SAMPLE PATH AND ASYMPTOTIC PROPERTIES OF SPACE-TIME MODELS
By
Yun Xue

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Statistics
2011

ABSTRACT
SAMPLE PATH AND ASYMPTOTIC PROPERTIES OF SPACE-TIME
MODELS
By
Yun Xue

Spatio-temporal models are widely used for inference in statistics and many applied areas. In
such contexts interests are often in the geometric nature (e.g. anisotropy), and the statistical
properties of these models. This dissertation has two parts. The first part focuses on the
sample path properties of space-time models. We apply the theory of Yaglom (1957) to construct a large class of space-time models with stationary increments (also called intrinsically
stationary random fields) and study their statistical and geometric properties. We derive
upper and lower bounds for the prediction errors, establish criteria for the mean-square
and sample path differentiability, all in terms of the parameters of the models explicitly.
Moreover, it is shown that when the random fields are not smooth, we can generate various
kinds of random fractals and the related Hausdorff dimensions are computed. Our main
results show that the statistical and geometric properties of the Gaussian random fields we
propose are very different from those obtained by deformation from any isotropic random
field; and they can be applied to analyze more general Gaussian intrinsic random functions,
convolution-based space-time Gaussian models [Higdon (2002), Calder and Cressie (2007)]
and the spatial processes in Fuentes (2002, 2005).
The second part of the dissertation pertains to equivalence of Gaussian measures and
asymptotically optimal predictions of intrinsically stationary random fields. We extend the
methods which Ibragimov and Rozanov (1978) use for stationary processes to study intrinsically stationary random fields. We describe the relationships among three corresponding

Hilbert spaces: the random variable space generated by the random field, the corresponding reproducing kernel Hilbert space, and the complex function space spanned by certain
analytic functions using the spectral measure. Criteria for equivalence and orthogonality
of intrinsically stationary Gaussian random fields are delivered in terms of their spectral
measures and the structures of their reproducing kernel Hilbert spaces. Our results are
different from those for stationary processes [see Ibragimov and Rozanov (1978)]. Given
the equivalence of two Gaussian measures, the asymptotic optimality of linear predictions
of intrinsically stationary random fields and the convergence rates are established in this
part. Moreover, the asymptotic efficient prediction of non-stationary, anisotropic space-time
models with a misspecified probability distribution is studied. The main results show that
under the equivalence of two Gaussian measures, the prediction based on the incorrect distribution is asymptotically optimal and efficient relative to the prediction under the correct
distribution, as the points of observations become increasingly dense in the study domain.
Our results extend those of Stein (1988, 1990, 1999a, 1999b) which were concerned with
isotropic and stationary Gaussian random fields.

This dissertation is dedicated to my parents for all their love and support.

iv

ACKNOWLEDGMENT

I would like to express my sincere gratitude to my advisor Professor Yimin Xiao for his invaluable guidance, intellect, support and continuing encouragement throughout my research
and the writing of this dissertation. This thesis could not have been finished without the
uncountable number of hours he spent sharing his knowledge and discussing various ideas
throughout the study. I would also like to express my heartfelt thanks to him for his patience.
In addition. I would like to thank all professors in my dissertation committee: Sasha
Kravchenko, Chae Young Lim, V.S. Mandrekar, Mark Meerschaert. I appreciate their time
taken out of busy schedules to serve on my committee and for their constructive feedback
through the dissertation process.
I give my special thanks to Professor James Stapleton. He is always ready to help me
like a father. Over the past five years, he has given me a lot of help and encouragement on
my study and life. I also thank all the other professors and my friends for their care and
help.
Last but not least, I would like to extend my thanks to my family. I thank my dear
husband, Linkan, for his tremendous love and support. His patience and encouragement are
highly appreciated. I thank my parents Xiuqin and Shoutuan, who have blessed me with
their unending love and support- and their financial assistance to a poor student. I thank
my parents-in-law Linying and Youxing for their love, encouragement and understandings.
Without their support and love, this dissertation would not have been completed.

v

TABLE OF CONTENTS

1 Introduction and preliminaries
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Definition and preliminaries . . . . . . . . . . . . . . .
1.2.1 Random fields . . . . . . . . . . . . . . . . . . .
1.2.2 Equivalence and orthogonality of two measures
1.2.3 Reproducing kernel Hilbert spaces . . . . . . . .
1.2.4 Hausdorff dimension . . . . . . . . . . . . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

2 Sample path properties of space-time models
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Anisotropic Gaussian models with stationary increments . . . . .
2.3 Prediction error for anisotropic Gaussian models . . . . . . . . . .
2.4 Smoothness properties of anisotropic Gaussian models . . . . . . .
2.4.1 Distributional properties of mean square partial derivatives
2.4.2 Criterion for mean square differentiability . . . . . . . . .
2.4.3 Criterion for sample path differentiability . . . . . . . . . .
2.5 Fractal properties of anisotropic Gaussian models . . . . . . . . .
2.6 Applications to some stationary space-time models . . . . . . . .
2.6.1 Stationary covariance models . . . . . . . . . . . . . . . .
2.6.2 Stationary spectral density models . . . . . . . . . . . . .
2.7 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.

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

3 Criteria for equivalence and asymptotically optimal predictions
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Three corresponding Hilbert spaces and equivalence . . . . . . . . .
3.3 Some conditions for equivalence of two Gaussian measures . . . . .
3.3.1 Case I: Same covariance function . . . . . . . . . . . . . . .
3.3.2 Case II: Same mean function . . . . . . . . . . . . . . . . . .
3.4 Asymptotic optimality of linear predictions . . . . . . . . . . . . . .
3.5 Explicit bounds with equal covariance functions . . . . . . . . . . .
3.5.1 One-dimensional Processes . . . . . . . . . . . . . . . . . . .
3.5.2 Two-dimensional random fields . . . . . . . . . . . . . . . .
3.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Conclusion and future work

.
.
.
.
.
.

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

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.

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

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.

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

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.

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

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.

1
1
6
6
8
9
10

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

12
12
14
19
21
22
26
28
29
34
34
36
37

.
.
.
.
.
.
.
.
.
.

56
56
59
63
65
68
76
80
81
84
86
105

vi

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

vii

Chapter 1
Introduction and preliminaries

1.1

Introduction

Spatio-temporal models are widely used for inference in statistics and many applied areas
such as meteorology, climatology, geophysical science, agricultural sciences, environmental
sciences, epidemiology, hydrology. Such models presume, on Rd × R, where d is the spatial
dimension, a collection of random variables X(x, t) at location x and time t. The family
{X(x, t) : (x, t) ∈ Rd × R} is referred to as a spatio-temporal random field or a space-time
model.
Many authors have constructed various stationary space-time models and the topic has
been under rapid development in recent years. See, for example, Jones and Zhang (1997),
Cressie and Huang (1999), de Iaco, Myers and Posa (2001, 2002, 2003), Gneiting (2002),
Gneiting, et al. (2009), Kolovos, et al. (2004), Kyriakidis and Journel (1999), Ma (2003a,
2003b, 2004, 2005a, 2005b, 2007, 2008), Stein (2005) and their combined references for
further information on constructions of space-time models and their applications.
There has also been increasing demand for non-stationary space-time models. For ex1

ample, in the analysis of spatio-temporal data of environmental studies, sometimes there is
little reason to expect stationarity under the spatial covariance structures, and it is more advantageous to have a space-time model whose variability changes with location and/or time.
Henceforth, the construction of non-stationary space-time models has become an attractive
topic and several approaches have been developed recently. These include the deforming of
the coordinates of an isotropic and stationary random field to obtain a rich class of nonstationary random fields [see Schmidt and O’Hagan (2003), Anderes and Stein (2008)], or
the use of convolution-based methods [cf. Higdon, Swall and Kern (1999), Higdon (2002),
Paciorek and Schervish (2006), Calder and Cressie (2007)] or spectral methods [Fuentes
(2002, 2005)].
In this dissertation, firstly, we apply the theory of Yaglom (1957) to construct a class
of space-time Gaussian models with stationary increments and study their statistical and
geometric properties. The main feature of this class of space-time models is that they are
anisotropic in time and space, and may have different smoothness and geometric properties
along different directions. Such flexible properties make them potentially useful as stochastic
models in various areas. By applying tools from Gaussian random fields, fractal geometry
and Fourier analysis, we derive upper and lower bounds for the prediction errors, establish
criteria for the mean-square and sample path differentiability and determine the Hausdorff
dimensions of the sample surfaces, all in terms of the parameters of the models explicitly.
Our main results show that the statistical and geometric properties of the Gaussian random
fields in this dissertation are very different from those obtained by deformation from any
isotropic random field. It is also worthwhile to mention that the methods developed in this
dissertation may be applied to analyze more general Gaussian intrinsic random functions,
convolution-based space-time Gaussian models [Higdon (2002), Calder and Cressie (2007)]
2

