cid:101)ΣΓ,n(Ui)
(cid:19)(cid:62)(cid:18)((cid:98)Xn(sk, t), t) − Ui

(cid:19)(cid:19)

hn

hn

(cid:18)

− 1
2

Algorithm 3.4: Noise Function

38

C1((s, t), (s∗, t)), s, s∗ ∈ [0, S], t ∈ [0, T ], the covariance function of the limiting Gaussian

process in Theorem 2.4.1., can be employed via Algorithm 3.5.

Input:

Initialize (cid:98)C1((s0, t), (s0, t)) = 0d×d at t ∈ [0, T ].
while sk+1 ≤ S do
(cid:98)C1((sk+1, t), (sk+1, t)) ≈ (cid:98)C1((sk, t), (sk, t))
v((cid:98)Dn((cid:98)Xn(sk, t), t))
(cid:16) ∂
+ δ(cid:98)C1((sk, t), (sk, t))
+ δΨ(v((cid:98)Dn((cid:98)Xn(sk, t), t)))
×(cid:104)(cid:98)Dn((cid:98)Xn(sk, t), t)(cid:98)D(cid:62)
×(cid:16) d
(cid:17)(cid:62)
v((cid:98)Dn((cid:98)Xn(sk, t), t))

d
dD

+ δ

dD

∂
∂x

(cid:98)Dn((cid:98)Xn(sk, t), t)(cid:98)C1((sk, t), (sk, t))
(cid:17)(cid:62)(cid:16) d
(cid:98)Dn((cid:98)Xn(sk, t), t)
v((cid:98)Dn((cid:98)Xn(sk, t), t))
(cid:105)
n ((cid:98)Xn(sk, t), t) +(cid:98)Σn((cid:98)Xn(sk, t), t)

d
dD

dD

∂x

(cid:17)(cid:62)
v((cid:98)Dn((cid:98)Xn(sk, t), t))

end
For Gaussian kernel with d = 3, Ψ(v(·)) = 1
√

8π

.

π

Algorithm 3.5: Covariance Function

By calling previously defined Algorithms, Algorithm 3.6 provides the numerical approxi-
mation of the 100(1− α)% confidence ellipsoid for the true fiber trajectory given the param-
eter time t ∈ [0, T ].

39

while sk ≤ S do

Let

(cid:113)

nhd
n

(cid:16)(cid:98)Xn(sk, t) − x(sk, t)
(cid:17)

.

y1(sk, t) =

Then the 100(1 − α)% Confidence Ellipsoid of x(sk, t) given t ∈ [0, T ] is as

follows:

P (

(cid:12)(cid:12)(cid:12)((cid:98)C1((sk, t), (sk, t)))−1/2(y1(sk, t) −(cid:98)µβ1

(cid:12)(cid:12)(cid:12) ≤ Rα) ≈ 1 − α,

(sk, t))

where P (|Z| ≤ Rα) = 1 − α for a standard normal vector Z in Rd.

end

Algorithm 3.6: Fiber Trajectory with Confidence Ellipsoids

3.2 Theorem 2.5.1.

In this section, let us first introduce Simpson’s rule to approximate the definite integral of

the covariance function in Theorem 2.5.1. under the null hypothesis. Simpson’s rule provides

an accurate (almost exact) numerical approximation of the definite integral with few data

points. For instance, suppose that we have five time points, i.e., nt = 5. Then Simpson’s

rule approximates the definite integral of f over the interval [t1, t5] as follows:

(cid:90) t5

t1

(cid:110)

f (t)dt ≈ t5 − t1

12

f (t1) + 4f (t2) + 2f (t3) + 4f (t4) + f (t5)

.

(cid:111)

For the sample size of nt ≥ 9, the extended Simpson’s rule based on Press and Vetterling

(1989) is as follows:

40

(cid:90) tnt

t1

f (t)dt ≈ tnt − t1
48(nt − 1)

(cid:110)

17f (t1) + 59f (t2) + 43f (t3) + 49f (t4) + 48

+ 49f (tnt−3) + 43f (tnt−2) + 59f (tnt−1) + 17f (tnt)

.

nt−4(cid:88)

i=5

f (ti)

(cid:111)

Returning back to the covariance function in Theorem 2.5.1. under H0, when d = 3,

a = t1, b = tnt, and w(t) = 1(cid:62)

3 , we have

C3,0(s, s∗) :=

1(cid:62)

3

1
4π

(cid:90) tnt

t1

C2,0((s, t), (s∗, t))dt13,

where

(cid:90) s∧s∗
×(cid:104)
×(cid:16) ∂

0

∂D

C2,0((s, t), (s∗, t)) :=

∂
∂D

G(s, ξ, t)

(cid:105)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)

v(D(x(ξ, t), t))

(cid:17)(cid:62)

v(D(x(ξ, t), t))

G(cid:62)(s∗, ξ, t)dξ.

Then we use Simpson’s rule to approximate(cid:82) tnt

C2,0((s, t), (s∗, t))dt.

t1

We shall provide the numerical implementation for C2,0((s, t), (s∗, t)) as well. For the

variance function,

∂
∂s

C2,0((s, t), (s, t)) =

∂
∂D

v(D(x(s, t), t))

D(x(s, t), t)C2,0((s, t), (s, t))

(cid:17)(cid:62)(cid:16) ∂

∂D

(cid:17)(cid:62)

+ C2,0((s, t), (s, t))

D(x(s, t), t)

v(D(x(s, t), t))

v(D(x(s, t), t))

(cid:105)
∂
∂D
D(x(s, t), t)D(cid:62)(x(s, t), t) + Γ(x(s, t), t)Γ(cid:62)(x(s, t), t)

+

×(cid:104)
×(cid:16) ∂

∂D

v(D(x(s, t), t))

.

∂
∂x

(cid:16) ∂

∂x

(cid:17)(cid:62)

41

For the covariance function, suppose s∗ = s + ∆s, where ∆s > 0. Then

∂
∂s

C2,0((s, t), (s + ∆s, t)) =

D(x(s, t), t)C2,0((s, t), (s + ∆s, t))

∂
∂D

v(D(x(s, t), t))

∂
∂x

(cid:17)(cid:62)

+ C2,0((s, t), (s + ∆s, t))

D(x(s + ∆s, t), t)

(cid:16) ∂
(cid:17)(cid:62)

∂x

(cid:17)(cid:62)

×(cid:16) ∂
×(cid:104)
×(cid:16) ∂

+

v(D(x(s + ∆s, t), t))

v(D(x(s, t), t))

∂D
∂
∂D
D(x(s, t), t)D(cid:62)(x(s, t), t) + Γ(x(s, t), t)Γ(cid:62)(x(s, t), t)

(cid:105)

v(D(x(s, t), t))

∂D

G(cid:62)(s + ∆s, s, t),

where Green’s function is

G(s + ∆s, s, t) = G(s, s, t) +

∂
∂u

G(u, s, t)du

(cid:90) s+∆s

s

∂
∂D

(cid:90) s+∆s

= Id +
≈ Id + ∆s

s

v(D(x(u, t), t))

∂
∂x

D(x(u, t), t)G(u, s, t)du

∂
∂D

v(D(x(s, t), t))

∂
∂x

D(x(s, t), t).

In general, for the covariance function between s and s + l∆s, l ≥ 1, we have

∂
∂s

C2,0((s, t), (s + l∆s, t)) =

D(x(s, t), t)C2,0((s, t), (s + l∆s, t))

∂
∂D

v(D(x(s, t), t))

∂
∂x

(cid:17)(cid:62)

+ C2,0((s, t), (s + l∆s, t))

D(x(s + l∆s, t), t)

(cid:16) ∂
(cid:17)(cid:62)

∂x

×(cid:16) ∂
×(cid:104)
×(cid:16) ∂

+

v(D(x(s + l∆s, t), t))

v(D(x(s, t), t))

∂D
∂
∂D
D(x(s, t), t)D(cid:62)(x(s, t), t) + Γ(x(s, t), t)Γ(cid:62)(x(s, t), t)

(cid:105)

v(D(x(s, t), t))

∂D

G(cid:62)(s + l∆s, s, t),

(cid:17)(cid:62)

42

where Green’s function can be obtained by

G(s + l∆s, s, t) ≈ l(cid:89)

(cid:104)Id + ∆s

∂
∂D

v(D(x(s + (j − 1)∆s, t), t))

∂
∂x

j=1

D(x(s + (j − 1)∆s, t), t)

(cid:105)

.

The following set of Algorithms are used to test the null hypothesis regarding the zero

rate of change in time of the true fiber trajectory. We make full use of Algorithms stated in

Section 3.1.

Input:

Fix a and b such that a = t1 and b = tnt.

Fix β1 > 0 such that nhd+4

n → β1 as n → ∞.

Let s0 = 0. Let δ > 0 be a size of step such that sk+1 = sk + δ, k = 0, 1, . . . .
Choose w(t) = 1(cid:62)

d , i.e, the constant weight function.

while sk+1 ≤ S do(cid:98)Xn(sk+1, tnt) ≈ (cid:98)Xn(sk, tnt) + δv((cid:98)Dn((cid:98)Xn(sk, tnt), tnt))
(cid:98)Xn(sk+1, t1) ≈ (cid:98)Xn(sk, t1) + δv((cid:98)Dn((cid:98)Xn(sk, t1), t1))
(cid:18)(cid:98)Xn(sk+1, tnt) − (cid:98)Xn(sk+1, t1)
(cid:19)
(cid:99)Wn,0(sk+1) ≈(cid:113)

n1(cid:62)

nhd

d

end

Algorithm 3.7: Statistic

43

Input:

Initialize (cid:98)C2,0((s0, t), (s0, t)) = 0d×d and (cid:98)C2,0((s0, t), (s0+l, t)) = 0d×d, l = 1, 2, . . . at

t ∈ [0, T ].

while sk+1 = S or sk+1+l = S do

∂x

j=1

∂D

+ δ

+ δ

∂
∂x

∂
∂D

∂
∂D

(cid:17)(cid:62)(cid:16) ∂

(cid:98)C2,0((sk+1, t), (sk+1, t)) ≈ (cid:98)C2,0((sk, t), (sk, t))
(cid:98)Dn((cid:98)Xn(sk, t), t)(cid:98)C2,0((sk, t), (sk, t))
(cid:98)Dn((cid:98)Xn(sk, t), t)

v((cid:98)Dn((cid:98)Xn(sk, t), t))
(cid:17)(cid:62)
(cid:16) ∂
v((cid:98)Dn((cid:98)Xn(sk, t), t))
+ δ(cid:98)C2,0((sk, t), (sk, t))
v((cid:98)Dn((cid:98)Xn(sk, t), t))
×(cid:104)(cid:98)Dn((cid:98)Xn(sk, t), t)(cid:98)D(cid:62)
(cid:105)
n ((cid:98)Xn(sk, t), t) +(cid:98)Σn((cid:98)Xn(sk, t), t)
×(cid:16) ∂
(cid:17)(cid:62)
v((cid:98)Dn((cid:98)Xn(sk, t), t))
(cid:104)Id + δ
(cid:98)G(sk+l, sk, t) ≈ l(cid:89)
v((cid:98)Dn((cid:98)Xn(sk + (j − 1)δ, t), t))
(cid:105)
(cid:98)Dn((cid:98)Xn(sk + (j − 1)δ, t), t)
(cid:98)C2,0((sk+1, t), (sk+1+l, t)) ≈ (cid:98)C2,0((sk, t), (sk+l, t))
(cid:98)Dn((cid:98)Xn(sk, t), t)(cid:98)C2,0((sk, t), (sk+l, t))
(cid:98)Dn((cid:98)Xn(sk+l, t), t)

× ∂
∂x

∂
∂D

∂D

+ δ

∂
∂D

v((cid:98)Dn((cid:98)Xn(sk, t), t))
(cid:16) ∂
+ δ(cid:98)C2,0((sk, t), (sk+l, t))
v((cid:98)Dn((cid:98)Xn(sk, t), t))
×(cid:104)(cid:98)Dn((cid:98)Xn(sk, t), t)(cid:98)D(cid:62)
×(cid:16) ∂
v((cid:98)Dn((cid:98)Xn(sk, t), t))

∂
∂D

+ δ

∂x

∂
∂x

(cid:17)(cid:62)(cid:16) ∂

(cid:17)(cid:62)
v((cid:98)Dn((cid:98)Xn(sk+l, t), t))
(cid:105)
n ((cid:98)Xn(sk, t), t) +(cid:98)Σn((cid:98)Xn(sk, t), t)
(cid:17)(cid:62)(cid:98)G(cid:62)(sk+l, sk, t)

∂D

∂D

end

Algorithm 3.8: Nested Variance Function and Nested Covariance Function

44

For example, nt ≥ 9,

while sk = S or sk+l = S do

(cid:98)C3,0(sk, sk) = 1(cid:62)
(cid:90) tnt

d

(cid:90) tnt

t1

(cid:101)Ψt,0(v((cid:98)Dn((cid:98)Xn(sk, t), t)))C2,0((sk, t), (sk, t))dt1d

C2,0((sk, t), (sk, t))dt ≈ tnt − t1
48(nt − 1)

t1

17C2,0((sk, t1), (sk, t1))

+ 59C2,0((sk, t2), (sk, t2)) + 43C2,0((sk, t3), (sk, t3))

(cid:110)

nt−4(cid:88)

i=5

(cid:110)

nt−4(cid:88)

i=5

+ 49C2,0((sk, t4), (sk, t4)) + 48

C2,0((sk, ti), (sk, ti))

+ 49C2,0((sk, tnt−3), (sk, tnt−3)) + 43C2,0((sk, tnt−2), (sk, tnt−2))

+ 59C2,0((sk, tnt−1), (sk, tnt−1)) + 17C2,0((sk, tnt), (sk, tnt))

(cid:111)
(cid:101)Ψt,0(v((cid:98)Dn((cid:98)Xn(sk, t), t)))C2,0((sk, t), (sk+l, t))dt1d

(cid:90) tnt

(cid:98)C3,0(sk, sk+l) = 1(cid:62)
(cid:90) tnt

d

t1

C2,0((sk, t), (sk+l, t))dt ≈ tnt − t1
48(nt − 1)

t1

17C2,0((sk, t1), (sk+l, t1))

+ 59C2,0((sk, t2), (sk+l, t2)) + 43C2,0((sk, t3), (sk+l, t3))

+ 49C2,0((sk, t4), (sk+l, t4)) + 48

C2,0((sk, ti), (sk+l, ti))

+ 49C2,0((sk, tnt−3), (sk+l, tnt−3)) + 43C2,0((sk, tnt−2), (sk+l, tnt−2))

+ 59C2,0((sk, tnt−1), (sk+l, tnt−1)) + 17C2,0((sk, tnt), (sk+l, tnt))

(cid:111)

end

For Gaussian kernel with d = 3, (cid:101)Ψt,0(v(·)) = 1

4π .

Algorithm 3.9: Covariance Function

Then Algorithm 3.10 returns Theorem 2.5.1. as follows:

45

Let m when sm reaches S. Consider

(cid:99)Wn,0(·) :=



(cid:99)Wn,0(s1)
(cid:99)Wn,0(s2)
...(cid:99)Wn,0(sm)

 and (cid:98)C3,0(·,·) :=



(cid:98)C3,0(s1, s1)
(cid:98)C3,0(s2, s1)
(cid:98)C3,0(sm, s1)

...

. . .

. . .

. . .

. . .

 .

(cid:98)C3,0(s1, sm)
(cid:98)C3,0(s2, sm)
(cid:98)C3,0(sm, sm)

...

Then the Wald test of level α rejects H0 as in Theorem 2.5.1. if and only if

(cid:99)W(cid:62)
n,0(·)

(cid:104)(cid:98)C3,0(·,·)

(cid:105)−1(cid:99)Wn,0(·) > χ2

α,df =m,

where χ2

α,df =m is the upper-tail critical value of the limiting chi-square distribution

with m degrees of freedom.

Algorithm 3.10: Hypothesis Test

46

Chapter 4

Simulation and Application to Real

Data

4.1 Artificial data: semicircular trajectory over time

In this section, we attempt to replicate an artificial diffusion tensor where the true fiber

trajectory in 2D projection onto the xy-plane is semicircular at a fixed time point. A similar

example without considering time points can be found in Koltchinskii et al. (2007) and

Carmichael and Sakhanenko (2016). We consider a longitudinal DTI study for the case of
d = 3. The corresponding diffusion tensor D is a 6 × 1 vector due to its symmetry. D is
defined at u ∈ G = [0, 1]4, where X = [0, 1]3 and T = 1. The initial value condition for
the ODE is x0 = [0.5 cos(0.3) 0.5 sin(0.3) 0.5](cid:62) ∈ X . For the estimation procedure, we use

Xj, j = 1, 2, . . . , nx which is a value on a 3D grid and Tk = k/nt, k = 1, 2, . . . , nt which

belongs to a set of equally spaced values in [0, 1]. The number of magnetic field gradient

directions is 48, i.e., N = 48. The number of scans at each visit is 1, i.e, M = 1. A
48 × 6 tensor B is used corresponding to the uniformly distributed 48 gradient directions
on a unit sphere. A 48 × 1 additive noise tensor is normally distributed with mean 0
and standard deviation of 0.2236, i.e., 0.25 × 10/(SN R1.5), where the signal-to-noise ratio
(SNR) is 5. Then a 48 × 1 response tensor Y is generated as in (2.2). Thereafter, the

47

standard 3D Gaussian kernel for the Nadaraya-Watson kernel estimator is applied on the
OLS estimates. The bandwidth hn = (n/β1)−7 is chosen based on different sample sizes n
and β1 = 10−4. The bandwidths for the first and second derivatives are hn,1 = (n/β1)−1/9
and hn,2 = (n/β1)−1/10, respectively. Without loss of generality, we simulate equally spaced

time points tk = k/nt, k = 1, 2, . . . , nt. The constant weight function between a = t1 and

b = tnt is used. The size of each step δ is 0.015 and the number of steps m is 30. The null

hypothesis H0 is specified, followed by the alternative hypothesis such as HA1

, HA2

or HA3

.

Monte Carlo Simulations with a size of 100 were performed with different sample sizes.