and the spatial processes in Fuentes (2002, 2005).
On the other hand, optimal linear prediction has been widely used in spatial statistics
and geostatistics, where it is known as kriging. In kriging, to guarantee good linear predictors based on an estimated Gaussian probability measure, it is of great value to be able to
distinguish between two orthogonal probability measures and to determine when one can tell
which measure is correct and which is not. Many authors have created various criteria for
the equivalence and orthogonality of two Gaussian measures in a one-dimensional stochastic
process or Gaussian random field. The references include Gihman and Skorohod (1974),
Ibragimov and Rozanov (1978), Parzen (1963), Chatterji and Mandrekar (1978), Kallianpur
and Oodaira (1963), Yadrenko (1983), Stein (1999b), Du (2009) and so on. In fact, Parzen
(1963) developed an approach for equivalence of two Gaussian measures by using two concepts: the notion of probability spectral density function and the notion of a reproducing
kernel Hilbert space (RKHS, for short) of a time series. Chatterji and Mandrekar (1978)
also used the method of RKHS to find the sufficient and necessary conditions for the equivalence of two Gaussian measures in a general setting. It is worth noting that the approach
which uses RKHS has no constraints like stationarity or isotropy on the underlying process,
and the results are applicable to random fields. Ibragimov and Rozanov (1978) obtained
necessary and sufficient conditions for equivalence of two Gaussian measures involving the
entropy of distributions, and developed the conditions for stationary processes by associating a Hilbert space spanned by certain analytic functions. Moreover, given two equivalent
Gaussian processes, Kallianpur and Oodaira (1963) defined the notion of a non-anticipative
representation of one of the processes with respect to the other. Later, Yadrenko (1983)
extended the results of Ibragimov and Rozanov (1978) to stationary and isotropic random
fields. Du (2009) gave some reviews of the basic results for the equivalence and orthogonal3

ity of two Gaussian measures, and provided a detailed re-proof of Theorem 4 in Yadrenko
(1983), page 156, under the setting of stationary and isotropic random fields. In the literature, there are few explicit results available for the equivalence of two Gaussian measures in
a non-stationary random field, especially for anisotropic cases.
In the second part of this dissertation, we extend Ibragimov and Rozanov (1978)’s method
to study intrinsically stationary random fields. We describe the relationships among three
corresponding Hilbert spaces: the random variable space generated by the random field,
the reproducing kernel Hilbert space corresponding to the covariance kernel, and the complex function space spanned by certain analytic functions. Criteria for equivalence and
orthogonality of intrinsically stationary Gaussian random fields are given in terms of their
probability spectral density functions and the structures of their reproducing kernel Hilbert
spaces. The results we have obtained are different from those of stationary processes [see
Ibragimov and Rozanov (1978)]. Moreover, given the equivalence of two random fields, we
obtain a representation of one of the random fields with respect to the other. The advantage
of our representation over the original one is that it is much simpler with respect to some
prediction questions.
In practice, the true probability distribution of our Gaussian model is always unknown
and must be estimated from the gathered data. To this end, it is of great value to investigate the effect of using a fixed but incorrect probability distribution, especially, when
more sample data can be obtained by sampling the spatial or temporal domain increasingly densely (fix-domain asymptotics). We establish the asymptotic optimality of linear
predictions of intrinsically stationary Gaussian models and the convergence rates in this
dissertation. Moreover, the asymptotic efficient prediction of non-stationary, anisotropic
space-time models with a misspecified probability distribution is studied. The main results
4

show that under the equivalence of two Gaussian measures, the prediction based on the
incorrect distribution is asymptotically optimal and efficient relative to the prediction under
the correct distribution, as the points of observations become increasingly dense in the study
domain. The results extend those of Stein (1988, 1990, 1999a, 1999b) which were concerned
with isotropic and stationary Gaussian random fields.
The rest of this dissertation is organized as follows. In Section 2 of Chapter 1, we collect
definitions and some properties of Gaussian random fields, equivalence and orthogonality
of two measures, reproducing kernel Hilbert spaces and Hausdorff dimensions. Chapter 2
studies sample path properties of space-time models. We construct a class of space-time
Gaussian models with stationary increments, establish bounds on the prediction errors and
investigate smoothness properties and fractal properties of this class of Gaussian models.
The results are applied directly to analyze the stationary space-time models in Section 5.
Section 6 gives proofs of the main theorems and lemmas in this chapter. In Chapter 3, we
investigate asymptotic properties of space-time models. We extend the methods in Ibragimov
and Rozanov (1978) to study intrinsically stationary random fields. We obtain criteria for
equivalence and orthogonality of this class of random fields in Section 2. The asymptotic
optimality of linear predictions and the convergence rates are established in Sections 3 and
4. In Section 5, we present proofs of the main results in this chapter. Finally, we conclude
by describing some of our ongoing projects and future work in Chapter 4.

5

1.2

Definition and preliminaries

This section contains basic definitions and facts of stationary and intrinsically stationary
Gaussian random fields, equivalence and orthogonality of two measures, reproducing kernel
Hilbert spaces and Hausdorff dimensions, which will be used in subsequent chapters.
Throughout this dissertation, for simplicity of notation, we use RN (RN = [0, ∞)N ) or
+
Rd (in Chapter 3), instead of Rd × R, as the index set for random fields. We use | · | to
denote the Euclidean norm in RN . The inner product in RN is denoted by ·, · . A typical
coordinate, t ∈ RN is written as t = (t1 , . . . , tN ). For any s, t ∈ RN such that sj < tj
(j = 1, . . . , N ), [s, t] =

N
j=1 [sj , tj ]

is called a closed interval (or a rectangle). For a positive

number x, we use x to denote the integer part of x.
We will use c, c1 , c2 , . . . , to denote unspecified positive and finite constants which may
not be the same in each occurrence.

1.2.1

Random fields

A formal definition of random fields is as follows:
Definition 1.2.1. Let a probability space, (Ω, U, P ), an integer p ≥ 1, and an index set,
T , be given. A random field indexed by T with values in Rp is then a Rp -valued function
X(t, ω) on T × Ω such that for every fixed t ∈ T , X(t, ·) is a random vector in Rp .
In this dissertation, we will take T = RN , the N -dimensional Euclidean space, or a subset
of RN . In this case, X is simply referred to as an (N, p) random field. The dependency
on the underlying probability space will usually be suppressed throughout the text, i.e. we
write
X(t) = X(t, ω),
6

t ∈ RN .

For a fixed ω ∈ Ω, the function X(t, ω) : RN → Rp is a non-random function of t. This
deterministic function is usually called a sample path or a realization of the random field.
The variable t is called the coordinate or position by standard terminology. In this context,
the formal definition of random field simply means:
An (N, p) random field X(t) is a function whose values are random vectors in Rp for
every t ∈ RN .
When N = 1, the random field is usually called a stochastic process. The term “random field”
is usually used to stress that the dimension of the coordinate is higher than one. Random
fields in two and three dimensions are widely used as spatial or spatio-temporal models
in many applied areas such as meteorology, climatology, geophysical science, agricultural
sciences, environmental sciences, epidemiology and hydrology.
In this dissertation, we mainly focus on Gaussian random fields, and study their properties and prediction problems. Gaussian random fields play an important role for several
reasons: The specification of their finite-dimensional distributions is simple, and the model
is determined by the mean and covariance functions only. Moreover, they are reasonable
models for many natural phenomena.

Definition 1.2.2. A Gaussian random field is a random field where all the finite dimensional
distributions are multivariate normal distributions.

In this dissertation, we construct a class of intrinsically stationary Gaussian random
fields (i.e. Gaussian random fields with stationary increments). We study its nature and
properties, and compare our results with that of stationary Gaussian random fields. The
following are the definitions for stationary and intrinsically stationary Gaussian random
fields.
7

Definition 1.2.3. A Gaussian random field X(t), t ∈ RN is said to be stationary if its
mean function m(t) is constant, and the covariance function

K(s, t) = E (X(s) − m(s))(X(t) − m(t))

depends only on the difference s − t, for all s, t ∈ RN .
Definition 1.2.4. A Gaussian random field X(t), t ∈ RN is said to be intrinsically stationary if the increment process X(t + h) − X(t) is stationary for any fixed h ∈ RN . Or
equivalently, ∀ h ∈ RN

d

X(t + h) − X(t), t ∈ RN

= X(t) − X(0), t ∈ RN ,

d

where “ = ” means equality of all finite dimensional distributions.
In this dissertion, we call the pair (m, K) the second-order structure of the Gaussian
random field X(t). A random field X is said to be isotropic if, for all rotation R in RN ,
d

X ◦ R = X. Otherwise, X is said to be anisotropic.

1.2.2

Equivalence and orthogonality of two measures

Let {X(t), t ∈ T } be Gaussian random field on the probability space (Ω, U, P ). Let P1 be
another Gaussian measure on the σ-algebra U. P1 is said to be absolutely continuous with
respect to P if for all A ∈ U, P (A) = 0 implies P1 (A) = 0. It is known that the absolutely
continuous measure P1 can be represented as

P1 (A) =

p(ω)P (dω),
A

8

A ∈ U,

where p(ω) is a nonnegative function on Ω, which is the Randon-Nikodym derivative of P1
with respect to P , i.e. p(ω) = P1 (dω)/P (dω). We also call it a density of P1 with respect
to P . The two measures P and P1 are said to be equivalent if they are mutually absolutely
continuous. The measures P and P1 are said to be orthogonal if there exists A ∈ U such
that P (A) = 1 and P1 (A) = 0. In this case, we also have P (Ac ) = 0 and P1 (Ac ) = 1.
Lemma 1.2.5. Any two Gaussian measures P and P1 are either equivalent or orthogonal.
For the proof of this lemma, see page 77 of Ibragimov and Rozanov (1978) or page 117 of
Stein (1999b). In Chapter 3 of this dissertation, we will provide some criteria for equivalency
of two intrinsically stationary Gaussian random fields.

1.2.3

Reproducing kernel Hilbert spaces

Let K(s, t) be the covariance function of a real-valued random field X(t), t ∈ T ⊆ RN . For
each t ∈ T , let K(·, t) be the function on T whose value at s ∈ T is equal to K(s, t). It
may be shown [see Aronszajn (1950)] that there exists a unique Hilbert space, denoted as
RK (T ), with the following properties:

(1) The members of RK (T ) are real-valued functions on T [if K(s, t) were complex-valued,
they would be complex-valued functions].

(2) For every t ∈ T , K(·, t) ∈ RK (T ).
(3) For every t ∈ T and f ∈ RK (T ),

f (t) = f, K(·, t) R (T ) ,
K
9

where the inner product between two functions f and g in RK (T ) is written as
f, g R (T ) .
K

We call RK (T ) the reproducing kernel Hilbert space (RKHS, for short) of the random field
X(t) with reproducing kernel K(s, t), s, t ∈ T . In fact, the Hilbert space RK (T ) is the
closure of the subspace spanned by the functions K(·, t), t ∈ T .
In the end of Section 2 of Chapter 3, we will encounter another type of kernel. A kernel
b(s, t): T × T → R is of Volterra type if b(s, t) = 0 implies t ≤ s, where s, t ∈ T . Sottinen
and Tudor (2006) use this type of kernel to characterize a representation of a Gaussian sheet
which is equivalent in law to the Brownian sheet. We can see that the kernel b(s, t) is not a
covariance function. But we can use this kernel to get a covariance function K(s, t), s, t ∈ T ,
as follows:

K(s, t) = b(s, t) + b(t, s) −

b(s, u)b(u, t)du.
T

See Sottinen and Tudor (2006) for more details on kernels of Volterra type.

1.2.4

Hausdorff dimension

Let Θ be the class of functions φ : (0, δ) → (0, 1), which are right continuous, monotone
increasing with φ(0+) = 0 and satisfying the following “doubling” property:
There exists a finite constant c > 0, for which

φ(2s)
≤ c,
φ(s)

δ
0<s< .
2
10

For φ ∈ Θ, the φ-Hausdorff measure of A ⊆ RN is defined as

φ − m(A) = lim inf
→0





∞

φ(2rj ) :
j

A⊆

O(xj , rj ), rj <
j=1




,



where O(x, r) denotes the open ball of radius r, centered at x. φ − m is a metric outer
measure and every Borel set in RN is φ − m measurable [cf. Rogers (1970)].
The Hausdorff dimension of A is defined as

dimH A = inf{α > 0 :
= sup{α > 0 :

sα − m(A) = 0}
sα − m(A) = ∞}.

If 0 < sα − m(A) < ∞, then A is called an α-set. If there exists φ ∈ Θ with 0 < φ − m(A) <
∞, then φ is called an exact Hausdorff measure function for A.
The following are some basic properties of Hausdorff dimensions:
(1) Monotonicity: if A ⊆ B, then dimH A ≤ dimH B.
(2) Hausdorff dimension is σ-stable: dimH ( ∞ An ) = supn≥1 dimH An .
n=1
We refer to Falconer (1990) for more details in Hausdorff dimensions. In Chapter 2 of
this dissertation, we write the Hausdorff dimension as dim, instead of dimH for simplicity.

11

Chapter 2
Sample path properties of space-time
models

2.1

Introduction

Space-time models are widely used for inference in spatial statistics and geostatistics. Various
stationary space-time models have been constructed in the literature, and the topic has been
under rapid development in recent years. See, for example, Jones and Zhang (1997), Cressie
and Huang (1999), de Iaco, Myers and Posa (2001, 2002, 2003), Gneiting (2002), Gneiting,
et al. (2009), Kolovos, et al. (2004), Kyriakidis and Journel (1999), Ma (2003a, 2003b, 2004,
2005a, 2005b, 2007, 2008), Stein (2005) and their combined references for further information
on constructions of space-time models and their applications.
In the meantime, there has also been increasing demand for non-stationary space-time
models. For example, in the analysis of spatio-temporal data of environmental studies,
sometimes there is little reason to expect stationarity under the spatial covariance structures,
and it is more advantageous to have a space-time model whose variability changes with
12

location and/or time. Henceforth, the construction of non-stationary space-time models
has become an attractive topic and several approaches have been developed recently. These
include the deforming of the coordinates of an isotropic and stationary random field to obtain
a rich class of non-stationary random fields [see Schmidt and O’Hagan (2003), Anderes
and Stein (2008)], or the use of convolution-based methods [cf. Higdon, Swall and Kern
(1999), Higdon (2002), Paciorek and Schervish (2006), Calder and Cressie (2007)] or spectral
methods [Fuentes (2002, 2005)].
In this chapter, we apply the theory of Yaglom (1957) to construct a class of spacetime Gaussian models with stationary increments and study their statistical and geometric
properties. The main feature of this class of space-time models is that they are anisotropic in
time and space, and may have different smoothness and geometric properties along different
directions. Such flexible properties make them potentially useful as stochastic models in
various areas. By applying tools from Gaussian random fields, fractal geometry and Fourier
analysis, we derive upper and lower bounds for the prediction errors, establish criteria for
mean-square and sample path differentiability and determine the Hausdorff dimensions of
the sample surfaces, all in terms of the parameters of the models explicitly. Our main
results show that the statistical and geometric properties of the Gaussian random fields in
this dissertation are very different from those obtained by deformation from any isotropic
random field. It is also worth mentioning that the method in this dissertation may be applied
to analyze more general Gaussian intrinsic random functions, convolution-based space-time
Gaussian models [Higdon (2002), Calder and Cressie (2007)] and the spatial processes in
Fuentes (2002, 2005).
The rest of this chapter is organized as follows. In Section 2 we construct a class of spacetime intrinsically stationary Gaussian models by applying the theory of Yaglom (1957). Then
13

we establish upper and lower bounds for the prediction errors of this class of models in Section
3. In Section 4 we consider smoothness properties of the models and establish explicit criteria
for the existence of mean-square directional derivatives, mean-square differentiability and
sample path continuity of partial derivatives. In Section 5 we look into the fractal properties
of these models and determine the Hausdorff dimensions of the range, graph and level sets.
In Section 6, we apply the main results of Section 5 to some stationary space-time models,
such as those constructed by Cressie and Huang (1999), Gneiting (2002) and Stein (2005).
Finally, in Section 7, we provide the proofs of the main results in this chapter.