The power of the test was computed as the proportion of the time we rejected the null

hypothesis H0 when the alternative hypothesis was indeed true. We used the empirical 5%

critical value driven from the null 100 simulations to ensure a 5% of type I error and the

theoretical 5% critical value from the limiting chi-square distribution with 30 degrees of

freedom, i.e., χ2

0.05,30 = 43.7730. All analyses were performed using MATLAB R2019b with

C-subroutines.

satisfies |(cid:113)

Null Hypothesis 4.1.1. For the null hypothesis H0, 2D semicircular trajectories remain
the same over the time from t1 to tnt. At any fixed time point, a spatial point x ∈ [0, 1]3
2 − 0.5| < 0.05 and |x3 − 0.5| < 0.05. Then the corresponding diffusion
tensor is defined as D = U ΛU(cid:62), where the columns of U are orthonormal eigenvectors and

x2
1 + x2

Λ is the diagonal matrix associated with the eigenvalue, i.e,



x2(cid:113)
− x1(cid:113)

x2
1+x2
2

x2
1+x2
2
0

U =

 , Λ =



0

0

1

10

0

0

x1(cid:113)
x2(cid:113)

x2
1+x2
2

x2
1+x2
2
0

 .

0

0

1

0

2

0

48

In the following Figure 4.1, we used the 1st null Monte Carlo simulation with the sample

size of nx = 803 and nt = 5.

(a) 3D

(b) (a) is projected onto the xy-plane

Figure 4.1: A solid blue line indicates the estimated trajectory given the 5th time point
under H0, whereas blue dotted lines represent the pointwise 95% confidence ellipsoids along
the estimated trajectory.

At each time point our estimated trajectory was nearly the same as the true semicircular

pathway in its 2D projection onto the xy-plane, although the estimated trajectory slightly

varied along the z-axis.

In the following alternative hypotheses HA1

and HA2

, we fixed

nt = 5, but used either nx = 403 or nx = 803.

Alternative Hypothesis 4.1.1. For the alternative hypothesis HA1

curve into a semi-ellipse while we keep the four semicircular curves with the radius of 0.5.

At the 5th time point, a spatial point x ∈ [0, 1]3 satisfies |(cid:113)

2 − 0.5| < 0.05
and |x3 − 0.5| < 0.05, where the value of c is associated with the y-coordinate such as

, we change the 5th

1 +(cid:0) 0.5

c

(cid:1)2x2

x2

0.55, 0.525, 0.475, and 0.45. Depending on the number c, the last curve is either stretched or

squeezed into the semi-ellipse along the y direction.

49

Figure 4.2 is the plot of the 1st Monte Carlo simulation under HA1

with the sample size

of nx = 803 and nt = 5.

(a) c = 0.55

(b) c = 0.525

(c) c = 0.475

(d) c = 0.45

Figure 4.2: At each c value the 5th estimated trajectory and its 95% confidence ellipsoids
under HA1
, both colored in red, are overlaid with those of the reference value (c = 0.5)
under H0, colored in blue. All 3D figures are projected onto the xy-plane.

50

the power of the test

nx

nt

403

803

5

5

n

c = 0.55

c = 0.525

c = 0.475

c = 0.45

320, 000

1.00

2, 560, 000

1.00

0.80

1.00

0.62

1.00

1.00

1.00

(a) The upper 5th percentile of the simulated distribution under H0 is used as a

critical value.

nx

nt

403

803

5

5

the power of the test

n

c = 0.55

c = 0.525

c = 0.475

c = 0.45

320, 000

0.64

2, 560, 000

1.00

0.07

0.90

0.01

0.90

0.53

1.00

(b) The upper 5th percentile of the limiting chi-square distribution with 30 de-

grees of freedom is used as a critical value.

Table 4.1: Monte Carlo simulation-based power analysis when HA1

is true

Table 4.1 summarizes the power of the test using either the empirical or theoretical 5%

critical value. We observed that the empirical 5% critical value was lower than the theoretical

5% critical value, i.e., χ2

0.05,30 = 43.7730, in either case nx = 403 and nt = 5 or nx = 803

and nt = 5, resulting in higher power. We noticed two findings about the power of the test

regardless of whether we used either of the two critical values. First, the power of the test

increased as the value of c deviated from the reference value (c = 0.5). Second, the power

was also improved by increasing the size of nx. In other words, we expect a higher power for
the test using DTI images based on a matrix size of 256 × 256 with 48 slices than one using
a matrix size of 128 × 128 with 48 slices.

51

Alternative Hypothesis 4.1.2. For the alternative hypothesis HA2

, we change the radius

2 − r| < 0.05 and |x3 − r| < 0.05

of the 5th curve while the four semicircular curves have the same radius of 0.5. At the 5th

time point, a spatial point x ∈ [0, 1]3 satisfies |(cid:113)

1 + x2
x2
where r is the radius in the set of {0.55, 0.525, 0.475, 0.45}.

While HA1

studied a gradual change of the true trajectory from the beginning to the

end, HA2

addressed its radical change throughout the whole pathway. The entire change

of the true trajectory was linked with the drastic increase or decrease in (cid:99)Wn,0 in Theorem

2.5.1., which extremely contributed to the magnitude of the test statistic compared to the

incremental change of the true pathway. Table 4.2 shows that the highest power of the test

regardless of the type of critical value since (cid:99)Wn,0 is sensitive to the entire change of fiber

pathway compared to its incremental change.

the power of the testa

nx

nt

403

803

5

5

n

r = 0.55

r = 0.525

r = 0.475

r = 0.45

320, 000

1.00

2, 560, 000

1.00

1.00

1.00

1.00

1.00

1.00

100

a the power of the test was the same in either case using the empirical or

theoretical 5% critical value.

Table 4.2: Monte Carlo simulation-based power analysis when HA2

is true

The following Figure 4.3 is a plot obtained from the 1st Monte Carlo simulation under

HA2

with the sample size of nx = 803 and nt = 5.

52

(a) r = 0.55

(b) r = 0.525

(c) r = 0.475

(d) r = 0.45

Figure 4.3: At each r value the 5th estimated trajectory and its 95% confidence ellipsoids
under HA2
, both colored in red, are overlaid with those of the reference value (r = 0.5)
under H0, colored in blue. All 3D figures are projected onto the xy-plane.

In the following alternative hypothesis HA3

, we focused on the incremental change of the

true fiber trajectory. In HA3

, we used a reasonably larger sample size for time points in DTI

such as nt = 10 and nt = 20.

53

Alternative Hypothesis 4.1.3. The alternative hypothesis HA3

is akin to HA1

, however,

we divide the simulated curves by half so that one group of curves occurs at t1, t2, . . . , tnt/2

and the other group of curves occurs at tnt/2+1, tnt/2+2, . . . , tnt. We change the latter half

of curves into semi-ellipses while the former half of curves remains as semicircles with the

radius of 0.5. Depending on the number c, the latter half of curves is stretched or squeezed

into the semi-ellipses along the y direction.

The power of the test

nx

403

803

nt

10

20

10

20

n

c = 0.55

c = 0.525

c = 0.475

c = 0.45

640, 000

1, 280, 000

5, 120, 000

10, 240, 000

1.00

1.00

1.00

1.00

0.56

0.81

1.00

1.00

0.42

0.62

1.00

1.00

0.98

1.00

1.00

1.00

(a) The upper 5th percentile of the simulated distribution under H0 is used as a

critical value.

nx

403

803

nt

10

20

10

20

The power of the test

n

c = 0.55

c = 0.525

c = 0.475

c = 0.45

640, 000

1, 280, 000

5, 120, 000

10, 240, 000

0.63

0.92

1.00

1.00

0.05

0.17

0.83

0.97

0.01

0.15

0.91

0.96

0.56

0.86

1.00

1.00

(b) The upper 5th percentile of the limiting chi-square distribution with 30 degrees

of freedom is used as a critical value.

Table 4.3: Monte Carlo simulation-based power analysis when HA3

is true

54

Table 4.3 shows the power of test based on the value of c at the given size of nx and nt.

By increasing the size of nt given the fixed size of nx, we found the discrepancy between

the empirical and theoretical critical values can be reduced to almost zero. Furthermore,

the power of the test shows that the probability to detect small effects gets higher as nt

increases, which implies that for MRI scans with a fixed number of spatial points such as
128 × 128 × 48, increasing time points (i.e., the number of visits over the study period) is

critical to achieve a sufficient statistical power in order to detect small pathological changes

of fiber pathways over time.

4.2 Real longitudinal DTI data

Diffusion weighted imaging (DWI) scans were obtained on a healthy male brain 19 times

over a 4-year period from 2014 to 2018. A GE 3T Signa HDx MR scanner (GE Healthcare,

Waukesha, WI) with an 8-channel head coil was used to collect longitudinal DTI images of

the brain. DWI scans were acquired with a spin-echo echo-planar imaging (EPI) sequence for

12 minutes and 6 seconds by using the following imaging parameters: 48 contiguous 2.4mm
axial slices in an interleaved order, the field of view (FOV) = 22 × 22cm2, the number of
pixels (matrix size) = 128×128, the number of excitations (NEX) = 2, the echo time (TE) =

76.3ms, the repetition time (TR) = 13.7s, 25 diffusion-weighted volumes (one per gradient

direction) with b = 1000s/mm2, 1 volume with b = 0 and parallel imaging acceleration

factor = 2.

At each time point, the number of spatial locations was the same as nx = 128×128×48 =

786, 432. The number of time points was nt = 19, and hence the total sample size was

n = 14, 942, 208. For time points, we used an index of the events rescaled by the total

55

number of time points as follows: tk = k/19, k = 1, 2, . . . , 19, regardless of the elapsed time

between calendar dates.

In this study, ROIs were the anterior and posterior regions of the corpus callosum (CC).

The CC is often of interest in white matter tractography since it is the largest bundle of

white matter nerve fibers connecting the hemispheres of the brain. The initial point x0 was

chosen as a balancing point in each ROI between right and left hemispheres of the brain.

The estimation procedure of the posterior portion of the CC was found to be more robust

to the initial point selected than that of the anterior part of the CC. This difference can be

explained by the patient’s supine position with the head resting on a pillow during the MRI

scan. In fact, it is reasonable to assume that a resting head experiences more shifting at the

front.

In both ROIs, we fixed β1 = 10−8 to avoid over- or under-smoothing caused by too

wide or too narrow bandwidth choice. The standard 3D Gaussian kernel was used with the

corresponding bandwidth hn = 0.0068. The step size was δ = 0.003 and the number of

steps m was determined before the estimate of the covariance function grew too large. As a

result of larger confidence ellipsoids in early steps, m = 30 was used in the anterior part of

the CC as opposed to m = 70 in the posterior part of the CC. In Figure 4.4, the estimated

trajectory onto the xy-plane can be depicted as a slightly divergent U-shaped tube each

time, albeit shifting due to the head motion. Figure 4.6 shows that the estimated pathway

onto the xy-plane at each time point is seen as an omega-shaped tube which is widely and

deeply divergent from the initial point, although at some time points the estimated pathway

became inverted due to the limitations of DTI on branching fibers. The following Table 4.4

shows the test statistics computed from x0 through the left side of the curve and from x0

through the right side of the curve at given ROIs. Since the test statistic was considerably

56

smaller than the corresponding critical value, we failed to reject the null hypothesis at the

significance level of 5%, and hence we reached a conclusion that there was no sufficient

statistical evidence to detect the pathological change of the true fiber pathway in either of

two ROIs over the observed period of time.

anterior part of the CC

posterior part of the CC

χ2
0.05,30 = 43.7730

χ2
0.05,70 = 90.5312

left of x0

a

right of x0

left of x0

b

right of x0

0.1464

0.0020

3.5882

7.5200

a x0 = [0.5078 0.6563 0.5417](cid:62)
b x0 = [0.5156 0.4063 0.5208](cid:62)
Table 4.4: Result of test statistics in both ROIs

57

Figure 4.4: The estimated trajectory colored in red is projected onto the xy-plane over the
observed period from July 2014 to December 2018 in ascending order in the anterior part of
the CC.

58

00.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.51(a) The estimated trajectory colored in red

(b) The estimated trajectory colored in red

is projected onto the xy-plane.

along with the 95% confidence ellipsoids colored

in cyan are projected onto the xy-plane at every

5th step.

(c) 3D of (a)

(d) 3D of (b)

Figure 4.5: Anterior part of the CC scanned in December 2014

59

00.20.40.60.8100.20.40.60.8100.20.40.60.8100.20.40.60.8100.20.410.60.810.5100.5000.20.410.60.810.5100.50Figure 4.6: The estimated trajectory colored in red is projected onto the xy-plane over the
observed period from July 2014 to December 2018 in ascending order in the posterior part
of the CC.

60

00.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.5100.51(a) The estimated trajectory colored in red

(b) The estimated trajectory colored in red

is projected onto the xy-plane.

along with the 95% confidence ellipsoids colored

in cyan are projected onto the xy-plane at every

10th step.

(c) 3D of (a)

(d) 3D of (b)

Figure 4.7: Posterior part of the CC scanned in December 2018

61

00.20.40.60.8100.20.40.60.8100.20.40.60.8100.20.40.60.8100.20.410.60.810.5100.5000.20.410.60.810.5100.50(a) The estimated trajectory colored in red

(b) 3D of (a)

along with the 95% confidence ellipsoids col-

ored in cyan are projected onto the yz-plane

at every 10th step.

Figure 4.8: Left isthmus of the cingulate cortex scanned in July 2014

In addition to the CC, we performed tracing the fiber pathway in left isthmus of the

cingulate cortex. We kept β1 and the step size δ the same as we specified for the analysis of

the anterior and posterior portions of the CC. The number of steps m = 60 was used. The
initial point x0 = [0.5391 0.3672 0.5833](cid:62) was fixed. The computed test statistic was 7.8719
which was considerably smaller than the corresponding critical value χ2

0.05,60 = 79.0819. As

a result, we also failed to detect the pathological change of the true fiber pathway in this

ROI over the observed period of time at the significance level of 5%.

62

00.20.40.60.8100.20.40.60.810110.50.510.500Chapter 5

Conclusion and Discussion

This dissertation provides the comprehensive estimation procedure of the spatio-temporal

fiber pathway model, and further proposes the straightforward hypothesis test associated

with the rate of change in true fiber trajectory with respect to time. While many neu-

roimaging publications rely on existing statistical comparisons of the scalar measures such

as FA, MD, AD and RD in ROI between time points, this dissertation attempts to offer a

new statistical perspective on the degree of pathological change of fiber pathways over the

period of time. The proposed approach is computationally efficient and the power of the

test is much improved with increasing time points given fixed spatial points.

One limitation of this dissertation can be found as a boundary effect near an endpoint

of the support (in particular, the set of points in [0, T ]) which is inherent in kernel density

estimation. We shall caution that insufficient number of time points can lead to substantial

increases in bias and variance at boundary time points. Another limitation is that the

simulation study was performed with a constant weight function w(t) during the entire

period of time. An additional research on the weight function is needed to study its impacts

on our testing procedure in terms of numerical implementation and power performance.

Future studies can move further in several directions. First, the proposed method can

be extended and developed in HARDI. Such extension of the spatio-temporal fiber pathway

model to HARDI can overcome the limitations of DTI for complex fiber configurations such

63

as crossing, branching or kissing fibers.

Second, measurement errors during MRI image acquisitions can be divided into sys-

tematic error which has a consistent effect on measurement in the same direction and/or

magnitude and random error which does not. While B was specified as the known constant

tensor obtained by the MRI image acquisition in the model (2.2), B can be viewed as a

matrix distorted by such measurement errors. That is,

B = B0 + es + er,

where B0 is the true b-matrix of DTI, es is the systematic error and er is the random error.

For instance, suppose that a patient’s head is tilted in one particular direction repeatedly

over the study period, then this systemic error can cause distortion in the B matrix. As

two types of measurement errors are induced into the model, we can further validate the

robustness to violations of the model assumptions.

Third, we can further study the rate of change in noise level over time. We hypothesize

that the level of noise is fairly stable for a healthy normal brain. Based on the model of

(2.2), the hypothesis testing problem can be stated as follows:

H0 :

∂
∂t

σij(u) = 0 versus HA :

σij(u) > 0, ∀i, j ∈ {1, 2, . . . , N}, u ∈ G,

∂
∂t

where σij is the element in the ith row and jth column of the matrix Σ in (2.2). Such

future study can investigate the degree of elevation in the noise level as the stage of disease

progression develops.

Lastly, our methodology should be further developed in order to analyze a set of longitu-

64

dinal DTI data sets collected from a group of patients. Defining a white matter tractography

model with the marginal or “population-average” perspective can be a challenging problem,

however, it should be addressed to understand brain connectivity in both individual and

group level.

65

Chapter 6

Proofs of Theorem 2.4.1., Theorem

2.4.2., and Theorem 2.4.3.

Proofs of Theorem 2.4.1., Theorem 2.4.2. and Theorem 2.4.3. are provided as in Koltchin-

skii et al. (2007). In Section 6.1, we decompose the sequence of stochastic processes into the

sequence of stochastic processes which converges in distribution to the Gaussian process and

the sequence of remaining processes which converges to zero in probability. Mean and covari-

ance functions are presented in Section 6.2 and Section 6.3, respectively. The corresponding

weak convergence is proved in terms of the convergence of finite-dimensional distributions

in Section 6.4 and the asymptotic equicontinuity is established in Section 6.5. The proofs of

propostions can be found in Section 6.6. We refer to classical books of Vaart and Wellner

(1996) and Billingsley (1999) for the weak convergence topic.

6.1 Asymptotic representation

(i) Theorem 2.4.1.

Define y1(s, t) := (cid:98)Xn(s, t) − x(s, t). Then
y1(s, t) = (cid:98)Xn(s, t) − x(s, t)

(cid:90) s

0

=

(cid:110)
(cid:111)
v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − v(D(x(ξ, t), t))

dξ

66

(cid:90) s
(cid:90) s

0

0

=

+

∂
∂D
∂
∂D

(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)

v(D(x(ξ, t), t))

(cid:111)

dξ

v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)y1(ξ, t)dξ + r1(s, t),

and the remainder r1(s, t) is defined as

r1(s, t) :=

0

(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s

0

0

−

+

−

(cid:110)
(cid:111)
v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − v((cid:98)Dn(x(ξ, t), t))
(cid:110)
(cid:111)
v((cid:98)Dn(x(ξ, t), t)) − v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

∂
∂D

∂
∂x

D(x(ξ, t), t)y1(ξ, t)dξ

dξ

(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)

dξ

(cid:111)

dξ.