2.2

Anisotropic Gaussian models with stationary increments

We consider a special class of intrinsic random functions; namely, space-time models with
stationary increments (also called intrinsically stationary space-time models). We will further
restrict ourselves to Gaussian random fields for which powerful general Gaussian principles
can be applied. Many of the results in this chapter can be extended to non-Gaussian spacetime models (such as stable or more general infinitely divisible random fields), but their
proofs require different methods and go beyond the scope of this chapter. One can find some
information for stable random fields in Xiao (2011).
Let X = {X(t), t ∈ RN } be a real-valued, centered Gaussian random field with X(0) = 0.
We assume that X has stationary increments and continuous covariance function K(s, t) =
E[X(s)X(t)]. According to Yaglom (1957), K(s, t) can be represented as

K(s, t) =

RN

ei s,λ − 1 e−i t,λ − 1 F (dλ) + s, W t ,
14

(2.1)

where W is an N × N non-negative definite matrix and F (dλ) is a nonnegative symmetric
measure on RN \ {0} satisfying
|λ|2
F (dλ) < ∞.
2
RN 1 + |λ|

(2.2)

In analogy to the stationary case, the measure F is called the spectral measure of X. If F
is absolutely continuous with respect to the Lebesgue measure in RN , its density f will be
called the spectral density of X.

It follows from (2.1) that X has the following stochastic integral representation:

X(t), t ∈ RN

d

=

RN

ei t,λ − 1 Φ(dλ) + Y, t , t ∈ RN

,

(2.3)

d

where X1 = X2 means the processes X1 and X2 have the same finite dimensional distributions, Y is an N -dimensional Gaussian random vector with mean 0 and covariance matrix
W , Φ(dλ) is a centered complex-valued Gaussian random measure which is independent of
Y and satisfies
E Φ(A)Φ(B) = F (A ∩ B) and Φ(−A) = Φ(A)
for all Borel sets A, B ⊆ RN , with finite F -measure. The spectral measure F is called
the control measure of Φ. Since the linear term Y, t in (2.3) will not have any effect on
the problems considered in this dissertation, we will from now on assume Y = 0. This is
equivalent to assuming W = 0 in (2.1). Consequently, we have

v(h)

E X(t + h) − X(t)

2

=2
15

RN

1 − cos h, λ F (dλ).

(2.4)

It is important to note that the function v(h), called variogram in spatial statistics, is
a negative definite function in the sense of I. J. Schoenberg, which is determined by the
spectral measure F . See Berg and Forst (1975) for more information on negative definite
functions.
The above shows that various centered intrinsically stationary Gaussian random fields can
be constructed by choosing appropriate spectral measures F . For the well known fractional
Brownian motion B H = {B H (t), t ∈ RN } of Hurst index H ∈ (0, 1), its spectral measure
has a density function
fH (λ) = c(H, N )

1
|λ|2H+N

,

where c(H, N ) > 0 is a normalizing constant such that v(h) = |h|2H . Since v(h) depends
on |h| only, B H is isotropic. Other examples of isotropic Gaussian fields with stationary
increments can be found in Xiao (2007). We also remark that all centered stationary Gaussian
random fields can be treated using the above framework. In fact, if Z = {Z(t), t ∈ RN } is a
centered stationary Gaussian random field, it can be represented as Z(t) = RN ei t,λ Φ(dλ).
Thus the random field X defined by

X(t) = Z(t) − Z(0) =

RN

ei t,λ − 1 Φ(dλ),

∀ t ∈ RN

is Gaussian with stationary increments (intrinsically stationary Gaussian random field) and
X(0) = 0. Note that the spectral measure F of X in the sense of (2.4) is the same as the
spectral measure [in the ordinary sense] of the stationary random field Z.
In the following, we propose and investigate a class of centered, anisotropic, intrinsically
stationary Gaussian random fields, whose spectral measures are absolutely continuous with
respect to the Lebesgue measure in RN . More precisely, we assume that the spectral measure
16

F of X = {X(t), t ∈ RN } is absolutely continuous with density function f (λ) which satisfies
(2.2) and the following condition:
(C) There exist positive constants c1 , c2 , c3 , γ and (β1 , · · · , βN ) ∈ (0, ∞)N such that
N

γ>
j=1

1
βj

(2.5)

and

c2

c1

≤ f (λ) ≤
β γ
N
|λj | j
j=1

βj γ
N
j=1 |λj |

,

∀ λ ∈ RN with |λ| ≥ c3 .

(2.6)

The following proposition shows that (2.5) is needed to ensure f is a legitimate spectral
density function.
Proposition 2.2.1. Assume that f (λ) is a non-negative measurable function defined on RN .
If
|λ|2 f (λ)dλ < ∞
|λ|≤c3

and (2.6) holds, then f (λ) is a legitimate spectral density if and only if the parameters γ and
βj for j = 1, · · · , N satisfy (2.5).
Some remarks about Condition (C) are given in the following.
Remark 2.2.2
(1) There is an important connection between the random field models that satisfy Condition (C) and those considered in Xiao (2009). For j = 1, · · · , N , let
N
βj
1
Hj =
γ−
2
βi
i=1

17

(2.7)

N
1
j=1 Hj .

and let Q =

c4
N
j=1 |λj

H
| j

2+Q

Then (2.6) can be rewritten as

c5

≤ f (λ) ≤

N
j=1 |λj

H 2+Q
| j

∀ λ ∈ RN with |λ| ≥ c3 ,

,

(2.8)
where the positive and finite constants c4 and c5 depend on N , c1 , c2 , βj and γ only.
To verify this claim, we will make use of the following elementary fact: For any positive
numbers N and q, there exist positive and finite constants c4 and c5 such that
N

c4

N

q

N
q

≤

aj

aj ≤ c 5
j=1

j=1

q

(2.9)

aj
j=1

for all non-negative numbers a1 , . . . , aN . Note that
N

|λj

H
| j

j=1

and 1 γ −
2

N 1
i=1 βi

N

2+Q

1
βj · 2 γ−

|λj |

=

N 1
i=1 βi

2+Q

j=1

(2 + Q) = γ. We apply (2.9) with q = 1 γ −
2

N 1
i=1 βi

to see that

(2.6) and (2.8) are equivalent.
In turns out that the expression (2.8) is essential in this chapter and will be used
frequently. For simplicity of notation, from now on we take c3 = 1.
(2) It is also possible to consider intrinsically stationary Gaussian random fields whose
spectral measures are not absolutely continuous. Some examples of such covariance
space-time models can be found in Cressie and Huang (1999), Gneiting (2002), Ma
(2003a, 2003b). Since the mathematical tools for studying such random fields are
quite different [see Luan and Xiao (2010)], we will deal with them systematically in
the future.
18

(3) Non-stationary Gaussian random fields can be constructed through deformation of an
isotropic Gaussian random field. Refer to Anderes and Stein (2008) for more details.
One of the advantages of deformation is to closely connect a nonstationary and/or
anisotropic random field to a stationary and isotropic one for which existing statistical
techniques are available. However, there is also a disadvantage [from the point of view of
flexibility] associated with deformation. Let X(t) = Z(g −1 (t)), where {Z(t), t ∈ RN }
is an isotropic Gaussian model and g is a smooth bijection of RN . Since the function
g is bi-Lipschitz on compact intervals, the fractal dimensional properties of X are
the same as those of Z. Hence deformation of isotropic Gaussian models will not
generate anisotropic random fields with rich geometric structures as shown by the
models introduced in this chapter.

2.3

Prediction error for anisotropic Gaussian models

Suppose we observe an anisotropic Gaussian random field X on RN at t1 , . . . , tn and wish
to predict X(u), for u ∈ RN . Then the inference about X(u) will be based upon the
conditional distribution of X(u) given the observed values of X(t1 ), . . . , X(tn ). Refer to
Stein (1999b, Section 1.2) for the closed form of this conditional distribution. A statistical
analysis typically aims at the optimal linear predictor of this unobserved X(u), known as
simple kriging. The simple kriging predictor of X(u) is

X ∗ (u) = c(u)T Σ−1 Z,
19

(2.10)

where Z = X(t1 ), . . . , X(tn )

T

, c(u)T = Cov{X(u), Z} and Σ = Cov(Z, ZT ). The form

(2.10) minimizes the mean square prediction error, which then is given as Var(X(u)) −
c(u)T Σ−1 c(u). Since X is Gaussian, the simple kriging is the conditional expectation of
X(u) given Z, and the mean square prediction error is the conditional variance of X(u)
given the observations Z.
The main result of this section is Theorem 2.3.1 below, which gives lower and upper
bounds for the mean square prediction error for intrinsically stationary Gaussian random
fields which satisfy Condition (C). It shows that, similar to stationary Gaussian field models
[cf. Stein (1999b)], the prediction error of the models in this chapter only depends on the
high frequency behavior of the spectral density of X.

Theorem 2.3.1. Let X = {X(t), t ∈ RN } be a centered intrinsically stationary Gaussian
random field valued in R with spectral density f (λ) satisfying (2.6). Then, for any given
constant M > 0, there exist constants c6 > 0 and c7 > 0, such that for all integers n ≥ 1
and all u, t1 , · · · , tn ∈ [−M, M ]N ,

N
0≤k≤n

2Hj

|uj − tk |
j

c6 min

N

σj |uj − tk | ,
j

≤ Var X(u)|X(t1 ), · · · , X(tn ) ≤ c7 min

0≤k≤n

j=1

j=1

(2.11)
where Hj is given in (2.7), t0 = 0 and σj : R+ → R+ is defined by

 2Hj
 r




2
σj (r) =
 r | log r|



 2
 r

if 0 < Hj < 1,
if Hj = 1,

(2.12)

if Hj > 1.

If Hj < 1, for j = 1, · · · , N , then the two bounds in (2.11) match. When there is some
20

Hj > 1, that means, the random field X(t) is smoother in the j-th direction [see Corollaries
2.4.3, 2.4.7 and Theorem 2.4.8 below], then the upper and lower bounds are not the same any
more. This suggests that the prediction error may become larger as X(t) becomes smoother
in some directions.
The proof of Theorem 2.3.1, as well as those of Theorems 2.4.9, 2.5.1 and 2.5.2 relies
partially on the following lemma, which provides upper and lower bounds for the variogram
of the model.
Lemma 2.3.2. Let X = {X(t), t ∈ RN } be a centered intrinsically stationary Gaussian
random field valued in R with spectral density f (λ) satisfying (2.6). Then, for any given
constant M > 0, there exist constants c8 > 0 and c9 > 0, such that for s, t ∈ [−M, M ]N ,
N

σj |sj − tj | ≤ E X(s) − X(t)

c8
j=1

2

N

≤ c9

σj |sj − tj | ,

(2.13)

j=1

where the function σj is defined in (2.12).
The upper bound in (2.13) implies that X has a version whose sample functions are
almost surely continuous. Throughout this dissertation, without loss of generality, we will
assume that the sample function t → X(t) is almost surely continuous.

2.4

Smoothness properties of anisotropic Gaussian models

Regularity properties of sample path of random fields are of fundamental importance in probability and statistics. Many authors have studied mean square and sample path continuity
and differentiability of Gaussian processes and random fields. See Cram´r and Leadbetter
e
21

(1967), Alder (1981), Stein (1999b), Banerjee and Gelfand (2003), Adler and Taylor (2007).
In this section we provide explicit criteria for mean square and sample path differentiability
for the models introduced in Section 2.

2.4.1

Distributional properties of mean square partial derivatives

Banerjee and Gelfand (2003) studied the smoothness properties of stationary random fields
and some non-stationary relatives through directional derivative processes and their distributional properties. To apply their method to intrinsically stationary random fields, let us
first recall the definition of mean square directional derivatives.

Definition 2.4.1. Let u ∈ RN be a unit vector. A second order random field {X(t), t ∈ RN }
has mean square directional derivative Xu (t) at t ∈ RN in the direction u if, as h → 0,

Xu,h (t) =

X(t + hu) − X(t)
h

converges to Xu (t) in the L2 -sense. In this case, we write Xu (t) = l.i.m.h→0 Xu,h (t).

Let e1 , e2 , · · · , eN be an orthonormal basis for RN . If u = ej , then Xe (t) is the mean
j
square partial derivative in the j-th direction defined in Adler (1981), which will simply be
written as Xj (t). We will also write Xej ,h (t) as Xj,h (t).
For any second-order, centered random field {X(t), t ∈ RN }, similar to Theorem 2.2.2 in
Adler (1981), one can easily establish a criterion in terms of the covariance function K(s, t) =
E X(s)X(t) for the existence of mean square directional derivative Xu (t). Banerjee and
22

Gelfand (2003) further showed that the covariance function of Xu (t) is given by

Ku (s, t = lim lim E Xu,h (s)Xu,k (t)
h→0 k→0

K(s + hu, t + ku) − K(s + hu, t) − K(s, t + ku) + K(s, t)
.
hk
h→0 k→0

= lim lim

Extending their argument, one obtains the following theorem for intrinsically stationary
Gaussian random fields.
Theorem 2.4.2. Let X = {X(t), t ∈ RN } be a centered intrinsically stationary Gaussian
random field valued in R, then the mean square partial derivative Xj (t) exists for all t ∈ RN
if and only if the limit
v(hej ) + v(kej ) − v (h − k)ej
hk
h,k→0
lim

(2.14)

exists, where v(t) is defined in (2.4). Moreover, this later condition is equivalent to v(t) has
second-order partial derivatives at 0 in the j-th direction.
As a consequence, we obtain an explicit criterion for the existence of mean square partial
derivatives of Gaussian random fields in Section 2.
Corollary 2.4.3. Let X = {X(t), t ∈ RN } be a centered intrinsically stationary Gaussian
random field valued in R with spectral density f (λ) satisfying Condition (C). Then for every
j = 1, · · · , N , the mean square partial derivative Xj (t) exists if and only if
N

βj γ −
i=1

1
βi

> 2,

(2.15)

or equivalently Hj > 1 [cf. (2.7)].
Assume condition (2.14) of Theorem 2.4.2 holds so that the mean square partial derivative
23

Xj (t) exists for all t ∈ RN . We now consider the distributional properties of the random
field {Xj (t), t ∈ RN }.
Since E(X(t)) = 0 for all t ∈ RN , we have E(Xj,h (t)) = 0 and E(Xj (t)) = 0. Let
(h)

Kj (s, t) and Kj (s, t) denote the covariance functions of the random fields {Xj,h (t), t ∈ RN }
and {Xj (t), t ∈ RN }, respectively. Let ∆ = s − t, we immediately have

(h)

Kj (s, t) =

v(∆ + hej ) + v(∆ − hej ) − 2v(∆)
,
2h2

(2.16)

and Var(Xj,h (t)) = v(hej )/h2 , which only depends on the scalar h.

Theorem 2.4.4. Let X = {X(t), t ∈ RN } be a centered intrinsically stationary Gaussian
random field valued in R. Suppose that all second-order partial derivatives of the variogram
v(t) exist. Then the covariance function of Xj (t) is given by
1
Kj (s, t) = vj (s − t),
2

(2.17)

where vj (t) is the second-order partial derivative of v at t in the j-th direction. In particular,
{Xj (t), t ∈ RN } is a stationary Gaussian random field.

Proof

The desired result follows from (2.16).

It is also useful to determine the covariance of X(s) and Xj (t) for all s, t ∈ RN . Since
1
Cov X(s), Xj,h (t) =
v(t + hej ) − v(t) + v(∆) − v(∆ − hej ) ,
2h
24

where ∆ = s − t, we obtain

1
v(t + hej ) − v(t) − v(hej )
Cov X(t), Xj,h (t) =
2h

and
1
v(t + hej ) − v(t) + v(∆) − v(∆ − hej )
Cov X(s), Xj (t) = lim
h→0 2h
1
= vj (t) + vj (∆) ,
2

(2.18)

where vj (t) is the partial derivative of v at t in the j-th direction.
In particular, Cov X(t), Xj (t) = vj (t)/2, which is different from the stationary case. Recall that if Z(t) is a stationary Gaussian field with mean square partial derivative Zj (t), then
Z(t) and Zj (t) are uncorrelated (and thus independent). However, this is not always true
for non-stationary Gaussian random fields, which is one of the reasons why non-stationary
models are more difficult to study.

Next we consider the bivariate process

(h)

Yj



 X(t) 
(t) = 
.
Xj,h (t)

It can be verified that this process has mean 0 and cross-covariance matrix

Vj,h (s, t)


v(s) + v(t) − v(∆)

2
=
 v(s + hej ) − v(s) + v(∆) − v(∆ + hej )
2h
25


v(t + hej ) − v(t) + v(∆) − v(∆ − hej )

2h
.
v(∆ + hej ) + v(∆ − hej ) − 2v(∆) 
2h2

(h)

Because Yj

(t) is obtained by linear transformation of X(t), the above is a valid cross-

covariance matrix in RN . Since this is true for every h, letting h → 0 we see that


1
 2 v(s) + v(t) − v(∆)
Vj (s, t) = 
1
v (s) − vj (∆)
2 j


1
v (t) + vj (∆) 
2 j

1
v (∆)
2 j

is a valid cross-covariance matrix in RN . In fact, Vj is the cross-covariance matrix for the
bivariate process




 X(t) 
Yj (t) = 
.
Xj (t)

2.4.2

Criterion for mean square differentiability

Benerjee and Gelfand (2003) pointed out that the existence of all mean square directional
derivatives of a random field X does not even guarantee mean square continuity of X, and
they introduced a notion of mean square differentiability which has analogous properties of
total differentiability of a function in RN in the non-stochastic setting. We first recall their
definition.
Definition 2.4.5. A random field {X(t), t ∈ RN } is mean square differentiable at t ∈ RN
if there exists a (random) vector

X (t)

∈ RN such that for all scalar h > 0, all vectors

u ∈ SN = {t ∈ RN : |t| = 1}

X(t + hu) = X(t) + huT X (t) + r(t, hu),

(2.19)

where r(t, hu)/h → 0 in the L2 -sense as h → 0.
Refer to Definition 2.1 of Potthoff (2010) for more details on the definition of mean square
26

differentiability. In other words, for all vectors u ∈ SN , it is required that
X(t + hu) − X(t) − huT X (t) 2
= 0.
h

lim E

h→0

(2.20)

It can be seen that if X is mean square differentiable at t, then for all unit vectors u ∈ SN
l.i.m. X(t + hu) − X(t) l.i.m. huT X (t) + r(t, hu)
= h→0
Xu (t) = h→0
h
h
= uT X (t).
Hence it is necessary that

X (t)

= (X1 (t), . . . , XN (t)).