∂
∂D

0

v(D(x(ξ, t), t))

By the decomposition of y1(s, t) := z1(s, t) + δ1(s, t), z1(s, t) and δ1(s, t) are as follows:

(cid:90) s
(cid:90) s
(cid:90) s

0

0

0

∂
∂D
∂
∂D
∂
∂D

z1(s, t) =

+

δ1(s, t) =

(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111)

dξ

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

D(x(ξ, t), t)z1(ξ, t)dξ,

v(D(x(ξ, t), t))

D(x(ξ, t), t)δ1(ξ, t)dξ + r1(s, t).

∂
∂x
∂
∂x

For z1(s, t), we consider the first-order differential equation indexed by s ∈ [0, S] with
initial-value condition z1(0, t) = 0 given the parameter time t ∈ [0, T ], which is equivalent to

z1(s, t) =

:=

G(s, ξ, t)

g1(s, ξ, t)

∂
∂D

v(D(x(ξ, t), t))

(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111)

dξ,

(cid:111)
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)

dξ

(cid:90) s
(cid:90) S

0

0

where g1(s, ξ, t) := I{0≤ξ≤s}G(s, ξ, t) ∂

∂D v(D(x(ξ, t), t)) with a d × d matrix-valued Green’s

67

function G. Furthermore, g1(s, ξ, t) ∈ L, s ∈ [0, S], is almost everywhere continuous and
bounded on R, where L is a linear space with the support of g in [0, S].

Similarly, let us define y2(s, t) such that y2(s, t) = ∂

∂tx(s, t) := z2(s, t) + δ2(s, t).

∂t(cid:98)Xn(s, t)− ∂

(cid:111)

∂t

x(s, t)

v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − d

(ii) Theorem 2.4.2.

0

0

0

0

∂
∂t

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

(cid:98)Xn(s, t) − ∂
(cid:90) s
(cid:110) d
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s

dt
∂
∂D
∂
∂D
∂2
∂D2 v(D(x(ξ, t), t))
∂2
∂D2 v(D(x(ξ, t), t))
∂2
∂D2 v(D(x(ξ, t), t))
∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D
∂
∂D
∂
∂D

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

0

0

0

0

0

0

y2(s, t) =

=

=

+

+

+

+

+

+

+

+

(cid:111)
(cid:111) ∂

dξ

D(x(ξ, t), t)

dt

∂t

∂t

dξ

v(D(x(ξ, t), t))

(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂

∂x

∂x

D(x(ξ, t), t)

∂t

x(ξ, t)dξ

∂t

D(x(ξ, t), t)dξ

D(x(ξ, t), t)

∂
∂t

x(ξ, t)dξ

∂x

D(x(ξ, t), t)y1(ξ, t)dξ

D(x(ξ, t), t)y1(ξ, t)

∂
∂t

x(ξ, t)dξ

D(x(ξ, t), t)

D(x(ξ, t), t)

∂
∂t
∂
∂x

∂
∂x
∂
∂x
∂2
∂x∂t
∂2
∂x2 D(x(ξ, t), t)y1(ξ, t)
∂
∂x

D(x(ξ, t), t)y1(ξ, t)dξ

D(x(ξ, t), t)y2(ξ, t)dξ + r2(s, t),

∂
∂t

x(ξ, t)dξ

where the remainder r2(s, t) is defined as

(cid:110) ∂

(cid:90) s
(cid:90) s
(cid:90) s

0

0

v((cid:98)Dn(x(ξ, t), t)) − ∂

r2(s, t) =

−

+

∂D
∂2
∂D2 v(D(x(ξ, t), t))

v((cid:98)Dn((cid:98)Xn(ξ, t), t))

(cid:110) ∂

0

∂D

∂D

v(D(x(ξ, t), t))

(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)dξ
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
v((cid:98)Dn(x(ξ, t), t))
(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂

∂t

∂t

∂
∂t

∂D

D(x(ξ, t), t)dξ

68

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)y1(ξ, t)dξ

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

v(D(x(ξ, t), t))

∂
∂x
∂2
∂x∂t

∂D

∂D
∂2
∂D2 v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

D(x(ξ, t), t)y1(ξ, t)dξ

v((cid:98)Dn(x(ξ, t), t)) − ∂

(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
v((cid:98)Dn(x(ξ, t), t))
v((cid:98)Dn((cid:98)Xn(ξ, t), t))
(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂
(cid:98)Xn(ξ, t)dξ

(cid:111) ∂

∂
∂x

∂
∂t

∂D

∂x

∂x

D(x(ξ, t), t)

∂t

x(ξ, t)dξ

∂
∂t

x(ξ, t)dξ

(cid:111)

dξ

× ∂
∂t
−

−

+

−

+

0

0

0

(cid:110) ∂

(cid:98)Dn(x(ξ, t), t)
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t)
(cid:90) s
(cid:90) s
(cid:90) s

∂D

0

0

0

× ∂
∂x
−

0

−

(cid:110) ∂
∂D
× y2(ξ, t)dξ.

+

0

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

v(D(x(ξ, t), t))

v((cid:98)Dn(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)

∂
∂x
∂2
∂x2 D(x(ξ, t), t)y1(ξ, t)

(cid:98)Dn(x(ξ, t), t) − ∂

∂
∂t

∂
∂x

D(x(ξ, t), t)y1(ξ, t)

∂
∂t

x(ξ, t)dξ

x(ξ, t)dξ

v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)

(cid:111)

∂D

The unique solution z2(s, t) with z2(0, t) = 0 can be written as follows:

0

(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S

0

0

0

0

0

z2(s, t) =

+

+

+

+

+

g1(s, ξ, t)

g1(s, ξ, t)

g2(s, ξ, t)

g2(s, ξ, t)

g3(s, ξ, t)

D(x(ξ, t), t)

x(ξ, t)dξ

(cid:111)
(cid:111) ∂

dξ

∂t

D(x(ξ, t), t)

∂t

∂t

∂x

(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:90) S
(cid:90) S

g1(ξ, ζ, t)

∂x

0

∂t

∂x

D(x(ξ, t), t)

(cid:111)
(cid:110)(cid:98)Dn(x(ζ, t), t) − D(x(ζ, t), t)
(cid:110)(cid:98)Dn(x(ζ, t), t) − D(x(ζ, t), t)
(cid:111)

D(x(ξ, t), t)dξ

g4(s, ξ, t)

g1(ξ, ζ, t)

0

∂
∂t

x(ξ, t)dξ

dζdξ

dζ

∂
∂t

x(ξ, t)dξ,

69

where

g1(s, ξ, t) := I{0≤ξ≤s}G(s, ξ, t)

g2(s, ξ, t) := I{0≤ξ≤s}G(s, ξ, t)

g3(s, ξ, t) := I{0≤ξ≤s}G(s, ξ, t)

+

∂
∂D

v(D(x(ξ, t), t))

g4(s, ξ, t) := I{0≤ξ≤s}G(s, ξ, t)

+

∂
∂D

v(D(x(ξ, t), t))

(cid:110) ∂2
(cid:110) ∂2

v(D(x(ξ, t), t))

∂
∂D
∂2
∂D2 v(D(x(ξ, t), t))

∂D2 v(D(x(ξ, t), t))
∂2
∂x∂t

D(x(ξ, t), t)

∂D2 v(D(x(ξ, t), t))
∂2
∂x2 D(x(ξ, t), t)

.

(cid:111)
(cid:111)

∂
∂x

∂
∂x

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)

D(x(ξ, t), t)

∂
∂x

D(x(ξ, t), t)

Furthermore, the sequence of remaining processes δ2(s, t) is represented by

0

(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s

0

0

0

0

δ2(s, t) =

+

+

+

+

D(x(ξ, t), t)δ1(ξ, t)dξ

D(x(ξ, t), t)δ1(ξ, t)

∂
∂t

x(ξ, t)dξ

∂2
∂D2 v(D(x(ξ, t), t))
∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D
∂
∂D
∂
∂D

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

D(x(ξ, t), t)

D(x(ξ, t), t)

∂
∂t
∂
∂x

∂
∂x
∂
∂x
∂2
∂x∂t
∂2
∂x2 D(x(ξ, t), t)δ1(ξ, t)
∂
∂x

D(x(ξ, t), t)δ1(ξ, t)dξ

D(x(ξ, t), t)δ2(ξ, t)dξ + r2(s, t).

∂
∂t

x(ξ, t)dξ

(iii) Theorem 2.4.3.

Let 0 < a < b ≤ T. Suppose w is a positive vector-valued Lebesgue measurable function.

(cid:90) b
(cid:90) b

a

(cid:18) ∂
w(cid:62)(t)
w(cid:62)(t)y2(s, t)dt

∂t

a

(cid:98)Xn(s, t) − ∂

∂t

y3(s) :=

=

(cid:19)

x(s, t)

dt

70

where z3(s) =(cid:82) b

w(cid:62)(t)δ2(s, t)dt := z3(s) + δ3(s),

a w(cid:62)(t)δ2(s, t)dt, respectively.

a

a

=

(cid:90) b

w(cid:62)(t)z2(s, t)dt +

(cid:90) b
a w(cid:62)(t)z2(s, t)dt and δ3(s) =(cid:82) b
(cid:19)
(cid:18) ∂
(cid:90) s
(cid:90) s
(cid:90) s

(cid:98)Xn(s, t) − ∂
(cid:110) d
v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − d

v(D(x(ξ, t), t))

x(s, t)

∂t

∂t

dt

0

0

v(D(x(ξ, t), t))

dt
∂
∂D
∂
∂D

∂2
∂D2 v(D(x(ξ, t), t))

∂2
∂D2 v(D(x(ξ, t), t))

(cid:111)

D(x(ξ, t), t)

dt

∂t

∂t

dξdt

v(D(x(ξ, t), t))

(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂

∂x

∂x

∂t

D(x(ξ, t), t)

(cid:111)
(cid:111) ∂

dξdt

∂t

D(x(ξ, t), t)dξdt

D(x(ξ, t), t)

∂x

x(ξ, t)dξdt

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)
w(cid:62)(t)
(cid:90) s

0

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

0

0

(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s

0

0

0

0

a

0

× ∂
∂t

x(ξ, t)dξdt
w(cid:62)(t)

y3(s) =

=

=

+

+

+

a

a

a

a

a

(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b

a

a

a

a

a

(cid:90) b
(cid:90) b

a

+

+

+

+

+

D(x(ξ, t), t)y1(ξ, t)dξdt

D(x(ξ, t), t)y1(ξ, t)

∂
∂t

x(ξ, t)dξdt

∂2
∂D2 v(D(x(ξ, t), t))
∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D
∂
∂D
∂
∂D

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

D(x(ξ, t), t)

D(x(ξ, t), t)

∂
∂t
∂
∂x

∂
∂x
∂
∂x
∂2
∂x∂t
∂2
∂x2 D(x(ξ, t), t)y1(ξ, t)
∂
∂x

D(x(ξ, t), t)y1(ξ, t)dξdt

D(x(ξ, t), t)y2(ξ, t)dξdt + δ3(s),

∂
∂t

x(ξ, t)dξdt

where the remainder δ3(s) is defined as

(cid:110) ∂

(cid:90) s
(cid:90) s

0

v((cid:98)Dn(x(ξ, t), t)) − ∂

(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)dξdt
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂

v(D(x(ξ, t), t))

∂D

∂t

∂D
∂2
∂D2 v(D(x(ξ, t), t))

D(x(ξ, t), t)dξdt

∂t

δ3(s) =

−

w(cid:62)(t)

w(cid:62)(t)

a

0

71

v((cid:98)Dn(x(ξ, t), t)) − ∂

−

−

+

+

a

× ∂
∂t

0

∂D

∂D

0

0

0

a

a

dξdt

∂
∂x
∂2
∂x∂t

w(cid:62)(t)

w(cid:62)(t)

v(D(x(ξ, t), t))

(cid:110) ∂

x(ξ, t)dξdt
w(cid:62)(t)

v((cid:98)Dn((cid:98)Xn(ξ, t), t))

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

(cid:90) b
(cid:90) s
(cid:110) ∂
w(cid:62)(t)
(cid:111)
(cid:98)Dn(x(ξ, t), t)
(cid:90) s
(cid:90) b
w(cid:62)(t)
(cid:90) b
(cid:90) s
(cid:90) s
(cid:90) b
(cid:90) s
(cid:90) b
(cid:90) s
(cid:90) b
(cid:110) ∂
x(ξ, t)dξdt
v((cid:98)Dn((cid:98)Xn(ξ, t), t))
w(cid:62)(t)
(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)
(cid:98)Xn(ξ, t)dξdt
(cid:90) b
(cid:90) s
w(cid:62)(t)
(cid:90) s
(cid:90) b
(cid:90) b
(cid:90) s
(cid:90) s
(cid:90) b

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D
∂
∂D
∂
∂D

v((cid:98)Dn(x(ξ, t), t))

∂2
∂D2 v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

a

a

0

0

0

∂D

a

0

∂t

a

0

0

a

× ∂
∂t

−

a

× ∂
∂t

+

a

× ∂
∂x

−

−

+

−

(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂

∂D

∂
∂t

v((cid:98)Dn(x(ξ, t), t))

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)y1(ξ, t)dξdt

∂x

∂D

v(D(x(ξ, t), t))

D(x(ξ, t), t)y1(ξ, t)dξdt

(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂

v((cid:98)Dn(x(ξ, t), t))

∂x

D(x(ξ, t), t)

∂
∂x

∂D

D(x(ξ, t), t)y1(ξ, t)

∂
∂t

x(ξ, t)dξdt

x(ξ, t)dξdt

∂
∂x

D(x(ξ, t), t)

∂
∂x
∂2
∂x2 D(x(ξ, t), t)y1(ξ, t)
∂
∂x
∂
∂x

(cid:98)Dn(x(ξ, t), t)y2(ξ, t)dξdt

D(x(ξ, t), t)y2(ξ, t)dξdt.

∂
∂t

6.2 Mean function

(cid:90) S

0

(cid:90)

Rd+1

(cid:16)(x(ξ, t), t) − w

(cid:17)

hn

dwdξ

D(w)K

g1(s, ξ, t)

(i) Theorem 2.4.1.

E[z1(s, t)] =

−

1

(cid:90) S

hd+1
n

0

g1(s, ξ, t)D(x(ξ, t), t)dξ

72

by letting ψ = (x(ξ,t),t)−w

hn

(cid:90) S

0

(cid:110)

(cid:90)
(cid:90) S

0

=

g1(s, ξ, t)

Rd+1

× K(ψ)dψdξ −

(cid:90) S

0

=

h2
n
2

(cid:90)

by Taylor’s Theorem in a sufficiently small neighborhood of D(x(ξ, t), t),

(cid:16)

(cid:17)

D(x(ξ, t), t) + D((x(ξ, t), t) − hnψ) − D(x(ξ, t), t)

(cid:111)

g1(s, ξ, t)D(x(ξ, t), t)dξ

ψ(cid:62) ∂2
(cid:113)
(cid:90)

g1(s, ξ, t)

Rd+1

∂u2 D(x(ξ, t), t)ψK(ψ)dψdξ

1 + op(1)

.

(s, t) := limn→∞

nhd

nE[z1(s, t)] is defined as

g1(s, ξ, t)

ψ(cid:62) ∂2

∂u2 D(x(ξ, t), t)ψK(ψ)dψdξ,

Rd+1

Then mean function µβ1
√
β1
2

(s, t) =

µβ1

(cid:90) S

0

where β1 is a known fixed number such that nhd+4

(ii) Theorem 2.4.2.

(cid:90) S

(cid:90)

Rd+1

D(w)K

g1(s, ξ, t)

(1)
t

E[z2(s, t)] =

hd+2
n

0

n → β1 > 0 as n → ∞.
(cid:16)(x(ξ, t), t) − w

(cid:17)

dwdξ

hn

g1(s, ξ, t)

D(x(ξ, t), t)dξ

g1(s, ξ, t)

Rd+1

D(w)K

(1)
x

(cid:16)(x(ξ, t), t) − w

(cid:17)

hn

dw

∂
∂t

x(ξ, t)dξ

1

(cid:90) S

0
1

(cid:90) S

hd+2
n

0
1

(cid:90) S

hd+1
n

0
1

−

+

−

+

−

+

(cid:90) S

0

(cid:90) S

0

(cid:90) S

∂
∂t

∂
∂x

(cid:90)

(cid:90)

(cid:90)

g1(s, ξ, t)

D(x(ξ, t), t)

x(ξ, t)dξ

∂
∂t

(cid:16)(x(ξ, t), t) − w

g2(s, ξ, t)

D(w)K

Rd+1

hn

g2(s, ξ, t)D(x(ξ, t), t)

D(x(ξ, t), t)dξ

∂
∂t

hd+1
n

0

g2(s, ξ, t)

D(w)K

Rd+1

(cid:16)(x(ξ, t), t) − w

hn

73

(cid:17)

(cid:17)

dw

∂
∂t

D(x(ξ, t), t)dξ

dw

∂
∂x

D(x(ξ, t), t)

∂
∂t

x(ξ, t)dξ

by letting ψ = (x(ξ,t),t)−w

hn

−

+

−

+

−

=

−

+

−

+

−

+

−

+

−

+

(cid:90) S

0

0

0

0

1
hn

0
1
hn

(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S

0

0

0

0

0

∂
∂x

(cid:90)

(cid:90)

(cid:90) S
(cid:90) S
(cid:90) S

0

0

0

(cid:90) S

0
1

(cid:90) S

hd+1
n

0
1

(cid:90) S

hd+1
n

0

(cid:90) S

0

(cid:90) S

0

g4(s, ξ, t)

(cid:90) S

0

(cid:90) S

0

(cid:90) S

0

(cid:90) S

0

g2(s, ξ, t)D(x(ξ, t), t)

D(x(ξ, t), t)

x(ξ, t)dξ

∂
∂t

∂
∂t

g3(s, ξ, t)

g1(ξ, ζ, t)

D(w)K

Rd+1

g3(s, ξ, t)

g1(ξ, ζ, t)D(x(ζ, t), t)dζdξ

(cid:90)

(cid:90)

g4(s, ξ, t)

g1(ξ, ζ, t)

D(w)K

Rd+1

(cid:16)(x(ζ, t), t) − w

(cid:17)

dwdζdξ

hn

(cid:16)(x(ζ, t), t) − w

(cid:17)

hn

dwdζ

∂
∂t

x(ξ, t)dξ

g1(ξ, ζ, t)D(x(ζ, t), t)dζ

∂
∂t

x(ξ, t)dξ

(cid:90)

Rd+1

D((x(ξ, t), t) − hnψ)K

(1)
t

(ψ)dψdξ

g1(s, ξ, t)

∂
∂t

(cid:90)

g1(s, ξ, t)

D(x(ξ, t), t)dξ

g1(s, ξ, t)

D((x(ξ, t), t) − hnψ)K

(1)
x (ψ)dw

∂
∂t

x(ξ, t)dξ

Rd+1

D(x(ξ, t), t)

∂
∂t

x(ξ, t)dξ

g1(s, ξ, t)

g2(s, ξ, t)

D((x(ξ, t), t) − hnψ)K(ψ)dψ

∂
∂t

D(x(ξ, t), t)dξ

Rd+1

g2(s, ξ, t)D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)dξ

g2(s, ξ, t)

D((x(ξ, t), t) − hnψ)K(ψ)dψ

∂
∂x

D(x(ξ, t), t)

∂
∂t

x(ξ, t)dξ

Rd+1

g2(s, ξ, t)D(x(ξ, t), t)

D(x(ξ, t), t)

∂
∂t

x(ξ, t)dξ

g3(s, ξ, t)

g1(ξ, ζ, t)

Rd+1

D((x(ξ, t), t) − hnψ)K(ψ)dψdζdξ

g3(s, ξ, t)

g1(ξ, ζ, t)D(x(ζ, t), t)dζdξ

g4(s, ξ, t)

g1(ξ, ζ, t)

Rd+1

D((x(ξ, t), t) − hnψ)K(ψ)dψdζ

∂
∂t

(cid:90)

(cid:90)

0

× ∂
∂t

x(ξ, t)dξ

74

(cid:90) S

0

−

(cid:90) S

0

g4(s, ξ, t)

g1(ξ, ζ, t)D(x(ζ, t), t)dζ

∂
∂t

x(ξ, t)dξ.