The next theorem provides a sufficient condition for a intrinsically stationary Gaussian
random field to be mean square differentiable.

Theorem 2.4.6. Let X = {X(t), t ∈ RN } be a centered intrinsically stationary Gaussian
random field valued in R. If all the second-order partial and mixed derivatives of the variogram v(t) exist and are continuous, then X is mean square differentiable at every t ∈ RN .

As a consequence of Theorem 2.4.6 and Corollary 2.4.3 we obtain

Corollary 2.4.7. Let X = {X(t), t ∈ RN } be a centered intrinsically stationary Gaussian
random field valued in R with spectral density f (λ) satisfying Condition (C). Then X is
mean square differentiable at every t ∈ RN if and only if
N

βj γ −
i=1

1
βi

>2

for every j = 1, . . . , N.

27

(2.21)

2.4.3

Criterion for sample path differentiability

For many theoretical and applied purposes, one often needs to work with random fields
which have smooth sample functions. Refer to Adler (1981), Adler and Taylor (2007) and
the reference therein for more information. Since in general mean square differentiability does
not imply almost sure sample path differentiability, it is of interest to provide convenient
criteria for the latter.
For Gaussian random fields considered in this chapter, it turns out that under the same
condition as Corollary 2.4.3, the partial derivatives of X are almost surely continuous.
Theorem 2.4.8. Let X = {X(t), t ∈ RN } be a separable and centered intrinsically stationary Gaussian random field with values in R. We assume that X satisfies Condition (C).
(i). If
N

βj γ −
i=1

1
βi

>2

(i.e., Hj > 1),

(2.22)

for some j ∈ {1, · · · , N }, then X has a version X with continuous sample functions such
that its jth partial derivative Xj (t) is continuous almost surely.
(ii). If (2.22) holds for all j ∈ {1, · · · , N }, then X has a version X which is continuously
differentiable in the following sense: with probability 1,
X(t + hu) − X(t) − huT X (t)
lim
=0
h
h→0

for all u ∈ SN and t ∈ RN .

(2.23)

If condition (2.22) does not hold for some j ∈ {1, · · · , N }, then X(t) does not have mean
square partial derivatives along the j-th direction and the sample path of X(t) is usually a
random fractal. In this case, it is of interest to characterize the asymptotic behavior of X(t)
by its local and uniform moduli of continuity.
28

These problems for anisotropic Gaussian random fields have been considered in Xiao
(2009), Meerschaert, Wang and Xiao (2010). The methods there are applicable to X with a
little modification. For completeness, we state the following result which can be proved by
using Lemma 2.3.2 and general Gaussian methods. We omit its proof.
Theorem 2.4.9. Let X = {X(t), t ∈ RN } be as in Theorem 2.4.8. Then for every compact
interval I ⊂ RN , there exists a positive and finite constant c10 , depending only on I and
Hj , (j = 1, . . . , N ) such that

lim sup
|ε|→0

where ϕ(ε) =

N
j=1 σj (|εj |)

supt∈I, s∈[0,ε] |X(t + s) − X(t)|
ϕ(ε) log(1 + ϕ(ε)−1 )

≤ c10 ,

(2.24)

for all ε = (ε1 , . . . , εN ) ∈ RN , and the function σj is defined in

(2.12).

2.5

Fractal properties of anisotropic Gaussian models

The variations of soil, landform and geology are usually highly non-regular in form and can
be better approximated by a stochastic fractal. Hausdorff dimensions have been extensively
used in describing fractals. We refer to Kahane (1985) or Falconer (1990) for their definitions
and properties.
Let X = {X(t), t ∈ RN } be a real-valued, centered Gaussian random field. For any
integer p ≥ 1, we define an (N, p)-Gaussian random field X = {X(t), t ∈ RN } by

X(t) = X1 (t), . . . , Xp (t) ,

where X1 , . . . , Xp are independent copies of X.
29

t ∈ RN ,

(2.25)

In this section, under more general conditions on X than those in Sections 2–4, we
study the Hausdorff dimensions of the range X([0, 1]N ) = {X(t) : t ∈ [0, 1]N }, the graph
GrX([0, 1]N ) = {(t, X(t)) : t ∈ [0, 1]N } and the level set X−1 (x) = {t ∈ RN : X(t) = x}
(x ∈ Rp ). The results in this section can be applied to wide classes of Gaussian spatial or
space-time models (with or without stationary increments).

First, consider fractional Brownian motion B H = {B H (t), t ∈ RN } valued in Rp with
Hurst index H ∈ (0, 1). B H (t) is a special example of our model which, however, has
isotropic spectral density. It is known [cf. Kahane (1985)] that

dim Gr B H [0, 1]N = min N + (1 − H)p,

N
H

a.s.

Especially, when p = 1,

dim Gr B H [0, 1]N = N + 1 − H

a.s.

and moreover, for every x ∈ R,

dim (B H )−1 (x) = N − H,

a.s.

The fractal properties of B H have been applied by many statisticians to estimate the Hurst
index H and it is sufficient to choose p = 1. Refer to Hall and Wood (1993), Constantine
and Hall (1994), Kent and Wood (1997), Davis and Hall (1999), Chan and Wood (2000,
2004), Zhu and Stein (2002).

Let (H 1 , . . . , H N ) ∈ (0, 1]N be a constant vector. Without loss of generality, we assume
30

that they are ordered as
0 < H 1 ≤ H 2 ≤ · · · ≤ H N ≤ 1.

(2.26)

We assume the following conditions.
(D1) For any η > 0, there exist positive constants δ0 , c11 ≥ 1 such that for all s, t ∈ [0, 1]N
with |s − t| ≤ δ0
N

c−1
11

2H j +η

|sj − tj |

≤ E X(t) − X(s)

2

N

≤ c11

j=1

2H j −η

|sj − tj |

.

(2.27)

j=1

(D2) For any constant ε ∈ (0, 1), there exists a positive constant c12 such that for all u, t
∈ [ε, 1]N , we have
N

Var ( X(u) | X(t)) ≥ c12

uj − tj

2H j +η

.

(2.28)

j=1

The following theorems determine the Hausdorff dimensions of range, graph and level sets
of X. Because of anisotropy, these results are significantly different from the aforementioned
results for fractional Brownian motion or other isotropic random fields [cf. Xiao (2007)].
Even though Theorems 2.5.1 and 2.5.2 below are similar to Theorems 6.1 and 7.1 in Xiao
(2009), they have wider applicability. In particular, they can be applied to a random field
X which may be smooth in certain (or all) directions.
Theorem 2.5.1. Let X = {X(t), t ∈ RN } be an (N, p)-Gaussian random field defined by
(2.25). If the coordinate process X satisfies Condition (D1), then, with probability 1,
N

dim X [0, 1]N = min p;

31

1
,
Hj
j=1

(2.29)

and
k

dim GrX

where

Proof

0
1
j=1 H
j

[0, 1]N

= min

1≤k≤N

N

Hk
1
+ N − k + (1 − H k )p;
,
Hj
Hj
j=1
j=1

(2.30)

:= 0.

The right inequality in (2.27) and Theorem 2.4.9 show that X(t) satisfies a uniform

H¨lder condition on [0, 1]N which, in turn, implies the desired upper bounds in (2.29) and
o
(2.30).
The lower bounds for dim X [0, 1]N and dim GrX [0, 1]N can be derived from the left
inequality in (2.27) and a capacity argument. See the proof of Theorem 6.1 in Xiao (2009)
for details.
For the level sets of X, we have
Theorem 2.5.2. Let X = {X(t), t ∈ RN } be an (N, p)-Gaussian random field defined by
(2.25). If the coordinate process X satisfies Conditions (D1) and (D2), then the following
statements hold:
(i) If
(ii) If

N
1
p
−1
j=1 H < p, then for every x ∈ R \{0}, X (x) = ∅ a.s.
j
N
1 > p, then for any x ∈ Rp , with positive probability
j=1 H
j

k

dim X−1 (x)

Proof

= min

1≤k≤N

Hk
+ N − k − Hk p .
Hj
j=1

(2.31)

The results (i) and (ii) follow from the proof of Theorem 7.1 in Xiao (2009).

Let X = {X(t), t ∈ RN } be a centered intrinsically stationary Gaussian random field
valued in R with spectral density f (λ) satisfying (2.6). Let H1 , . . . , HN be defined by (2.7).
32

Then, by Theorem 2.3.1 and Lemma 2.3.2, we see that X satisfies (D1) for all H j = 1 ∧ Hj
(1 ≤ j ≤ N ). It also satisfies Condition (D2) with H j = Hj provided Hj ≤ 1 for all
j = 1, · · · , N . Hence one can apply Theorems 2.5.1 and 2.5.2 to derive the following result.
Corollary 2.5.3. Let X = {X(t), t ∈ RN } be a centered intrinsically stationary Gaussian
random field valued in Rp defined by (2.25). We assume that its coordinate process X has
spectral density f (λ) satisfying (2.6) and Hj (j = 1, · · · , N ) defined by (2.7) are ordered as
H1 ≤ H2 ≤ · · · ≤ HN . We have
(i) With probability 1, (2.29) and (2.30) hold with H j = 1 ∧ Hj (1 ≤ j ≤ N ).
(ii) If, in addition, Hj ≤ 1 (so H j = Hj ) for all j = 1, · · · , N and

N
1
j=1 Hj

> p, then

(2.31) holds with positive probability.
We believe that the above fractal properties can also be useful for estimating the parameters H1 , . . . , HN of our model. However this will be more subtle than the isotropic
case, where only the single parameter is involved, for the following two reasons. First, if a
parameter Hj > 1, then the sample function X(t) is smooth in the j-th direction and the
Hausdorff dimensions of X has nothing to do with Hj . In other words, based on fractal
dimensions, a parameter Hj can be explicitly estimated only when Hj < 1.
Secondly, if we let p = 1, then (2.30) gives dimGrX [0, 1]N = N + 1 − H 1 , which does
not give any information about the other parameters H2 , . . . , HN . This suggests that, in
order to estimate all the parameters of an anisotropic random field model, one has to work
with a multivariate random field X as defined by (2.25).

33

2.6

Applications to some stationary space-time models

The above results can be applied to the stationary space-time Gaussian fields constructed
by Cressie and Huang (1999), Gneiting (2002), de Iaco, Myers, and Posa (2002), Ma (2003a,
2003b) and Stein (2005).

2.6.1

Stationary covariance models

Extending the results of Cressie and Huang (1999), Gneiting (2002) showed that, for (x, t) ∈
Rd × R,
K(x, t) =

σ2
1 + a|t|2α

βd/2

exp

−

c|x|2γ
1 + a|t|2α

βγ

,

(2.32)

is a stationary space-time covariance function, where σ > 0, a > 0, c > 0, α ∈ (0, 1],
β ∈ (0, 1] and γ ∈ (0, 1] are constants. It can be verified that the corresponding spectral
measure is absolutely continuous in the space-variable x and discrete in the time-variable t.
See Ma (2003a, 2003b) for more examples of stationary covariance models.
In the following, we verify that the sample functions of these space-time models are
fractals. We will check Conditions (D1) and (D2) first, and then obtain the corresponding
Hausdorff dimension results from Theorems 2.5.1 and 2.5.2.

Proposition 2.6.1. Let X = {X(x, t), (x, t) ∈ Rd × R} be a centered stationary Gaussian
random field in R with covariance function as (2.32). Then for any M > 0, there exist
constants c13 > 0 and c14 > 0 such that

c13 |x − y|2γ + |t − s|2α ≤ E X(x, t) − X(y, s)
34

2

≤ c14 |x − y|2γ + |t − s|2α

(2.33)

and
Var X(x, t) X(y, s) ≥ c13 |x − y|2γ + |t − s|2α

(2.34)

for all (x, t) and (y, s) ∈ [−M, M ]d+1 .
Proposition 2.6.2. Let X = {X(x, t), (x, t) ∈ Rd × R} be a centered stationary Gaussian
random field in R with covariance function as (2.32), and let X be its associated (N, p)random field defined by (2.25). Then, with probability 1,

dim X [0, 1]d+1 = min p ;

1
d
+
.
γ α

(2.35)

And if 0 < α ≤ γ < 1, then

dim GrX [0, 1]d+1 =



 d + 1 + (1 − α)p





γ
 d + α + (1 − γ)p



 1 d
 +
α
γ

1
if p < α ,
1
1
d
if α ≤ p < α + γ ,

(2.36)

1
d
if p ≥ α + γ .

If 0 < γ ≤ α < 1, then


 d + 1 + (1 − γ)p




d+1 =
dα
dim GrX [0, 1]
 γ + 1 + (1 − α)p



 1 d
 +
α
γ

d
if p < γ ,
d
1
d
if γ ≤ p < α + γ ,

(2.37)

1
d
if p ≥ α + γ .

Remark 2.6.3 Applying the method in Luan and Xiao (2011), it is possible to further
determine the exact Hausdorff measure function for X [0, 1]d+1 .
Proposition 2.6.4. Let X = {X(x, t), (x, t) ∈ Rd × R} be a centered stationary Gaussian
random field in R with covariance function as (2.32), and let X be its associated (N, p)35

random field.
d
1
(i) When α + γ < p, then for every x ∈ Rp , X−1 (x) = ∅ a.s.
1
d
(ii) When α + γ > p, if 0 < α ≤ γ ≤ 1, then for any x ∈ Rp , with positive probability

=

1
if p < α ,


 d + γ − γp
α

dim X−1 (x)



 d + 1 − αp

1
α,

if p ≥

(2.38)

and if 0 < γ ≤ α ≤ 1, then for any x ∈ Rp , with positive probability

dim X−1 (x)

=



 d + 1 − γp

d
if p < γ ,

 dα

γ + 1 − αp

2.6.2

if p ≥

(2.39)

d
γ.

Stationary spectral density models

In Section 2.6.1, the stationary space-time models are constructed directly by covariance
functions, which are isotropic in the space variable. Stein (2005) showed that stationary
covariance functions which are anisotropic in space can be constructed by choosing
−ν

d+1

cj aj + |λj

f (λ) =

α
|2 j

,

∀λ ∈ Rd × R,

(2.40)

j=1

where ν > 0, cj > 0, aj > 0 and αj ∈ N for j = 1, · · · , d + 1 are constants such that
d+1
j=1

1
< 2ν.
αj

This last condition guarantees f ∈ L1 (Rd+1 ). Clearly f (λ) in (2.40) satisfies (2.6) with
βj = αj and γ = 2ν. Hence we may apply our results to analyze this class of models,
36

through their smoothness and fractal properties.

Proposition 2.6.5. Let X = {X(x, t), (x, t) ∈ Rd × R} be a centered stationary Gaussian
random field in R with spectral density as (2.40).
(i) If
d+1

2ν >
j=1

2
1
+
,
αj min1≤ ≤d+1 α

then X(x, t) is mean square differentiable and has a version X(x, t) which is sample path
differentiable almost surely.
(ii) X is a fractal [i.e. the sample path of X may have fractional Hausdorff dimension] if
and only if
d+1
j=1

1
< 2ν ≤
αj

d+1
j=1

1
2
+
.
αj min1≤ ≤d+1 α

The Hausdorff dimensions of various fractals generated by this kind of model can also
be computed using Corollary 2.5.3, with Hj = αj ν −

d+1 1
=1 2α

, H j = 1 ∧ Hj for j =

1, · · · , d + 1. We leave the details to an interested reader.

2.7

Proofs

Proof of Proposition 2.2.1
Note that (2.2) is equivalent to RN 1 ∧ |λ|2 f (λ)dλ < ∞. Since |λ|≤1 |λ|2 f (λ)dλ < ∞
is given, it is enough for us to show |λ|>1

dλ
N |λ |βj γ
j=1 j

< ∞ is equivalent to (2.5).