The rest of the proof is similar to (i) in Section 6.2. By Taylor’s Theorem in a suffi-

ciently small neighborhood of D(u) together with kernel Lt(ψ) = −ψtK
−ψxK
nhd+2

(s, t) := limn→∞

(ψ) and Lx(ψ) =
n E[z2(s, t)] as follows:

(1)
t

(cid:113)

µβ2

0

√

Rd+1

g2(s, ξ, t)

(1)
x (ψ), we have the mean function µβ2
ψ(cid:62) ∂2
ψ(cid:62) ∂2
(cid:90)
(cid:90)

(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S

(cid:90)
(cid:90)
(cid:90) S
(cid:90) S

β2
(s, t) =
√
2
β2
√
2
β2
√
2
β2
2

g1(ξ, ζ, t)

g1(ξ, ζ, t)

g2(s, ξ, t)

g4(s, ξ, t)

g3(s, ξ, t)

Rd+1

+

+

+

0

0

0

0

0

∂
∂u2 D(x(ξ, t), t)ψK(ψ)dψ
∂t
∂
∂x

∂u2 D(x(ξ, t), t)ψK(ψ)dψ

D(x(ξ, t), t)dξ

D(x(ξ, t), t)

∂
∂t

x(ξ, t)dξ

ψ(cid:62) ∂2
ψ(cid:62) ∂2

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζdξ
∂
∂t

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζ

x(ξ, t)dξ,

Rd+1

Rd+1

where β2 is a known fixed number such that nhd+6

n → β2 > 0 as n → ∞.

(iii) Theorem 2.4.3.

It is analogous to (ii) in Section 6.2.

µβ1

(s) =
√

√

w(cid:62)(t)

a

β1
2

(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b

a

a

0

(cid:90) S
w(cid:62)(t)
(cid:90) S
(cid:90) S
(cid:90) S

0

0

x(ξ, t)dξdt
w(cid:62)(t)

w(cid:62)(t)

a

0

x(ξ, t)dξdt,

+

β1
2
× ∂
√
∂t

+

β1
2
√
β1
2
× ∂
∂t

+

(cid:90)

(cid:90)
(cid:90) S
(cid:90) S

0

0

g2(s, ξ, t)

g2(s, ξ, t)

ψ(cid:62) ∂2

Rd+1

∂u2 D(x(ξ, t), t)ψK(ψ)dψ
∂
∂x

∂u2 D(x(ξ, t), t)ψK(ψ)dψ

ψ(cid:62) ∂2

Rd+1

∂
∂t

D(x(ξ, t), t)dξdt

D(x(ξ, t), t)

(cid:90)
(cid:90)

Rd+1

Rd+1

g1(ξ, ζ, t)

g1(ξ, ζ, t)

ψ(cid:62) ∂2
ψ(cid:62) ∂2

g3(s, ξ, t)

g4(s, ξ, t)

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζdξdt

∂u2 D(x(ζ, t), t)ψK(ψ)dψdζ

where β1 is a known fixed number such that nhd+4

n → β1 > 0 as n → ∞.

75

6.3 Covariance function

(i) Theorem 2.4.1.

0

0

=

nh

(cid:90) S

1
2(d+1)
n

(cid:90)
(cid:90) S
Cov[z1(s, t), z1(s∗, t∗)]
(cid:16)(x(η, t∗), t∗) − w
(cid:90)
(cid:90) S
(cid:16)(x(η, t∗), t∗) − w

×(cid:16)
×(cid:16)

1
2(d+1)
n

(cid:90) S

hn

nh

K

K

+

0

0

Rd+1

(cid:17)(cid:17)(cid:62)

Rd+1

(cid:17)(cid:17)(cid:62)

hn

(cid:16)(x(ξ, t), t) − w

(cid:17)

g1(s, ξ, t)D(w)K

D(cid:62)(w)g(cid:62)

hn
1 (s∗, η, t∗)dwdηdξ
(cid:16)(x(ξ, t), t) − w

(cid:17)

g1(s, ξ, t)Γ(w)K

hn

Γ(cid:62)(w)g(cid:62)

1 (s∗, η, t∗)dwdηdξ + o(1/n)

by change of variable η = ξ + τ hn

nh2d+1

n

0

1

−ξ/hn

(cid:90) S

(cid:90) (S−ξ)/hn

(cid:90)
(cid:16)(x(ξ + τ hn, t∗), t∗) − w
(cid:90)
(cid:16)(x(ξ + τ hn, t∗), t∗) − w

(cid:90) (S−ξ)/hn

(cid:90) S

−ξ/hn

hn

0

1

Rd+1

(cid:17)(cid:17)(cid:62)

Rd+1

(cid:17)(cid:17)(cid:62)

nh2d+1

n

=

×(cid:16)
×(cid:16)

+

K

K

hn

g1(s, ξ, t)D(w)K

D(cid:62)(w)g(cid:62)

(cid:17)
(cid:16)(x(ξ, t), t) − w
1 (s∗, ξ + τ hn, t∗)dwdτ dξ
(cid:16)(x(ξ, t), t) − w
(cid:17)

hn

g1(s, ξ, t)Γ(w)K

hn

Γ(cid:62)(w)g(cid:62)

1 (s∗, ξ + τ hn, t∗)dwdτ dξ + o(1/n)

hn

(cid:90)

(cid:90) S

by letting ψ = (x(ξ,t),t)−w
(cid:90) (S−ξ)/hn
−ξ/hn
(x(ξ + τ hn, t∗), t∗) − (x(ξ, t), t)

(cid:17)(cid:17)(cid:62)
hn
1 (s∗, ξ + τ hn, t∗)dψdτ dξ
× D(cid:62)((x(ξ, t), t) − hnψ)g(cid:62)

×(cid:16)

1
nhd
n

Rd+1

(cid:16)

ψ +

K

=

0

g1(s, ξ, t)D((x(ξ, t), t) − hnψ)K(ψ)

76

(cid:90) S
(cid:16)

0

(cid:90)

(cid:90) (S−ξ)/hn
−ξ/hn
(x(ξ + τ hn, t∗), t∗) − (x(ξ, t), t)

Rd+1

(cid:17)(cid:17)(cid:62)

1
nhd
n

+

×(cid:16)

K

ψ +

g1(s, ξ, t)Γ((x(ξ, t), t) − hnψ)K(ψ)

hn
1 (s∗, ξ + τ hn, t∗)dψdτ dξ + o(1/n).
× Γ(cid:62)((x(ξ, t), t) − hnψ)g(cid:62)

If t (cid:54)= t∗, the covariance function is close to infinity as n → ∞ under any density kernel

function. For the example of the Gaussian kernel function,

(cid:18)
(cid:17) ∝ exp
(cid:19)(cid:62)(cid:18)(x(ξ + τ hn, t∗), t∗) − (x(ξ, t∗), t∗)
(cid:18)(x(ξ + τ hn, t∗), t∗) − (x(ξ, t∗), t∗)
(cid:19)(cid:62)(cid:18)(x(ξ, t∗), t∗) − (x(ξ, t), t)
(cid:18)(x(ξ, t∗), t∗) − (x(ξ, t), t)

(x(ξ + τ hn, t∗), t∗) − (x(ξ, t), t)
(cid:18)
(cid:18)

(cid:19)(cid:19)

− 1
2

ψ(cid:62)ψ

(cid:19)

hn

hn

hn

(cid:19)(cid:19)

(cid:16)

K

ψ +

× exp

× exp

− 1
2
− 1
2

hn

hn

.

Note that t and t∗ are fixed scalars in [0, T ]. Since hn → 0 as n → ∞,
(cid:19)(cid:19)

(cid:19)(cid:62)(cid:18)(x(ξ, t∗), t∗) − (x(ξ, t), t)

(cid:18)(x(ξ, t∗), t∗) − (x(ξ, t), t)

(cid:18)

hn

hn

exp

− 1
2

→ ∞ as n → ∞.

Due to the limit behavior of this kernel term, the covariance function becomes infinitely
larger when t (cid:54)= t∗.

However, If t = t∗, then we have

(x(ξ + τ hn, t), t) − (x(ξ, t), t)

hn

→ (τ v(D(ξ, t), t), 0) as n → ∞.

and hence under the Gaussian kernel function,

77

(cid:16)

K

(cid:18)
(cid:18)

ψ +

× exp

→ exp

(x(ξ + τ hn, t), t) − (x(ξ, t), t)

(cid:17) ∝ exp
(cid:18)(x(ξ + τ hn, t), t) − (x(ξ, t), t)

hn

(cid:19)

(cid:18)
(cid:19)(cid:62)(cid:18)(x(ξ + τ hn, t), t) − (x(ξ, t), t)

− 1
2

ψ(cid:62)ψ

(cid:19)(cid:19)

(cid:19)

ψ(cid:62)ψ

exp

− 1
2
− 1
2

(cid:18)

hn
− 1
2

(cid:19)
hn
(τ v(D(ξ, t), t), 0)(cid:62)(τ v(D(ξ, t), t), 0)

as n → ∞.

To sum up, the covariance function for all pairs of spatial points (s, s∗) ∈ [0, S] given the

time point t ∈ [0, T ] is defined as follows:

(cid:90) S
×(cid:104)

0

C1((s, t), (s∗, t)) =

Ψ(v(D(x(ξ, t), t)))g1(s, ξ, t)

D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)

(cid:105)
g(cid:62)
1 (s∗, ξ, t)dξ,

where Ψ(v(D(x(ξ, t), t))) :=(cid:82)

(ii) Theorem 2.4.2.

(a) = Cov

g(s, ξ, t)

R

Rd+1 K(ψ)K(ψ + (τ v(D(x(ξ, t), t)), 0))dψdτ .

∂t

(cid:82)
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:98)Dn(x(η, t∗), t∗) − ∂
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:98)Dn(x(η, t∗), t∗) − ∂
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:98)Dn(x(η, t∗), t∗) − ∂

∂x

∂x

∂t

∂t

∂x

(cid:110) ∂

∂t

(cid:110) ∂

∂x

(cid:110) ∂

∂x

,

dη

dξ,

D(x(ξ, t), t)

(cid:111)
(cid:111)
∂t
D(x(η, t∗), t∗)

(cid:105)
(cid:111) ∂
(cid:111) ∂
∂x
∂t
x(η, t∗)dη
D(x(η, t∗), t∗)
(cid:111)
(cid:111) ∂
∂t
D(x(η, t∗), t∗)

x(η, t∗)dη

D(x(ξ, t), t)

D(x(ξ, t), t)

dξ,

∂t

(cid:105)

,

(cid:105)

.

x(ξ, t)dξdt,

∂t

0

(cid:104)(cid:90) S
(cid:104)(cid:90) S
(cid:104)(cid:90) S

0

g(s∗, η, t∗)

g(s∗, η, t∗)

0

g(s∗, η, t∗)

(cid:90) S

0

(cid:90) S

0

(cid:90) S

0

(b) = Cov

g(s, ξ, t)

(c) = Cov

g(s, ξ, t)

For t (cid:54)= t∗, these covariance functions under any density kernel function diverge to infinity
n Cov[z2(s, t), z2(s∗, t∗)] are close to zero as
as n → ∞. Also, the remaining terms of
n → ∞ when t = t∗. In what follows, smaller order terms are omitted.

nhd+2

(cid:113)

78

(a) =

0

0

Rd+1

(cid:90)

(cid:90) S

(cid:90) S
(cid:16)(x(η, t∗), t∗) − w
(cid:90) S
(cid:16)(x(η, t∗), t∗) − w

(cid:90) S

Rd+1

(cid:90)

hn

0

0

1
2(d+2)
n

nh

K

(1)
t

1
2(d+2)
n

nh

×(cid:16)
×(cid:16)

+

K

(1)
t

hn

by change of variable η = ξ + τ hn

(cid:16)(x(ξ, t), t) − w

(cid:17)

g1(s, ξ, t)D(w)K

(1)
t

(cid:17)(cid:17)(cid:62)

(cid:17)(cid:17)(cid:62)

D(cid:62)(w)g(cid:62)

hn
1 (s∗, η, t∗)dwdηdξ
(cid:16)(x(ξ, t), t) − w

(cid:17)

g1(s, ξ, t)Γ(w)K

(1)
t

hn
1 (s∗, η, t∗)dwdηdξ

Γ(cid:62)(w)g(cid:62)

0

(cid:90)

Rd+1

−ξ/hn

(cid:90) (S−ξ)/hn

(cid:90) S
(cid:16)(x(ξ + τ hn, t∗), t∗) − w
(cid:90) S
(cid:16)(x(ξ + τ hn, t∗), t∗) − w

(cid:90) (S−ξ)/hn

−ξ/hn

Rd+1

(cid:90)

hn

0

(cid:17)(cid:17)(cid:62)

(cid:17)(cid:17)(cid:62)

=

×(cid:16)
×(cid:16)

+

1

nh2d+3

n

K

(1)
t

1

nh2d+3

n

K

(1)
t

hn

hn

(1)
t

(cid:17)
(cid:16)(x(ξ, t), t) − w
1 (s∗, ξ + τ hn, t∗)dwdτ dξ
(cid:16)(x(ξ, t), t) − w
(cid:17)
1 (s∗, ξ + τ hn, t∗)dwdτ dξ

(1)
t

hn

g1(s, ξ, t)D(w)K

D(cid:62)(w)g(cid:62)

g1(s, ξ, t)Γ(w)K

Γ(cid:62)(w)g(cid:62)

0

n

1

=

K

hn

(1)
t

(cid:90)

Rd+1

nhd+2

×(cid:16)

by letting ψ = (x(ξ,t),t)−w
(cid:90) S
(cid:16)
(cid:90) S
(cid:16)

(cid:90) (S−ξ)/hn
(cid:17)(cid:17)(cid:62)
−ξ/hn
(x(ξ + τ hn, t∗), t∗) − (x(ξ, t), t)
(cid:90) (S−ξ)/hn
(cid:90)
1 (s∗, ξ + τ hn, t∗)dψdτ dξ
× D(cid:62)((x(ξ, t), t) − hnψ)g(cid:62)
(cid:17)(cid:17)(cid:62)
−ξ/hn
(x(ξ + τ hn, t∗), t∗) − (x(ξ, t), t)
1 (s∗, ξ + τ hn, t∗)dψdτ dξ.

× Γ(cid:62)((x(ξ, t), t) − hnψ)g(cid:62)

×(cid:16)

nhd+2

Rd+1

(1)
t

ψ +

ψ +

hn

hn

K

+

1

n

0

79

g1(s, ξ, t)D((x(ξ, t), t) − hnψ)K

(1)
t

(ψ)

g1(s, ξ, t)Γ((x(ξ, t), t) − hnψ)K

(1)
t

(ψ)

In a similar manner to (a), we have

(b) =

0

n

1

K

∂t

ψ +

(1)
x

(cid:90)

Rd+1

nhd+2

(cid:90) S

×(cid:16) ∂

(cid:90) (S−ξ)/hn
(cid:16)
(cid:17)(cid:62)(cid:16)
−ξ/hn
x(ξ + τ hn, t∗)
hn
(cid:90) (S−ξ)/hn
(cid:90) S
(cid:90)
1 (s∗, ξ + τ hn, t∗)dψdτ dξ
× D(cid:62)((x(ξ, t), t) − hnψ)g(cid:62)
(cid:16)
(cid:17)(cid:62)(cid:16)
hn
× Γ(cid:62)((x(ξ, t), t) − hnψ)g(cid:62)
1 (s∗, ξ + τ hn, t∗)dψdτ dξ,

−ξ/hn
x(ξ + τ hn, t∗)

×(cid:16) ∂

nhd+2

Rd+1

(1)
x

ψ +

∂t

K

+

1

n

0

∂
∂t

(cid:17)(cid:17)(cid:62)

∂
∂t

(cid:17)(cid:17)(cid:62)

g1(s, ξ, t)D((x(ξ, t), t) − hnψ)K

(1)
x (ψ)

x(ξ, t)

(x(ξ + τ hn, t∗), t∗) − (x(ξ, t), t)

g1(s, ξ, t)Γ((x(ξ, t), t) − hnψ)K

(1)
x (ψ)

x(ξ, t)

(x(ξ + τ hn, t∗), t∗) − (x(ξ, t), t)

and

(c) =

0

n

1

K

∂t

ψ +

(1)
x

(cid:90)

Rd+1

nhd+2

(cid:90) S

×(cid:16) ∂

(cid:90) (S−ξ)/hn
(cid:17)(cid:62)(cid:16)
(cid:16)
−ξ/hn
x(ξ + τ hn, t∗)
hn
(cid:90) (S−ξ)/hn
(cid:90) S
(cid:90)
× D(cid:62)((x(ξ, t), t) − hnψ)g(cid:62)
1 (s∗, ξ + τ hn, t∗)dψdτ dξ
(cid:17)(cid:62)(cid:16)
(cid:16)
hn
1 (s∗, ξ + τ hn, t∗)dψdτ dξ.
× Γ(cid:62)((x(ξ, t), t) − hnψ)g(cid:62)

−ξ/hn
x(ξ + τ hn, t∗)

×(cid:16) ∂

nhd+2

Rd+1

(1)
x

ψ +

∂t

K

+

1

n

0

g1(s, ξ, t)D((x(ξ, t), t) − hnψ)K

(1)
t

(x(ξ + τ hn, t∗), t∗) − (x(ξ, t), t)

(ψ)

(cid:17)(cid:17)(cid:62)

g1(s, ξ, t)Γ((x(ξ, t), t) − hnψ)K

(1)
t

(x(ξ + τ hn, t∗), t∗) − (x(ξ, t), t)

(ψ)

(cid:17)(cid:17)(cid:62)

Premultiplying by nhd+2

the results of (a)-(c), we have the following limiting covariance
function for all pairs of spatial points (s, s∗) ∈ [0, S] given the time point t ∈ [0, T ] as n → ∞:

n

80

C2((s, t), (s∗, t)) =

0

Ψt(v(D(ξ, t), t))g1(s, ξ, t)

(cid:90) S
(cid:105)
×(cid:104)
(cid:90) S
1 (s∗, ξ, t)dξ
g(cid:62)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)
(cid:105)
×(cid:104)
(cid:90) S
1 (s∗, ξ, t)dξ
g(cid:62)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)
(cid:105)
×(cid:104)
D(x(ξ, t), t)D(cid:62)(x(ξ, t), t) + Γ(x(ξ, t), t)Γ(cid:62)(x(ξ, t), t)

Ψtx(v(D(ξ, t), t))g1(s, ξ, t)

Ψx(v(D(ξ, t), t))g1(s, ξ, t)

+

+

0

0

1 (s∗, ξ, t)dξ.
g(cid:62)

(iii) Theorem 2.4.3.

We only provide main terms of covariance function, whereas smaller order terms are omitted:

(a) = Cov

g1(s, ξ, t)

D(x(ξ, t), t)

dξdt,

(b) = Cov

g1(s, ξ, t)

D(x(ξ, t), t)

x(ξ, t)dξdt,

(c) = Cov

g1(s, ξ, t)

D(x(ξ, t), t)

dξdt,

(cid:90) b

a

(cid:90) b

a

(cid:90) b

a

∂λ

0

0

0

0

a

a

w(cid:62)(λ)

w(cid:62)(λ)

(cid:90) S

(cid:104)(cid:90) b
w(cid:62)(t)
(cid:90) S
(cid:104)(cid:90) b
w(cid:62)(t)
(cid:90) S
(cid:104)(cid:90) b
w(cid:62)(t)
(cid:90) S

(cid:110) ∂
g1(s∗, η, λ)
(cid:90) S
(cid:110) ∂
g1(s∗, η, λ)
(cid:90) S
(cid:110) ∂
w(cid:62)(λ)
g1(s∗, η, λ)
(cid:90) b
(cid:90) b
(cid:90) S
(cid:90) S
(cid:90)
(cid:17)(cid:17)(cid:62)
(cid:16)(x(η, λ), λ) − w
(cid:90)
(cid:90) S
(cid:90) b
(cid:90) b

(cid:90) S

hn

a

a

a

0

0

0

0

∂x

∂x

a

0

a

0

Rd+1

(a) =

1
2(d+2)
n

nh

×(cid:16)

K

(1)
λ

+

1
2(d+2)
n

nh

D(x(η, λ), λ)

dηdλ

,

∂t

∂t

∂λ

(cid:98)Dn(x(ξ, t), t) − ∂

(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:98)Dn(x(η, λ), λ) − ∂
(cid:110) ∂
(cid:98)Dn(x(η, λ), λ) − ∂
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:98)Dn(x(η, λ), λ) − ∂

∂x

∂x

∂t

∂t

∂x

∂x

(cid:111)

(cid:105)
(cid:111) ∂

∂t

(cid:111)

(cid:111) ∂
(cid:111)
(cid:111) ∂

∂λ

∂λ

D(x(η, λ), λ)

x(η, λ)dηdλ

,

D(x(η, λ), λ)

x(η, λ)dηdλ

.

(cid:105)

(cid:105)

(cid:16)(x(ξ, t), t) − w

(cid:17)

hn

(1)
t

(cid:16)(x(ξ, t), t) − w

(cid:17)

hn

w(cid:62)(t)g1(s, ξ, t)D(w)K

Rd+1
D(cid:62)(w)g1

(cid:62)(s∗, η, λ)w(λ)dwdηdλdξdt

w(cid:62)(t)g1(s, ξ, t)Γ(w)K

(1)
t

81

×(cid:16)

K

(1)
λ

(cid:16)(x(η, λ), λ) − w

(cid:17)(cid:17)(cid:62)

hn

Γ(cid:62)(w)g1

(cid:62)(s∗, η, λ)w(λ)dwdηdλdξdt

nhd+3

by letting ψ = (x(ξ,t),t)−w
(cid:90) S
(cid:90) b

hn

0

0

a

a

n

1

hn

ψ +

(1)
λ

(cid:90)

Rd+1

(cid:90) S

(cid:90) b
(cid:16)
K
(cid:90) b
(cid:62)(s∗, η, λ)w(λ)dwdηdλdξdt
(cid:16)
K
(cid:62)(s∗, η, λ)w(λ)dwdηdλdξdt

(cid:90) S

(cid:90) S

(cid:90) b

Rd+1

(cid:90)

(1)
λ

ψ +

hn

1

n

a

0

a

0

=

×(cid:16)

× g1

+

×(cid:16)

× g1

nhd+3

(cid:17)(cid:17)(cid:62)

(cid:17)(cid:17)(cid:62)

w(cid:62)(t)g1(s, ξ, t)D((x(ξ, t), t) − hnψ)K

(1)
t

(ψ)

(x(η, λ), λ) − (x(ξ, t), t)

D(cid:62)((x(ξ, t), t) − hnψ)

w(cid:62)(t)g1(s, ξ, t)Γ((x(ξ, t), t) − hnψ)K

(1)
t

(ψ)

(x(η, λ), λ) − (x(ξ, t), t)

Γ(cid:62)((x(ξ, t), t) − hnψ)

by change of variables η = ξ + τ hn and λ = t + γhn,

1
nhd
n

=

×(cid:16)

K

× g1
1
nhd
n

×(cid:16)

K

+

0

a

ψ +

(1)
γ

(cid:90)

Rd+1

−ξ/hn

(cid:90) S

(a−t)/hn
(x(ξ + τ hn, t + γhn), t + γhn) − (x(ξ, t), t)

(cid:90) (b−t)/hn

(cid:90) (S−ξ)/hn

(cid:90) b
(cid:16)
(cid:90) b
(cid:62)(s∗, ξ + τ hn, t + γhn)w(t + γhn)dwdτ dγdξdt
(cid:16)
(cid:62)(s∗, ξ + τ hn, t + γhn)w(t + γhn)dwdτ dγdξdt.

(cid:90) (S−ξ)/hn

(cid:90) (b−t)/hn

(a−t)/hn
(x(ξ + τ hn, t + γhn), t + γhn) − (x(ξ, t), t)

(cid:90) S

−ξ/hn

Rd+1

(cid:90)

(1)
γ

ψ +

hn

hn

a

0

× g1

w(cid:62)(t)g1(s, ξ, t)D((x(ξ, t), t) − hnψ)K
D(cid:62)((x(ξ, t), t) − hnψ)

(cid:17)(cid:17)(cid:62)

(1)
t

(ψ)

w(cid:62)(t)g1(s, ξ, t)Γ((x(ξ, t), t) − hnψ)K
Γ(cid:62)((x(ξ, t), t) − hnψ)

(cid:17)(cid:17)(cid:62)

(1)
t

(ψ)

82

w(cid:62)(t)g1(s, ξ, t)D((x(ξ, t), t) − hnψ)K

(1)
x (ψ)

x(ξ + τ hn, t + γhn)
∂γ
(x(ξ + τ hn, t + γhn), t + γhn) − (x(ξ, t), t)

(cid:17)(cid:17)(cid:62)

D(cid:62)((x(ξ, t), t) − hnψ)

Likewise,

(b) =

1
nhd
n

(cid:90) b

a

(cid:90) S
(cid:16) ∂

0

(cid:90) (b−t)/hn

(a−t)/hn

(cid:90) (S−ξ)/hn
(cid:90)
(cid:17)(cid:62)

−ξ/hn

Rd+1

× ∂
∂t

×(cid:16)

x(ξ, t)

K

(1)
x

ψ +

+

1
nhd
n

× ∂
∂t

×(cid:16)

x(ξ, t)

K

(1)
x

ψ +

(cid:16)
(cid:90) b

a

(cid:16)

× g(cid:62)(s∗, ξ + τ hn, t + γhn)w(t + γhn)dψdτ dγdξdt

(cid:90) (b−t)/hn

(a−t)/hn

(cid:90) S
(cid:16) ∂

0

hn

(cid:90)
(cid:90) (S−ξ)/hn
(cid:17)(cid:62)

−ξ/hn

w(cid:62)(t)g1(s, ξ, t)Γ((x(ξ, t), t) − hnψ)K

(1)
x (ψ)

Rd+1

x(ξ + τ hn, t + γhn)
∂γ
(x(ξ + τ hn, t + γhn), t + γhn) − (x(ξ, t), t)

(cid:17)(cid:17)(cid:62)

Γ(cid:62)((x(ξ, t), t) − hnψ)

× g(cid:62)(s∗, ξ + τ hn, t + γhn)w(t + γhn)dψdτ dγdξdt,

hn

and

(c) =

1
nhd
n

×(cid:16) ∂

∂γ

+

1
nhd
n

×(cid:16) ∂

∂γ

x(ξ + τ hn, t + γhn)

ψ +

(cid:90) b

(cid:90) S

(cid:90) (b−t)/hn

a

0

(a−t)/hn

(cid:90) b

(cid:90) S

(cid:90) (b−t)/hn

a

0

(a−t)/hn

(cid:90)

(cid:90)

K

(1)
x

−ξ/hn

(cid:90) (S−ξ)/hn
(cid:16)
(cid:17)(cid:62)(cid:16)
(cid:90) (S−ξ)/hn
(cid:17)(cid:62)(cid:16)
(cid:16)

−ξ/hn

K

(1)
x

x(ξ + τ hn, t + γhn)

ψ +

× D(cid:62)((x(ξ, t), t) − hnψ)g(cid:62)(s∗, ξ + τ hn, t + γhn)w(t + γhn)dψdτ dγdξdt

Rd+1

w(cid:62)(t)g1(s, ξ, t)D((x(ξ, t), t) − hnψ)K
(x(ξ + τ hn, t + γhn), t + γhn) − (x(ξ, t), t)

(1)
t

(ψ)

(cid:17)(cid:17)(cid:62)

Rd+1

w(cid:62)(t)g1(s, ξ, t)Γ((x(ξ, t), t) − hnψ)K

(1)
t

(x(ξ + τ hn, t + γhn), t + γhn) − (x(ξ, t), t)

(ψ)

(cid:17)(cid:17)(cid:62)

hn

hn

× Γ(cid:62)((x(ξ, t), t) − hnψ)g(cid:62)(s∗, ξ + τ hn, t + γhn)w(t + γhn)dψdτ dγdξdt.

83

Note that

(x(ξ + τ hn, t + γhn), t + γhn) − (x(ξ, t), t)

hn

(x(ξ + τ hn, t + γhn), t + γhn) − (x(ξ, t + γhn), t + γhn)
(x(ξ, t + γhn), t + γhn) − (x(ξ, t), t)

hn

hn

=

+

→ (τ v(D(x(ξ, t), t)) + γ

x(ξ, t), γ) as n → ∞.

∂
∂t

From (a) to (c), covariance function for all pairs of points (s, s∗) ∈ [0, S] is shown as the one
in Theorem 2.4.3. as n → ∞.

6.4 Convergence of finite-dimensional distributions

(i) Theorem 2.4.1.

finite-dimensional distributions of the sequence of stochastic processes {(cid:113)

The multivariate central limit theorem using Lyapunov’s condition requires to check that
nz1(s, t), s ∈
[0, S], t ∈ [0, T ]} converge to finite-dimensional distributions of the sequence of the limiting
Gaussian process {GP1(s, t), s ∈ [0, S], t ∈ [0, T ]} with mean function µβ1
ance function C1((s, t), (s∗, t)).

(s, t) and covari-

nhd

Let us consider

(cid:90) S

0

(cid:16)(x(ξ, t), t) − Ui

(cid:17)

hn

dξ, s ∈ [0, S], t ∈ [0, T ].

η1i :=

g1(s, ξ, t)(D(Ui) + Γ(Ui))K

η1, η1i’s are i.i.d. d × 1 random vectors in Rd satisfying

84

(cid:113)

nhd
n

(cid:16)

(cid:17)
z1(s, t) − E[z1(s, t)]

=

1(cid:113)

nhd+2

n

n(cid:88)
(η1i − E[η1i]).

i=1

Note that

E[|η1|4] ≤ chd+4

n

(cid:90) S

0

(cid:90)

(cid:90)

(cid:90)

R

R

R

|g1(s, ξ, t)|4dξ

Λn(τ1, τ2, τ3)dτ1dτ2dτ3,

where the limiting distribution of Λn(τ1, τ2, τ3) is defined as

(cid:90)

n(cid:88)

i=1

n→∞ Λn(τ1, τ2, τ3) :=
lim

Rd+1

K(ψ)K(ψ + (τ1v(D(x(ξ, t), t)), 0))

× K(ψ + (τ2v(D(x(ξ, t), t)), 0))K(ψ + (τ3v(D(x(ξ, t), t)), 0))dψ.

Thus, given the parameter time t ∈ [0, T ], we have

(cid:16)

(cid:17) → 0 as n → ∞.

1 + op(1)

E[|η1i − E[η1i]|4] ≤ C
nhd
n

1
2(d+2)
n2h
n

(ii) Theorem 2.4.2.

Let us define

(cid:90) S
(cid:90) S

0

0

η2i :=

g1(s, ξ, t)(D(Ui) + Γ(Ui))K

+

g1(s, ξ, t)(D(Ui) + Γ(Ui))K

(cid:16)(x(ξ, t), t) − Ui
(cid:17)
(cid:17) ∂
(cid:16)(x(ξ, t), t) − Ui

hn

dξ

hn

∂t

(1)
t

(1)
x

x(ξ, t)dξ, s ∈ [0, S], t ∈ [0, T ].

Then η2, η2i’s are i.i.d. random vectors in Rd satisfying

(cid:113)

(cid:16)

(cid:17)
z2(s, t) − E[z2(s, t)]

nhd+2

n

(cid:113)

1 + op(1)
nhd+2

n

=

n(cid:88)

i=1

(η2i − E[η2i]),

85

Note that

E[|η2|4] ≤ chd+4

n

(cid:90) S

0

(cid:90)

(cid:90)

(cid:90)

R

R

R

(cid:12)(cid:12)Λn,x,t(τ1, τ2, τ3)(cid:12)(cid:12)dτ1dτ2dτ3,

|g1(s, ξ, t)|4dξ

where the limiting distribution of Λn,x,t(τ1, τ2, τ3) is defined as

n→∞Λn,x,t(τ1, τ2, τ3)
lim
(1)
t

:=

K

Rd+1

(ψ)K

(1)
t

(ψ + (τ1v(D(x(ξ, t), t)), 0))

(cid:17)(cid:62)
(cid:17)(cid:62)

∂
∂t

(cid:17)(cid:62)

86

Rd+1

Rd+1

+ 6

+ 4

× K

× K

× K

(cid:90)
(cid:90)
(cid:90)
×(cid:16) ∂
(cid:90)
×(cid:16) ∂
(cid:90)
×(cid:16) ∂

× K

+ 4

∂t
× ∂
∂t

∂t

+

∂t

(cid:17)(cid:62)(cid:16)
(cid:17)(cid:62)(cid:16)

(1)
t

(cid:17)(cid:62)(cid:16)

(1)
t

(ψ + (τ2v(D(x(ξ, t), t)), 0))K

(1)
t

(ψ + (τ3v(D(x(ξ, t), t)), 0))dψ

K

(1)
t

(ψ)K

(1)
t

(ψ + (τ1v(D(x(ξ, t), t)), 0))

(1)
t

(ψ + (τ2v(D(x(ξ, t), t)), 0))K

(1)
x (ψ + (τ3v(D(x(ξ, t), t)), 0))

∂
∂t

x(ξ, t)dψ

K

(1)
t

(ψ)K

(1)
t

(ψ + (τ1v(D(x(ξ, t), t)), 0))

(1)
x (ψ + (τ2v(D(x(ξ, t), t)), 0))

∂
∂t

x(ξ, t)

x(ξ, t)

K

(1)
x (ψ + (τ3v(D(x(ξ, t), t)), 0))

dψ

K

(ψ)K

(1)
x (ψ + (τ1v(D(x(ξ, t), t)), 0))

x(ξ, t)

Rd+1

x(ξ, t)

K

(1)
x (ψ + (τ2v(D(x(ξ, t), t)), 0))

x(ξ, t)K

(1)
x (ψ + (τ3v(D(x(ξ, t), t)), 0))dψ

(cid:16) ∂

∂t

(cid:17)(cid:62)(cid:16)

K

(1)
x (ψ)

∂
∂t

x(ξ, t)

Rd+1

x(ξ, t)

K

(1)
x (ψ + (τ1v(D(x(ξ, t), t)), 0))

(cid:17)(cid:62)

(1)
x (ψ + (τ2v(D(x(ξ, t), t)), 0))

∂
∂t

x(ξ, t)

x(ξ, t)

K

(1)
x (ψ + (τ3v(D(x(ξ, t), t)), 0))

dψ.

Then given the parameter time t ∈ [0, T ], we have

n(cid:88)

i=1

1 + op(1)
2(d+2)
n2h
n

E[|η2i − E[η2i]|4] ≤ C
nhd
n
(cid:113)

(cid:16)

(cid:17) → 0 as n → ∞.

1 + op(1)

Since Lyapunov’s condition is satisfied, we state that finite-dimensional distributions
of the stochastic process {
z2(s, t), s ∈ [0, S], t ∈ [0, T ]} converge in the space of
C([0, S], Rd) to finite-dimensional distributions of the limiting Gaussian process {GP2(s, t),
(s, t) and covariance function C2((s, t), (s∗, t)).
s ∈ [0, S], t ∈ [0, T ]} with mean function µβ2

nhd+2

n

Let us define for 0 < a < b ≤ T and s ∈ [0, S]

(iii) Theorem 2.4.3.

(cid:90) b
(cid:90) b

a

(cid:90) S
w(cid:62)(t)
(cid:90) S

0

w(cid:62)(t)

a

0

η3i :=

+

g1(s, ξ, t)(D(Ui) + Γ(Ui))K

dξdt

g1(s, ξ, t)(D(Ui) + Γ(Ui))K

x(ξ, t)dξdt.

(1)
t

(1)
x

(cid:17)
(cid:16)(x(ξ, t), t) − Ui
(cid:17) ∂
(cid:16)(x(ξ, t), t) − Ui

hn

hn

∂t

n(cid:88)
(η3i − E[η3i]).

i=1

(cid:113)

1 + op(1)
nhd+4

n

Then η3, η3i are d × 1 random vectors in Rd such that

(cid:113)

nhd

n(z3(s) − E[z3(s)]) =

Note that

(cid:90) b
(cid:90) S
(cid:90)
(cid:90)
(cid:90)
(cid:90)
|g1(s, ξ, t)|4dξdt

a

0

(cid:90)
E[|η3|4] ≤ chd+8

(cid:90)

n

×

R

R

R

R

R

R

(cid:12)(cid:12)Λn,x,t(τ1, γ1, τ2, γ2, τ3, γ3)(cid:12)(cid:12)dτ1dγ1dτ2dγ2dτ3dγ3,

where the limiting distribution of Λn,x,t(τ1, γ1, τ2, γ2, τ3, γ3) is defined as

87

n→∞Λn,x,t(τ1, γ1, τ2, γ2, τ3, γ3)
lim

:=

Rd+1

K

(1)
t

(ψ)K

(1)
t

ψ + (τ1v(D(x(ξ, t), t)) + γ1

ψ + (τ2v(D(x(ξ, t), t)) + γ2

x(ξ, t), γ2)

ψ + (τ3v(D(x(ξ, t), t)) + γ3

x(ξ, t), γ3)

dψ

K

(1)
t

(ψ)K

(1)
t

Rd+1

ψ + (τ1v(D(x(ξ, t), t)) + γ1

ψ + (τ2v(D(x(ξ, t), t)) + γ2

x(ξ, t), γ2)

(cid:17)

x(ξ, t), γ1)

∂
∂t

(cid:17)

x(ξ, t), γ1)

∂
∂t

∂
∂t
∂
∂t

∂
∂t
∂
∂t

∂
∂t

∂
∂t

(cid:17)
(cid:17)
(cid:17)
(cid:17) ∂
(cid:17) ∂

∂t

∂
∂t

∂t
∂
∂t

∂
∂t

(cid:17) ∂

∂t

∂
∂t

(cid:17) ∂

∂t
∂
∂t

(cid:90)

(cid:16)
(cid:16)
(cid:16)
(cid:16)
(cid:16)

(cid:16)

(cid:16)

× K
× K

(1)
t

(1)
t

(cid:90)

+ 4

(1)
t

(1)
x

+ 6

× K
× K

(cid:90)
×(cid:16) ∂
(cid:90)

× K

+ 4

∂t

(1)
x

(1)
x

+

× ∂
∂t
× K

(cid:90)
×(cid:16) ∂
×(cid:16) ∂

∂t
× K
(1)
x

∂t

(cid:17)(cid:62)(cid:16)
(cid:16) ∂

(1)
t

K

∂t

(cid:17)(cid:62)(cid:16)
(cid:17)(cid:62)(cid:16)

(cid:16)

(cid:16)

(cid:16)

(cid:16)
(cid:16)

ψ + (τ3v(D(x(ξ, t), t)) + γ3

x(ξ, t), γ3)

x(ξ, t)dψ

K

(1)
t

(ψ)K

(1)
t

Rd+1

ψ + (τ1v(D(x(ξ, t), t)) + γ1

x(ξ, t), γ1)

ψ + (τ2v(D(x(ξ, t), t)) + γ2

x(ξ, t), γ2)

x(ξ, t)

x(ξ, t)

K

(1)
x

ψ + (τ3v(D(x(ξ, t), t)) + γ3

x(ξ, t), γ3)

dψ

Rd+1

(ψ)K

ψ + (τ1v(D(x(ξ, t), t)) + γ1

x(ξ, t), γ1)

x(ξ, t)

x(ξ, t)

ψ + (τ2v(D(x(ξ, t), t)) + γ2

x(ξ, t), γ2)

ψ + (τ3v(D(x(ξ, t), t)) + γ3

∂
∂t

x(ξ, t), γ3)

x(ξ, t)dψ

(cid:17)(cid:17)(cid:62)

∂
∂t

(cid:16)
(cid:16)
(cid:17)(cid:62)(cid:16)

(1)
x

(cid:16)

K

(1)
x

K

(1)
x (ψ)

∂
∂t

Rd+1

x(ξ, t)

x(ξ, t)

K

(1)
x

ψ + (τ1v(D(x(ξ, t), t)) + γ1

x(ξ, t), γ1)

ψ + (τ2v(D(x(ξ, t), t)) + γ2

x(ξ, t), γ2)

x(ξ, t)

x(ξ, t)

K

(1)
x

ψ + (τ3v(D(x(ξ, t), t)) + γ3

x(ξ, t), γ3)

dψ.

(cid:17)
(cid:17)(cid:17)(cid:62)
(cid:17)

(cid:17)(cid:17)(cid:62)
(cid:17)(cid:17)(cid:62)

Then given the parameter time t ∈ [0, T ], we have

n(cid:88)

i=1

1 + op(1)
2(d+4)
n2h
n

E[|η3i − E[η3i]|4] ≤ C
nhd
n

88

(cid:16)

(cid:17) → 0 as n → ∞.

1 + op(1)

6.5 Asymptotic equicontinuity

(i) Theorem 2.4.1.

For all pairs of points ((s, t), (s∗, t∗)) ∈ [0, S] × [0, T ], we define ˜η1i as follows:
(cid:17)

(cid:16)(x(ξ, t), t) − Ui
(cid:17)
(cid:16)(x(ξ, t∗), t∗) − Ui

g1(s, ξ, t)(D(Ui) + Γ(Ui))K
g1(s∗, ξ, t∗)(D(Ui) + Γ(Ui))K

(cid:90) S
(cid:90) S

˜η1i :=

−

dξ.

hn

dξ

0

hn

0

Note that ˜η1, ˜η1i’s are i.i.d. random vectors in Rd.

E(cid:104)(cid:12)(cid:12)(cid:12)(cid:12)(cid:113)

nhd
n

(cid:110)(cid:0)z1(s, t) − E[z1(s, t)](cid:1) −(cid:0)z1(s∗, t∗) − E[z1(s∗, t∗)](cid:1)(cid:111)(cid:12)(cid:12)(cid:12)(cid:12)4(cid:105)
(cid:111)
(cid:110)n(n − 1)
+ nE[|˜η1 − E[˜η1]|4]

(cid:16)E[|˜η1 − E[˜η1]|2]
(cid:17)2

.

=

1
2(d+2)
n2h
n

If t = t∗, then we can readily derive

(cid:90)

2

(cid:90) S

0

λn(τ )dτ,

E[|˜η1|2] ≤ chd+2

n

|g1(s, ξ, t) − g1(s∗, ξ, t)|2dξ

0

where the limiting distribution of λn(τ ) is defined as limn→∞ λn(τ ) :=(cid:82)
(τ v(D(x(ξ, t), t)), 0)(cid:62)(cid:1)dψ, and
(cid:90) S

(cid:90)

R

E[|˜η1|4] ≤ chd+4

n

|g1(s, ξ, t) − g1(s∗, ξ, t)|4dξ

λn(τ1, τ2, τ3)dτ,

R

Rd+1 K(ψ)K(cid:0)ψ +

where the limiting distribution of λn(τ1, τ2, τ3) is as in (i) of Section 6.4. When t (cid:54)= t∗ under
any density kernel function, limn→∞ hd+2

n E[|˜η1|2] = ∞ and limn→∞ hd+4

n E[|˜η1|4] = ∞.

Therefore, when t = t∗, due to continuity of g1 function, we have, for any ε > 0,

89

(cid:16)

lim sup
n→∞ P

(cid:12)(cid:12)(cid:12)(cid:12)(cid:113)

sup

s,s∗∈[0,S],
|s−s∗|<δ

(cid:8)(z1(s, t) − E[z1(s, t)]) − (z1(s∗, t) − E[z1(s∗, t)])(cid:9)(cid:12)(cid:12)(cid:12)(cid:12) > ε

(cid:17) → 0,

nhd
n

as δ → 0.

{(cid:113)

It shows the asymptotic equicontinuity condition of the stochastic process
n(z1(s, t)− E[z1(s, t)]), s ∈ [0, S], t ∈ [0, T ]} is met in the space of C([0, S], Rd). In the

nhd

space of C([0, S] × [0, T ], Rd), this condition is no longer satisfied.

(ii) Theorem 2.4.2.

0

˜η2i :=

g1(s, ξ, t)(D(Ui) + Γ(Ui))K

For all pairs of points ((s, t), (s∗, t∗)) ∈ [0, S] × [0, T ], let us define ˜η2i:
(cid:17)
(cid:16)(x(ξ, t), t) − Ui
(cid:16)(x(ξ, t), t) − Ui
(cid:17) ∂
(cid:16)(x(ξ, t∗), t∗) − Ui
(cid:16)(x(ξ, t∗), t∗) − Ui

g1(s, ξ, t)(D(Ui) + Γ(Ui))K
g1(s∗, ξ, t∗)(D(Ui) + Γ(Ui))K
g1(s∗, ξ, t∗)(D(Ui) + Γ(Ui))K

(cid:90) S
(cid:90) S
(cid:90) S
(cid:90) S

(1)
t

(1)
t

(1)
x

(1)
x

hn

hn

+

−

−

hn

dξ

∂t

(cid:17)
(cid:17) ∂

dξ

0

0

0

x(ξ, t)dξ

x(ξ, t∗)dξ.

hn

∂t

Note that ˜η2, ˜η2i’s are i.i.d. random vectors in Rd.

E(cid:104)(cid:12)(cid:12)(cid:12)(cid:12)(cid:113)

nhd+2

n

=

1 + op(1)
2(d+2)
n2h
n

(cid:110)(cid:0)z2(s, t) − E[z2(s, t)](cid:1) −(cid:0)z2(s∗, t∗) − E[z2(s∗, t∗)](cid:1)(cid:111)(cid:12)(cid:12)(cid:12)(cid:12)4(cid:105)
(cid:110)n(n − 1)
(cid:111)
+ nE[|˜η2 − E[˜η2]|4]

(cid:16)E[|˜η2 − E[˜η2]|2]

(cid:17)2

.

Then the rest of the proof is analogous to the one in (i) in Section 6.5. If t = t∗,

2

(cid:90) S

0

E[|˜η2|2] ≤ chd+2

n

|g1(s, ξ, t) − g1(s∗, ξ, t)|2dξ

(cid:90)

R

(cid:12)(cid:12)λn,x,t(τ )(cid:12)(cid:12)dτ,

where the limiting distribution of λn,x,t(τ ) is defined as

90

lim
n→∞ Λn,x,t(τ ) :=

(cid:90)

Rd+1

K

(1)
t

(ψ)K

K

(1)
t

(ψ)K

(1)
x

Rd+1

(1)
t

(cid:0)ψ + (τ v(D(x(ξ, t), t)), 0)(cid:1)
(cid:0)ψ + (τ v(D(x(ξ, t), t)), 0)(cid:1) ∂
(cid:16) ∂

(cid:17)(cid:62)(cid:16)

K

(1)
x (ψ)

∂
∂t

Rd+1

x(ξ, t)

x(ξ, t)

∂t

K

(1)
x

(cid:90)

+ 2

(cid:90)

+

∂t

x(ξ, t)

(cid:0)ψ + (τ v(D(x(ξ, t), t)), 0)(cid:1)(cid:17)(cid:62)

dψ,

and

E[|˜η2|4] ≤ chd+4

n

(cid:90) S

0

|g1(s, ξ, t) − g1(s∗, ξ, t)|4dξ

(cid:90)

(cid:90)

(cid:90)

R

R

R

(cid:12)(cid:12)λn,x,t(τ1, τ2, τ3)(cid:12)(cid:12)dτ1τ2τ3,

where the limiting distribution of λn,x,t(τ1, τ2, τ3) is as in (ii) of Section 6.4. Under any den-
sity kernel function, for t (cid:54)= t∗, we have limn→∞ hd+2
n E[|˜η2|4] =
∞.

n E[|˜η2|2] = ∞ and limn→∞ hd+4

Therefore, when t = t∗, due to continuity of g1 function, we have, for any ε > 0,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:113)

nhd+2

n

(cid:8)(z2(s, t) − E[z2(s, t)]) − (z2(s∗, t) − E[z2(s∗, t)])(cid:9)(cid:12)(cid:12)(cid:12)(cid:12) > ε

(cid:17) → 0,

(cid:16)

lim sup
n→∞ P

sup

s,s∗∈[0,S],
|s−s∗|<δ

as δ → 0. For the stochastic process {
(z2(s, t)−E[z2(s, t)]), s ∈ [0, S], t ∈ [0, T ]}, the
asymptotic equicontinuity is satisfied in the space of C([0, S], Rd). However, this condition
is no longer satisfied in the space of C([0, S] × [0, T ], Rd).

nhd+2

n

(cid:113)

For all pairs of points (s, s∗) ∈ [0, S], let us define ˜η3i:

(iii) Theorem 2.4.3.

(cid:90) b
(cid:90) b

a

(cid:90) S
w(cid:62)(t)
(cid:90) S

0

w(cid:62)(t)

a

0

˜η3i :=

+

g1(s, ξ, t)(D(Ui) + Γ(Ui))K

g1(s, ξ, t)(D(Ui) + Γ(Ui))K

x(ξ, t)dξdt

(cid:16)(x(ξ, t), t) − Ui
(cid:17)
(cid:16)(x(ξ, t), t) − Ui
(cid:17) ∂

hn

hn

∂t

(1)
t

(1)
x

dξdt

91

(cid:90) b
(cid:90) b

a

a

−

−

(cid:90) S
(cid:90) S

0

0

w(cid:62)(t)

w(cid:62)(t)

g1(s∗, ξ, t)(D(Ui) + Γ(Ui))K
g1(s∗, ξ, t)(D(Ui) + Γ(Ui))K

(1)
t

(1)
x

(cid:16)(x(ξ, t), t) − Ui
(cid:16)(x(ξ, t), t) − Ui

hn

(cid:17)
(cid:17) ∂

dξdt

hn

∂t

Note that ˜η3, ˜η3i’s are i.i.d. random vectors in Rd.

E(cid:104)(cid:12)(cid:12)(cid:12)(cid:12)(cid:113)

nhd
n

(cid:110)(cid:0)z3(s) − E[z3(s)](cid:1) −(cid:0)z3(s∗) − E[z3(s∗)](cid:1)(cid:111)(cid:12)(cid:12)(cid:12)(cid:12)4(cid:105)
(cid:110)n(n − 1)

(cid:16)E[|˜η3 − E[˜η3]|2]

(cid:17)2

+ nE[|˜η3 − E[˜η3]|4]

=

1 + op(1)
2(d+4)
n2h
n

2

x(ξ, t)dξdt.

(cid:111)

.

Here

E[|˜η3|2] ≤ chd+4

n

(cid:90) b

(cid:90) S

a

0

|g1(s, ξ, t) − g1(s∗, ξ, t)|2dξdt

(cid:90)

(cid:90)

R

R

(cid:12)(cid:12)λn,x,t(τ, γ)(cid:12)(cid:12)dτ dγ,

where the limiting distribution of λn,x,t(τ, γ) is defined as

lim
n→∞ Λn,x,t(τ, γ) :=

Rd+1

K

(1)
t

(ψ)K

(1)
t

∂
∂t

ψ + (γ

x(ξ, t) + τ v(D(x(ξ, t), t)), γ)

(cid:16)
(cid:16)

(cid:90)
(cid:90)
(cid:90)
×(cid:16) ∂

+ 2

+

Rd+1

K

Rd+1

x(ξ, t)

∂t

x(ξ, t)

∂
∂t

(1)
x (ψ)

(cid:17)(cid:62)(cid:16)

(cid:16)

K

(1)
x

K

(1)
t

(ψ)K

(1)
x

ψ + (τ v(D(x(ξ, t), t)) + γ

∂
∂t

x(ξ, t), γ)

x(ξ, t)dψ

ψ + (τ v(D(x(ξ, t), t)) + γ

(cid:17)(cid:17)(cid:62)

x(ξ, t), γ)

dψ

∂
∂t

(cid:17)
(cid:17) ∂

dψ

∂t

and

(cid:90) S
(cid:90) b
|g1(s, ξ, t) − g1(s∗, ξ, t)|4dξdt
(cid:90)
(cid:90)
(cid:90)
(cid:90)

a

0

(cid:12)(cid:12)λn,x,t(τ1, γ1, τ2, γ2, τ3, γ3)(cid:12)(cid:12)dτ1dγ1dτ2dγ2dτ3dγ3,

(cid:90)
E[|˜η3|4] ≤ chd+8

(cid:90)

n

×

R

R

R

R

R

R

92

where the limiting distribution of λn,x,t(τ1, γ1, τ2, γ2, τ3, γ3) is as in (iii) of Section 6.4. Due

to continuity of g1 function, we have, for any ε > 0,

(cid:16)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:113)

nhd
n

(cid:8)(cid:0)z3(s) − E[z3(s)](cid:1) −(cid:0)z3(s∗) − E[z3(s∗)](cid:1)(cid:9)(cid:12)(cid:12)(cid:12)(cid:12)4

(cid:17) → 0,

> ε

lim sup
n→∞ P

sup

s,s∗∈[0,S],
|s−s∗|<δ

as δ → 0. The stochastic process {(cid:113)

nhd

n(z3(s)−E[z3(s)]), s ∈ [0, S]} satisfies the asymptotic

equicontinuity in the space of C([0, S], Rd).

6.6 Propositions

To complete the proof of Theorem 2.4.1, we need the following proposition:

Proposition 6.6.1. The sequence of remaining processes {δ1(s, t), s ∈ [0, S], t ∈ [0, T ]} in

the proof of Theorem 2.4.1 satisfies

sup

s∈[0,S],t∈[0,T ]

|δ1(s, t)| = op

(cid:17)

.

(cid:16)

1(cid:113)

nhd
n

Proof of Proposition 6.6.1.

(cid:90) s

0

δ1(s, t) =

∂
∂D

v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)δ1(ξ, t)dξ + r1(s, t).

By the Gronwall-Bellman inequality, we have

|δ1(s, t)| ≤

sup

s∈[0,S],t∈[0,T ]

|r1(s, t)|

sup

s∈[0,S],t∈[0,T ]

exp

(cid:20)(cid:90) s

0

(cid:12)(cid:12)(cid:12)(cid:12) ∂

∂D

(cid:12)(cid:12)(cid:12)(cid:12)dξ

D(x(ξ, t), t)

(cid:21)

v(D(x(ξ, t), t))

∂
∂x

≤ c

sup

s∈[0,S],t∈[0,T ]

|r1(s, t)|,

since the exponent is bounded.

93

r1(s, t) =

dξ

dξ

∂
∂x

x=x(ξ,t)

0

=

+

0

0

0

0

0

0

∂
∂D

−

+

−

∂
∂D

dξ −

(cid:90) s

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

D(x(ξ, t), t)y1(ξ, t)dξ

(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111)

(cid:110)
(cid:111)
v((cid:98)Dn((cid:98)Xn(ξ, t), t)) − v((cid:98)Dn(x(ξ, t), t))
(cid:110)
(cid:111)
v((cid:98)Dn(x(ξ, t), t)) − v(D(x(ξ, t), t))

(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:12)(cid:12)(cid:12)(cid:12)x=(cid:98)Xn(ξ,t)
(cid:90) s
(cid:90) s
v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t)) − ∂
×(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111)
(cid:90) s
v((cid:98)Dn(λ(cid:98)Xn(ξ, t) + (1 − λ)x(ξ, t), t))
(cid:90) s
v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t)) − ∂
×(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111)
(cid:16)
dξ
|y1(s, t)| + sup
u∈Gδ

(cid:12)(cid:12)(cid:12)2(cid:17)
(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)

s∈[0,S],t∈[0,T ]

0

∂D

v((cid:98)Dn(x, t))
(cid:90) 1
(cid:110) ∂
(cid:90) 1
(cid:110) ∂
(cid:90) 1
(cid:110) ∂

∂D

0

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

− ∂
∂D

dλy1(ξ, t)dξ

D(x(ξ, t), t)

∂
∂D

dξ.

∂
∂x

0

0

∂D

=

0

= O

sup

∂
∂x

dξ

(cid:111)

+

D(x(ξ, t), t)y1(ξ, t)dξ

v(D(x(ξ, t), t))

dλ

∂D

(cid:111)

(cid:98)Dn(λ(cid:98)Xn(ξ, t) + (1 − λ)x(ξ, t), t)

∂
∂x

(cid:111)

v(D(x(ξ, t), t))

dλ

∂D

From Lemma 2.3.1. and Lemma 2.3.3., we simply have

sup

s∈[0,S],t∈[0,T ]

|r1(s, t)| = op

This implies that

sup

s∈[0,S],t∈[0,T ]

|δ1(s, t)| = op

(cid:16)

sup

s∈[0,S],t∈[0,T ]

|y1(s, t)|(cid:17)

.

(cid:16)

sup

s∈[0,S],t∈[0,T ]

|y1(s, t)|(cid:17)

.

Due to the equation y1(s, t) = z1(s, t) + δ1(s, t), the proof is complete.

94

The following proposition is required for Theorem 2.4.2:

Proposition 6.6.2. The sequence of remaining processes {δ2(s, t), s ∈ [0, S], t ∈ [0, T ]} in

the proof of Theorem 2.4.2 satisfies

sup

s∈[0,S],t∈[0,T ]

|δ2(s, t)| = op

(cid:16)

1(cid:113)

nhd+2

n

(cid:17)

.

Proof of Proposition 6.6.2.

0

(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s

0

0

0

0

δ2(s, t) =

+

+

+

+

∂2
∂D2 v(D(x(ξ, t), t))
∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D
∂
∂D
∂
∂D

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

D(x(ξ, t), t)

D(x(ξ, t), t)

∂
∂t
∂
∂x

∂
∂x
∂
∂x
∂2
∂x∂t
∂2
∂x2 D(x(ξ, t), t)δ1(ξ, t)
∂
∂x

D(x(ξ, t), t)δ1(ξ, t)dξ

D(x(ξ, t), t)δ2(ξ, t)dξ + r2(s, t).

D(x(ξ, t), t)δ1(ξ, t)dξ

D(x(ξ, t), t)δ1(ξ, t)

∂
∂t

x(ξ, t)dξ

∂
∂t

x(ξ, t)dξ

Gronwall-Bellman inequality with bounded exponent yields

sup

s∈[0,S],t∈[0,T ]

sup

s∈[0,S],t∈[0,T ]

|r2(s, t)| + c2

|δ1(s, t)|.
(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)dξ
(cid:111) ∂
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
v((cid:98)Dn(x(ξ, t), t))
(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂

v(D(x(ξ, t), t))

∂D

∂t

∂t

∂
∂t

∂D

v((cid:98)Dn(x(ξ, t), t)) − ∂

∂D
∂2
∂D2 v(D(x(ξ, t), t))

v((cid:98)Dn((cid:98)Xn(ξ, t), t))

D(x(ξ, t), t)dξ

0

|δ2(s, t)| ≤ c1
(cid:90) s
(cid:110) ∂
(cid:90) s
(cid:90) s
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t)

∂D

0

0

× ∂
∂t

(cid:111)

dξ

r2(s, t) =

−

+

95

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)y1(ξ, t)dξ

x(ξ, t)dξ

∂
∂t

x(ξ, t)dξ

−

−

+

−

+

0

0

0

(cid:110) ∂

(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t)
(cid:90) s
(cid:90) s
(cid:90) s

∂D

0

0

0

0

× ∂
∂x
−

−

(cid:110) ∂
∂D
× y2(ξ, t)dξ

+

0

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

v(D(x(ξ, t), t))

∂
∂x
∂2
∂x∂t

∂D

∂D
∂2
∂D2 v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

D(x(ξ, t), t)y1(ξ, t)dξ

v((cid:98)Dn(x(ξ, t), t)) − ∂

(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
v((cid:98)Dn(x(ξ, t), t))
v((cid:98)Dn((cid:98)Xn(ξ, t), t))
(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂
(cid:98)Xn(ξ, t)dξ

(cid:111) ∂

∂
∂x

∂
∂t

∂D

∂x

∂x

D(x(ξ, t), t)

∂t

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

v(D(x(ξ, t), t))

v((cid:98)Dn(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)

∂
∂x
∂2
∂x2 D(x(ξ, t), t)y1(ξ, t)

(cid:98)Dn(x(ξ, t), t) − ∂

∂
∂t

∂
∂x

D(x(ξ, t), t)y1(ξ, t)

∂
∂t

x(ξ, t)dξ

x(ξ, t)dξ

v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)

(cid:111)

∂D

:= (a) + (b) + (c) + (d) + (e), where

(a) =

0

0

=

∂t

∂2

−

∂D

v(D(x(ξ, t), t))

∂D
∂2
∂D2 v(D(x(ξ, t), t))

(cid:90) s
(cid:110) ∂
(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)dξ
v((cid:98)Dn(x(ξ, t), t)) − ∂
(cid:90) s
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:90) 1
(cid:90) s
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
∂D2 v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t))dλ
×(cid:110) ∂
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:90) 1
(cid:90) s
(cid:110) ∂2
∂D2 v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t)) − ∂2
×(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:12)(cid:12)(cid:12)(cid:12)(cid:17)
(cid:12)(cid:12)(cid:12)(cid:12) ∂
(cid:12)(cid:12)(cid:12) × sup
(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:16)
(cid:16)

(cid:111)
(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:12)(cid:12)(cid:12)2(cid:17)

(cid:98)Dn(u) − ∂

∂D2 v(D(x(ξ, t), t))

D(x(ξ, t), t)dξ

D(x(ξ, t), t)dξ

D(x(ξ, t), t)

(cid:111)

D(u)

= O

+ O

dλ

dξ

∂t

∂t

∂t

∂t

∂t

∂t

+

0

0

0

0

,

(cid:111)

sup
u∈Gδ

sup
u∈Gδ

u∈Gδ

96

(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂

∂D

v((cid:98)Dn(x(ξ, t), t))

∂
∂t

(cid:111)
(cid:98)Dn(x(ξ, t), t)

dξ

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)y1(ξ, t)dξ

D(x(ξ, t), t)y1(ξ, t)dξ

(cid:12)(cid:12)(cid:12)(cid:12)x=(cid:98)Xn(ξ,t)

x=x(ξ,t)

dξ

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)y1(ξ, t)dξ

(b) =

0

0

0

0

=

−

−

∂
∂t

∂
∂t

v(D(x(ξ, t), t))

(cid:110) ∂

∂
∂x
∂2
∂x∂t

(cid:98)Dn(x, t)

v((cid:98)Dn(x, t))

v((cid:98)Dn((cid:98)Xn(ξ, t), t))

∂D
∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) 1
(cid:98)Dn(λ(cid:98)Xn(ξ, t) + (1 − λ)x(ξ, t), t)
× ∂
∂t
(cid:90) 1
(cid:90) s
− ∂2

∂
∂D
∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

∂D2 v(D(x(ξ, t), t))

∂
∂x
∂2
∂x∂t

v(D(x(ξ, t), t))

D(x(ξ, t), t)

∂
∂x

−

−

=

0

0

0

0

+

0

0

D(x(ξ, t), t)y1(ξ, t)dξ

(cid:110) ∂2
∂D2 v((cid:98)Dn(λ(cid:98)Xn(ξ, t) + (1 − λ)x(ξ, t), t))
(cid:111)

∂
∂x

(cid:98)Dn(λ(cid:98)Xn(ξ, t) + (1 − λ)x(ξ, t), t)

∂t

∂t

∂D

sup

sup

D(u)

∂
∂t

sup
u∈Gδ

(cid:111)

sup
u∈Gδ

D(x(ξ, t), t)

D(x(ξ, t), t)

∂2
∂x∂t

dλy1(ξ, t)dξ

v(D(x(ξ, t), t))

(cid:12)(cid:12)(cid:12) ×

s∈[0,S],t∈[0,T ]

s∈[0,S],t∈[0,T ]

(cid:98)Dn(λ(cid:98)Xn(ξ, t) + (1 − λ)x(ξ, t), t)

dλy1(ξ, t)dξ
∂2
∂x∂t

(cid:110) ∂
v((cid:98)Dn(λ(cid:98)Xn(ξ, t) + (1 − λ)x(ξ, t), t))
(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
|y1(s, t)|(cid:17)
(cid:12)(cid:12)(cid:12)(cid:12) ∂
(cid:12)(cid:12)(cid:12)(cid:12) ×
|y1(s, t)|(cid:17)
(cid:98)Dn(u) − ∂
(cid:111) ∂
(cid:110) ∂
(cid:98)Dn(x(ξ, t), t)
v((cid:98)Dn(x(ξ, t), t)) − ∂
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:90) 1
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
∂D2 v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t))dλ
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:90) 1
(cid:110) ∂2
∂D2 v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t)) − ∂2

∂D
∂2
∂D2 v(D(x(ξ, t), t))

∂D2 v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

(cid:111) ∂

D(x(ξ, t), t)

D(x(ξ, t), t)

x(ξ, t)dξ

x(ξ, t)dξ

x(ξ, t)dξ

(cid:111)

(cid:111)

∂
∂t

∂
∂t

∂D

∂2

∂x

∂x

∂x

dλ

∂t

0

,

+ O

= O

− ∂
∂D

(cid:16)
(cid:16)
(cid:90) s
(cid:90) s
(cid:90) s
×(cid:110) ∂
(cid:90) s

−

=

0

0

0

+

∂x

0

0

(c) =

97

∂
∂t

x(ξ, t)dξ

v((cid:98)Dn(x(ξ, t), t))

∂D

(d) =

∂t

= O

∂
∂x

∂
∂x

∂x

,

∂
∂D

0

∂D

−

=

0

0

0

0

0

0

sup
u∈Gδ

y2(ξ, t)dξ

∂
∂x

∂
∂x

D(x(ξ, t), t)

× ∂
∂x
−

(cid:98)Dn(x, t)
(cid:98)Dn(x, t)

(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂

(cid:12)(cid:12)(cid:12)(cid:12)x=(cid:98)Xn(ξ,t)
(cid:12)(cid:12)(cid:12)(cid:12)x=(cid:98)Xn(ξ,t)

x=x(ξ,t)

v((cid:98)Dn(x, t)
v((cid:98)Dn(x, t)

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

v(D(x(ξ, t), t))

D(x(ξ, t), t)

∂
∂x
∂2
∂x2 D(x(ξ, t), t)y1(ξ, t)

×(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:12)(cid:12)(cid:12)2(cid:17)
(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:16)
(cid:90) s
(cid:110) ∂
v((cid:98)Dn((cid:98)Xn(ξ, t), t))
(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)
(cid:98)Xn(ξ, t)dξ
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) 1
∂D2 v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t))
(cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)dλy1(ξ, t)y2(ξ, t)dξ
(cid:90) s
(cid:90) 1
(cid:110) ∂2
∂D2 v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t))
(cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)
× ∂
∂x
(cid:90) 1
(cid:90) s
− ∂2
(cid:90) 1
(cid:90) s
+ (1 − λ)x(ξ, t), t)dλy1(ξ, t)y2(ξ, t)dξ

v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t))
v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t))

∂
∂D
∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

∂
∂x
∂2
∂x2 D(x(ξ, t), t)y1(ξ, t)

∂D2 v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

× ∂
∂x

D(x(ξ, t), t)

D(x(ξ, t), t)

+

−

−

D(x(ξ, t), t)

∂
∂t

∂
∂x

+

+

=

+

x(ξ, t)dξ

∂
∂x

∂2

0

0

∂
∂x

∂
∂D

∂
∂D

0

0

0

0

x=x(ξ,t)

0

0

98

∂
∂t

x(ξ, t)dξ

∂
∂t

x(ξ, t)dξ

D(x(ξ, t), t)y1(ξ, t)

∂
∂t

x(ξ, t)dξ

D(x(ξ, t), t)y1(ξ, t)

∂
∂t

x(ξ, t)dξ

(cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)
(cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)

∂
∂x

∂
∂x

∂2

∂
∂t

x(ξ, t)dξ

dλy1(ξ, t)

(cid:111)
∂x2(cid:98)Dn(λXn(ξ, t)
(cid:110) ∂2
∂x2(cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)

(cid:110) ∂2
∂x2(cid:98)Dn(x(ξ, t), t)

(cid:111)

v(D(x(ξ, t), t))

0

0

∂
∂t

∂
∂D

x(ξ, t)dξ

dλy1(ξ, t)

∂x2 D(x(ξ, t), t)

(cid:111)
∂x2(cid:98)Dn(x(ξ, t), t)
(cid:90) s
(cid:90) 1
− ∂2
v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t))dλ
(cid:111)
+
(cid:90) 1
(cid:90) s
− ∂2
(cid:110) ∂
v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)) − ∂
|y2(s, t)|(cid:17)
|y1(s, t)|(cid:17)

∂
∂x2 D(x(ξ, t), t)dλy1(ξ, t)
∂t
|y1(s, t)| ×

s∈[0,S],t∈[0,T ]

s∈[0,S],t∈[0,T ]

x(ξ, t)dξ

x(ξ, t)dξ

y1(ξ, t)

∂
∂t

sup

sup

∂D

0

∂D

s∈[0,S],t∈[0,T ]

sup

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:12)(cid:12)(cid:12) ×
(cid:12)(cid:12)(cid:12)(cid:12) ∂
(cid:98)Dn(u) − ∂

∂x

∂x

sup
u∈Gδ

sup
u∈Gδ

0

= O

+
× ∂2
(cid:16)
(cid:16)
(cid:16)

+ O

+ O

D(u)

sup

s∈[0,S],t∈[0,T ]

(cid:12)(cid:12)(cid:12)(cid:12) ×

|y1(s, t)|(cid:17)

,

(cid:111)

∂
∂x

D(x(ξ, t), t)

v(D(x(ξ, t), t))

(cid:111)

y2(ξ, t)dξ

D(x(ξ, t), t)

D(x(ξ, t), t)y2(ξ, t)dξ

(cid:111)

D(x(ξ, t), t)

y2(ξ, t)dξ

Dn(x(ξ, t), t) − D(x(ξ, t), t)

(cid:111)

and

(e) =

0

(cid:90) s
(cid:110) ∂
∂D
(cid:90) s
× y2(ξ, t)dξ
(cid:90) s
(cid:110) ∂
(cid:90) s
(cid:90) 1
(cid:90) s

∂D
∂
∂D

∂
∂D

+

=

=

0

0

0

+

0

0

× ∂
∂x

(cid:16)

= O

sup
u∈Gδ

∂x

∂D

∂
∂x

(cid:98)Dn(x(ξ, t), t) − ∂
(cid:98)Dn(x(ξ, t), t) − ∂

v((cid:98)Dn(x(ξ, t), t))
(cid:110) ∂
v((cid:98)Dn(x(ξ, t), t))
v((cid:98)Dn(x(ξ, t), t)) − ∂
(cid:110) ∂
v((cid:98)Dn(x(ξ, t), t))
(cid:110)
∂D2 v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t))dλ
(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:12)(cid:12)(cid:12) ×
|y2(s, t)|(cid:17)

(cid:98)Dn(x(ξ, t), t) − ∂

v(D(x(ξ, t), t))

(cid:111) ∂

sup

∂D

∂2

∂x

∂x

∂x

∂x

.

s∈[0,S],t∈[0,T ]

D(x(ξ, t), t)y2(ξ, t)dξ

99

From Lemma 2.3.1. to Lemma 2.3.4., we can easily derive

sup

s∈[0,S],t∈[0,T ]

|r2(s, t)| = op

sup

s∈[0,S],t∈[0,T ]

(cid:16)

|y2(s, t)|(cid:17)

.

Along with Proposition 6.6.1., this also implies

sup

s∈[0,S],t∈[0,T ]

|δ2(s, t)| = op

(cid:16)

|y2(s, t)|(cid:17)

.

sup

s∈[0,S],t∈[0,T ]

The proof is complete using the fact that y2(s, t) = z2(s, t) + δ2(s, t).

Lastly, Proposition 6.6.3. completes the proof of Theorem 2.4.3..

Proposition 6.6.3. The remainder {δ3(s), s ∈ [0, S]} in the proof of Theorem 2.4.3 satisfies

|δ3(s)| = op

sup
s∈[0,S]

(cid:17)

.

(cid:16)

1(cid:113)

nhd
n

Proof of Proposition 6.6.3.

δ3(s) =

−

+

0

a

a

0

0

a

(cid:110) ∂

w(cid:62)(t)

w(cid:62)(t)

(cid:90) s
(cid:90) b
(cid:90) s
(cid:90) b
(cid:90) s
(cid:90) b
(cid:110) ∂
w(cid:62)(t)
(cid:111)
(cid:98)Dn(x(ξ, t), t)
(cid:90) b
(cid:90) s
w(cid:62)(t)
(cid:90) s
(cid:90) b
(cid:90) s
(cid:90) b

w(cid:62)(t)

(cid:110) ∂

a

a

0

0

w(cid:62)(t)

a

0

∂D

× ∂
∂t

−

−

+

v((cid:98)Dn(x(ξ, t), t)) − ∂

∂D
∂2
∂D2 v(D(x(ξ, t), t))

v((cid:98)Dn((cid:98)Xn(ξ, t), t))

∂D

∂
∂t

∂D

v(D(x(ξ, t), t))

(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)dξdt
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂
v((cid:98)Dn(x(ξ, t), t))

∂t

∂t

∂D

D(x(ξ, t), t)dξdt

dξdt

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

v(D(x(ξ, t), t))

∂
∂x
∂2
∂x∂t

v((cid:98)Dn(x(ξ, t), t)) − ∂

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)y1(ξ, t)dξdt

D(x(ξ, t), t)y1(ξ, t)dξdt

v(D(x(ξ, t), t))

∂D

(cid:111) ∂

∂x

(cid:98)Dn(x(ξ, t), t)

100

(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂

∂x

∂
∂x

∂D

D(x(ξ, t), t)

v((cid:98)Dn(x(ξ, t), t))

× ∂
∂t

−

× ∂
∂t

× ∂
∂x

0

0

a

a

∂D

x(ξ, t)dξdt
w(cid:62)(t)

∂2
∂D2 v(D(x(ξ, t), t))

(cid:90) s
(cid:90) b
(cid:90) b
(cid:90) s
(cid:110) ∂
x(ξ, t)dξdt
v((cid:98)Dn((cid:98)Xn(ξ, t), t))
w(cid:62)(t)
(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)
(cid:98)Xn(ξ, t)dξdt
(cid:90) s
(cid:90) b
w(cid:62)(t)
(cid:90) b
(cid:90) s
(cid:90) s
(cid:90) b
(cid:90) s
(cid:90) b

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D
∂
∂D
∂
∂D

v((cid:98)Dn(x(ξ, t), t))

v(D(x(ξ, t), t))

v(D(x(ξ, t), t))

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

a

0

∂t

a

a

a

0

0

0

+

−

−

+

−

:= (a) + (b) + (c) + (d) + (e), where

D(x(ξ, t), t)y1(ξ, t)

∂
∂t

x(ξ, t)dξdt

x(ξ, t)dξdt

∂
∂x

D(x(ξ, t), t)

∂
∂x
∂2
∂x2 D(x(ξ, t), t)y1(ξ, t)
∂
∂x
∂
∂x

(cid:98)Dn(x(ξ, t), t)y2(ξ, t)dξdt

D(x(ξ, t), t)y2(ξ, t)dξdt

∂
∂t

(a) =

a

a

a

=

0

0

∂t

∂2

−

∂D

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

v(D(x(ξ, t), t))

∂D
∂2
∂D2 v(D(x(ξ, t), t))

(cid:90) s
(cid:90) b
(cid:110) ∂
(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)dξdt
v((cid:98)Dn(x(ξ, t), t)) − ∂
(cid:90) s
(cid:90) b
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:90) b
(cid:90) s
(cid:90) 1
∂D2 v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t))dλ
×(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111)(cid:110) ∂
(cid:111)
(cid:98)Dn(x(ξ, t), t) − ∂
(cid:90) b
(cid:90) s
(cid:90) 1
(cid:110) ∂2
∂D2 v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t)) − ∂2
(cid:111) ∂
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:12)(cid:12)(cid:12)(cid:12) ∂
(cid:12)(cid:12)(cid:12) × sup
(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:16)

(cid:98)Dn(u) − ∂

D(x(ξ, t), t)dξdt,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:17)

D(x(ξ, t), t)

w(cid:62)(t)

× dλ

(cid:16)

D(u)

= O

+ O

dξ

∂t

∂t

∂t

∂t

∂t

0

0

0

0

∂t

+

a

∂D2 v(D(x(ξ, t), t))

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:12)(cid:12)(cid:12)2(cid:17)

,

D(x(ξ, t), t)dξdt

sup
u∈Gδ

sup
u∈Gδ

u∈Gδ

(cid:111)

101

v((cid:98)Dn((cid:98)Xn(ξ, t), t))

(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂

∂D

∂
∂t

v((cid:98)Dn(x(ξ, t), t))

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)y1(ξ, t)dξdt

D(x(ξ, t), t)y1(ξ, t)dξ

(cid:12)(cid:12)(cid:12)(cid:12)x=(cid:98)Xn(ξ,t)

x=x(ξ,t)

dξdt

D(x(ξ, t), t)

∂
∂t

D(x(ξ, t), t)y1(ξ, t)dξdt

∂
∂t

x(ξ, t)dξdt

D(x(ξ, t), t)

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

v(D(x(ξ, t), t))

v((cid:98)Dn(x, t))

∂
∂t

∂
∂D
∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

v(D(x(ξ, t), t))

∂
∂x
∂2
∂x∂t

(cid:98)Dn(x, t)

∂
∂x
∂2
∂x∂t

∂D

dξdt

(b) =

× ∂
∂t

0

0

a

a

(cid:90) b
(cid:90) s
(cid:110) ∂
w(cid:62)(t)
(cid:111)
(cid:98)Dn(x(ξ, t), t)
(cid:90) s
(cid:90) b
w(cid:62)(t)
(cid:90) b
(cid:90) s
(cid:90) s
(cid:90) b
(cid:90) s
(cid:90) b
(cid:90) b
(cid:90) s
(cid:90) s
(cid:90) b

(cid:90) 1

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

a

a

a

a

0

0

0

0

w(cid:62)(t)

a

−

−

=

−

−

=

∂
∂t

+ (1 − λ)x(ξ, t), t)
(cid:90) b
− ∂2

∂D2 v(D(x(ξ, t), t))

w(cid:62)(t)

+

0

a

∂D
+ (1 − λ)x(ξ, t), t) − ∂
∂D

0

0

0

∂
∂t

∂
∂x

(cid:98)Dn(λ(cid:98)Xn(ξ, t)

(cid:98)Dn(λ(cid:98)Xn(ξ, t)

∂
∂x

D(x(ξ, t), t)

D(x(ξ, t), t)

D(x(ξ, t), t)

∂2
∂x∂t

∂2
∂x∂t

v(D(x(ξ, t), t))

dλy1(ξ, t)dξdt

D(x(ξ, t), t)y1(ξ, t)dξ

(cid:110) ∂2
∂D2 v((cid:98)Dn(λ(cid:98)Xn(ξ, t) + (1 − λ)x(ξ, t), t))
(cid:98)Dn(λ(cid:98)Xn(ξ, t) + (1 − λ)x(ξ, t), t)
(cid:111)
(cid:90) s
(cid:90) 1
(cid:110) ∂
v((cid:98)Dn(λ(cid:98)Xn(ξ, t) + (1 − λ)x(ξ, t), t))
(cid:111)
(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:12)(cid:12)(cid:12) ×
|y1(s, t)|(cid:17)
(cid:12)(cid:12)(cid:12)(cid:12) ∂
|y1(s, t)|(cid:17)
(cid:98)Dn(u) − ∂
(cid:90) s
(cid:110) ∂
(cid:90) s
(cid:90) s

(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)
(cid:111) ∂
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
∂D2 v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t))dλ

v((cid:98)Dn(x(ξ, t), t)) − ∂

∂D
∂2
∂D2 v(D(x(ξ, t), t))

s∈[0,S],t∈[0,T ]

s∈[0,S],t∈[0,T ]

(cid:12)(cid:12)(cid:12)(cid:12) ×

dλy1(ξ, t)dξdt

v(D(x(ξ, t), t))

(cid:90) 1

D(u)

sup

sup

∂D

∂2

∂x

∂x

∂t

∂t

0

0

,

= O

+ O

sup
u∈Gδ

sup
u∈Gδ

(c) =

−

w(cid:62)(t)

w(cid:62)(t)

(cid:16)
(cid:16)
(cid:90) b
(cid:90) b
(cid:90) b

a

a

× ∂
∂t

=

x(ξ, t)dξdt
w(cid:62)(t)

a

0

0

102

x(ξ, t)dξdt

∂t

∂D2 v(D(x(ξ, t), t))

(cid:111)

v((cid:98)Dn(x(ξ, t), t))

∂D

∂
∂t

x(ξ, t)dξdt

∂
∂t

x(ξ, t)dξdt

(cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)dλy1(ξ, t)

(d) =

a

∂t

0

0

∂
∂t

∂
∂x

∂x

,

∂x

∂x

0

∂D

= O

sup
u∈Gδ

w(cid:62)(t)

−

−

=

+

a

× dλ

w(cid:62)(t)

D(x(ξ, t), t)

D(x(ξ, t), t)

D(x(ξ, t), t)

x(ξ, t)dξdt

× ∂
∂x

(cid:98)Dn(x(ξ, t), t) − ∂

(cid:98)Dn((cid:98)Xn(ξ, t), t) − ∂

∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

v(D(x(ξ, t), t))

×(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:111)(cid:110) ∂
(cid:90) 1
(cid:90) s
(cid:90) b
(cid:110) ∂2
∂D2 v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t)) − ∂2
(cid:110)(cid:98)Dn(x(ξ, t), t) − D(x(ξ, t), t)
(cid:111) ∂
(cid:12)(cid:12)(cid:12)2(cid:17)
(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:16)
(cid:90) s
(cid:90) b
(cid:110) ∂
v((cid:98)Dn((cid:98)Xn(ξ, t), t))
w(cid:62)(t)
(cid:111) ∂
(cid:98)Dn(x(ξ, t), t)
(cid:98)Xn(ξ, t)dξdt
(cid:90) b
(cid:90) s
w(cid:62)(t)
(cid:90) s
(cid:90) b
(cid:90) s
(cid:90) b
(cid:90) b
(cid:90) s
(cid:90) s
(cid:90) b
(cid:90) s
(cid:90) b
(cid:90) s
(cid:90) b
w(cid:62)(t)
(cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)
× ∂
∂x
(cid:90) b
(cid:90) s
× y2(ξ, t)dξdt
w(cid:62)(t)
(cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)

(cid:110) ∂2
∂D2 v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t))

∂D2 v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t))

∂
∂D
∂2
∂D2 v(D(x(ξ, t), t))
∂
∂D

v(D(x(ξ, t), t))

∂
∂x
∂2
∂x2 D(x(ξ, t), t)y1(ξ, t)

∂
∂x

D(x(ξ, t), t)

∂
∂x
∂2
∂x2 D(x(ξ, t), t)y1(ξ, t)

v((cid:98)Dn(x, t)
v((cid:98)Dn(x, t)

(cid:12)(cid:12)(cid:12)(cid:12)x=(cid:98)Xn(ξ,t)
(cid:12)(cid:12)(cid:12)(cid:12)x=(cid:98)Xn(ξ,t)

x=x(ξ,t)

x=x(ξ,t)

(cid:98)Dn(x, t)
(cid:98)Dn(x, t)

(cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)

D(x(ξ, t), t)y1(ξ, t)

D(x(ξ, t), t)y1(ξ, t)

y2(ξ, t)dξdt

(cid:90) 1

(cid:90) 1

+

−

−

w(cid:62)(t)

w(cid:62)(t)

a

a

a

a

a

a

a

w(cid:62)(t)

∂
∂x

∂
∂x

0

0

0

0

0

0

w(cid:62)(t)

x(ξ, t)dξdt

x(ξ, t)dξdt

x(ξ, t)dξdt

∂
∂D

∂2

0

0

∂
∂t

∂
∂t

∂
∂x

∂
∂t

0

0

=

+

a

× ∂
∂x
− ∂2

∂D2 v(D(x(ξ, t), t))

∂
∂x

D(x(ξ, t), t)

D(x(ξ, t), t)

dλy1(ξ, t)

∂
∂t

x(ξ, t)dξdt

∂
∂x

∂
∂x
∂
∂x

103

(cid:111)

0

0

a

+

∂2

∂
∂D

w(cid:62)(t)

w(cid:62)(t)

(cid:90) b
v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t))
(cid:90) b
+ (1 − λ)x(ξ, t), t)dλy1(ξ, t)y2(ξ, t)dξdt
v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t))
∂
(cid:111)
+
∂x2(cid:98)Dn(x(ξ, t), t)
∂D
(cid:90) b
+ (1 − λ)x(ξ, t), t) − ∂2
v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t))dλ
+
(cid:90) b
− ∂2
(cid:110) ∂
v((cid:98)Dn(λXn(ξ, t) + (1 − λ)x(ξ, t), t)) − ∂

(cid:90) 1
(cid:90) 1
(cid:90) 1
(cid:111)
(cid:90) 1

(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s

∂x2 D(x(ξ, t), t)

∂x2(cid:98)Dn(λXn(ξ, t)
(cid:110) ∂2
∂x2(cid:98)Dn(λXn(ξ, t)
(cid:110) ∂2
∂x2(cid:98)Dn(x(ξ, t), t)

w(cid:62)(t)

w(cid:62)(t)

dλy1(ξ, t)

x(ξ, t)dξdt

x(ξ, t)dξdt

y1(ξ, t)

∂
∂D

∂
∂t

∂
∂t

(cid:111)

v(D(x(ξ, t), t))

a

a

0

0

0

0

∂D

0

0

∂D

a

= O

+
× ∂2
(cid:16)
(cid:16)
(cid:16)

+ O

+ O

∂x2 D(x(ξ, t), t)dλy1(ξ, t)

x(ξ, t)dξdt

∂
∂t

|y3(s)|(cid:17)

s∈[0,S],t∈[0,T ]

|y1(s, t)| × sup
s∈[0,S]

sup

(cid:12)(cid:12)(cid:12) ×
(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)
(cid:12)(cid:12)(cid:12)(cid:12) ∂
(cid:98)Dn(u) − ∂

∂x

∂x

sup
u∈Gδ

sup
u∈Gδ

sup

s∈[0,S],t∈[0,T ]

(cid:12)(cid:12)(cid:12)(cid:12) ×

D(u)

sup

s∈[0,S],t∈[0,T ]

|y1(s, t)|(cid:17)

|y1(s, t)|(cid:17)

,

and

(e) =

−

=

+

=

+

a

a

(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b
(cid:90) b

a

a

a

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

w(cid:62)(t)

0

0

(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s
(cid:90) s

0

0

0

a

0

0

∂
∂D
∂
∂D
∂
∂D

(cid:110) ∂
(cid:90) 1

∂D
∂
∂D

v(D(x(ξ, t), t))

∂
∂x
∂
∂x

D(x(ξ, t), t)y2(ξ, t)dξdt

(cid:98)Dn(x(ξ, t), t) − ∂

v((cid:98)Dn(x(ξ, t), t))
(cid:98)Dn(x(ξ, t), t)y2(ξ, t)dξdt
(cid:110) ∂
v((cid:98)Dn(x(ξ, t), t))
v((cid:98)Dn(x(ξ, t), t)) − ∂
(cid:110) ∂
v((cid:98)Dn(x(ξ, t), t))
∂D2 v(λ(cid:98)Dn(x(ξ, t), t) + (1 − λ)D(x(ξ, t), t))dλ

(cid:98)Dn(x(ξ, t), t) − ∂

v(D(x(ξ, t), t))

(cid:111) ∂

∂D

∂2

∂x

∂x

∂x

∂x

∂x

(cid:111)

(cid:111)

D(x(ξ, t), t)

y2(ξ, t)dξdt

D(x(ξ, t), t)y2(ξ, t)dξdt

D(x(ξ, t), t)

y2(ξ, t)dξdt

104

×(cid:110)

= O

(cid:16)

Dn(x(ξ, t), t) − D(x(ξ, t), t)

(cid:12)(cid:12)(cid:12)(cid:98)Dn(u) − D(u)

sup
u∈Gδ

(cid:111) ∂
(cid:12)(cid:12)(cid:12) × sup

s∈[0,S]

∂x

|y3(s)|(cid:17)

.

D(x(ξ, t), t)y2(ξ, t)dξdt

Then from Lemma 2.3.1. to Lemma 2.3.4. along with Proposition 6.6.1., we have

(cid:16)

|y3(s)|(cid:17)

.

|δ3(s)| = op

sup
s∈[0,S]

sup
s∈[0,S]

105

BIBLIOGRAPHY

106

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