For this purpose, we appeal to the following fact: Given positive constants β and γ, there
37

exists a finite constant c15 such that for all a > 0,

∞
0


1

 c15 a−(γ− β ) if βγ > 1,

dx
=

(a + xβ )γ
 +∞

(2.41)

if βγ ≤ 1.
1

To verify this, we make a change of variable x = a β y to obtain
∞
0

−(γ− 1 ) ∞
dy
dx
β
=a
.
β )γ
β γ
(a + x
0 (1 + y )

Thus (2.41) follows.
First we assume (2.5) holds. Since |λ| > 1 implies that |λj0 | > √1 for some j0 ∈
N
{1, . . . , N }. Without loss of generality we assume j0 = 1. Then by using (2.41) (N − 1)
times we obtain

|λ|>1

≤ 2N
βj γ
N
j=1 |λj |

∞

∞

∞

dλ

dλ1

1
√
N

because β1 γ −

N
1
j=2 βj

1
β γ− β
N −1
N
|λj | j
j=1

0

0
N −2

∞

≤c

dλ2 · · · dλN −1

···

dλ1

1
√
N

γ−

|λ1 |β1

< ∞,

1
N
j=2 βj

> 1. This proves the sufficiency of (2.5).

To prove the converse, we assume (2.5) does not hold. Then there is a unique integer
τ ∈ {1, . . . , N } such that

τ −1 1
i=1 βi

dλ

|λ|>1

<γ≤

≥
β γ
N
0
|λj | j
j=1

τ
1
i=1 βi .
∞

Note that
∞

∞

···
0

1

dλ1 · · · dλN
βj γ
N
j=1 |λj |

.

N −1

By using (2.41) and integrating dλ1 · · · dλτ , we see that the last integral is divergent. This
38

completes the proof.

Proof of Lemma 2.3.2

For any s, t ∈ [−M, M ]N , denote s0 = t, s1 = (s1 , t2 , · · · , tN ), s2 = (s1 , s2 , t3 , · · · , tN ),
ˆ
ˆ
ˆ
· · · , sN −1 = (s1 , · · · , sN −1 , tN ) and sN = s. Let h = s − t
ˆ
ˆ

(h1 , · · · , hN ). By Jensen’s

inequality, (2.4) and (2.8) we can write

E X(s) − X(t)

2

N

2

X(ˆk ) − X(ˆk−1 )
s
s

=E
k=1
N

E X(ˆk ) − X(ˆk−1 )
s
s

≤N

2

k=1
N

= 2N

N
k=1 R
N

≤ 2N
k=1 |λ|≤1

1 − cos(hk λk ) f (λ)dλ
(2.42)
1 − cos(hk λk ) f (λ)dλ
N

+ 2N c5

k=1 |λ|>1

1 − cos(hk λk )

dλ
N
H Q+2
i=1 |λi | i

I1 + I2 .
By using the inequality 1 − cos x ≤ x2 we have
N

|λ|2 f (λ)dλ

h2
k

I1 ≤ 2N

|λ|≤1

k=1

(2.43)

≤ c16 |s − t|2
for some positive and finite constant c16 , which depends on M . To bound the kth integral
in I2 , we note that, when |λ| > 1, either |λk | > √1 or there is j0 = k such that |λj0 | > √1 .
N

39

N

We break the integral according to these two possibilities.

|λ|>1

dλ

1 − cos(hk λk )

N
H Q+2
i=1 |λi | i

∞

≤2

1 − cos(hk λk ) dλk

1
√
N

dλ1 · · · dλk−1 dλk+1 · · · dλN

0

1 − cos(hk λk ) dλk

dλj0

1
√
N

(2.44)

dλ∨
k,j

∞

1

+4

N
H Q+2
i=1 |λi | i

RN −1

0
N
Hi Q+2
i=1 |λi |

RN −2

I3 + I4 ,
where dλ∨ denotes integration in λi (i = k, j0 ).
k,j0
By using (2.41) repeatedly [N − 1 times], we obtain
∞

I3 ≤ c

1 − cos(hk λk )

1
√
N

|λk |2Hk +1


≤c

1
|hk |
1
√
N

dλk

h2 λ2
k k dλ
2Hk +1 k
λk



∞

+

1

2Hk +1
1
|hk | λk

(2.45)

dλk 

≤ c σk (|hk |),
where σk is defined as in (2.12).
Similarly, we use (2.41) N − 2 times to get
∞

1

I4 ≤ c

1 − cos(hk λk ) dλk

0

1
√
N

0

H
λk k

1
1
Hj 2+ H + H
j0
0
k
+ λj
0

∞

dλj0

1
√
N

Hj
2Hj +1+ H 0
0
k
λj
0

1

≤c

dλj0

1 − cos(hk λk ) dλk

≤ c |hk

(2.46)

|2 .

Combining (2.42)–(2.46) yields the upper bound in (2.13) holds for all s, t ∈ [−M, M ]N .
40

Next we prove the lower bound in (2.13). By (2.4) and (2.8) we have

E X(s) − X(t)

where ρ(λ) =

Hj
N
j=1 |λj | ,

2

≥ c4

1 − cos s − t, λ
|λ|>1

dλ
,
ρ(λ)Q+2

(2.47)

2

λ ∈ RN . So, for the lower bound of E X(s) − X(t) , it is

enough to show that for every j = 1, · · · , N and all h ∈ RN , we have

1 − cos h, λ
|λ|>1

dλ
≥ c σj (|hj |),
ρ(λ)Q+2

(2.48)

where c is a positive constant.

We only prove (2.48) for j = 1, and the other cases are similar. Fix h ∈ RN with |h1 | > 0
[otherwise there is nothing to prove] and we make a change of variables
H −1

y = ρ(h)

∀ = 1, · · · , N.

λ,

We consider a subset of the integration region defined by

−1

D(h) = y ∈ RN : |y1 | ∈ [ρ(h)H1 , 1], |y | ≤ 1 and y h > 0 for 1 ≤ ≤ N .

Since ρ(λ) = ρ(y)/ρ(h), we have

1 − cos h, λ
|λ|>1

dλ
≥ ρ(h)2
ρ(λ)Q+2
D(h)

1 − cos

N
=1 h

ρ(h)

N
H
=1 |y |

−H −1

Q+2

y

dy. (2.49)

By using the inequality 1 − cos x ≥ c x2 for all |x| ≤ N , where c > 0 is a constant, and the
fact that h y > 0 for all 1 ≤

≤ N , we derive that the last integral is at least [up to a
41

constant]
N
=1 h

ρ(h)2

ρ(h)

−H −1

y

2

dy

N
H Q+2
=1 |y |
1
1
1
− 2 2
ρ(h)2
h2 ρ(h) H1 y1 dy1
···
H −1 1
ρ(h) 1
0
0
D(h)

≥

dy2 · · · dyN
N
H
=1 |y |

N −1

≥

2
2− H
1
c ρ(h)

h2
1

Q+2

(2.50)

2
y1 dy1

1
H −1
ρ(h) 1

1

+2
H
y1 1 H1

= c σ1 (|h1 |).
This proves (2.48) and hence Lemma 2.3.2.
In order to prove Theorem 2.3.1, we need use the following lemma which implies that the
prediction error of X is determined by the behavior of the spectral density f (λ) at infinity.

Lemma 2.7.1. Assume (2.6) is satisfied, then for any fixed constant M > 0, there exists a
positive and finite constant c17 such that for all functions g of the form
n

g(λ) =

k
ak ei t ,λ − 1 ,

k=1

where ak ∈ R and tk ∈ [−M, M ]N , we have

|g(λ)| ≤ c17 |λ|

RN

for all λ ∈ RN that satisfy |λ| ≤ 1.

42

|g(ξ)|2 f (ξ)dξ

1/2

(2.51)

Proof

By (2.6), we can find positive constants C and η, such that

f (λ) ≥

C
,
|λ|η

∀λ ∈ RN with |λ| large enough.

Then the desired result follows from the proof of Lemma 2.2 in Xiao (2007).

Proof of Theorem 2.3.1
First, let’s prove the upper bound in (2.11). By Lemma 2.3.2 we have

2

Var X(u)|X(t1 ), · · · , X(tn ) ≤ min E X(u) − X(tk )
0≤k≤n

(2.52)

N

σj |uj − tk |
j

≤ c9 min

0≤k≤n

.

j=1

In order to prove the lower bound for the conditional variance in (2.11), we assume that
u, t1 , · · · , tn ∈ [−M, M ]N are arbitrary and denote r ≡ min

0≤k≤n

N
j=1 |uj

H
− tk | j . Working
j

in the Hilbert space setting, the conditional variance is just the square of L2 (P)-distance of
X(u) from the subspace generated by {X(t1 ), · · · , X(tn )}, so it is sufficient to prove that
for all ak ∈ R, 1 ≤ k ≤ n,
n

ak X(tk )

E X(u) −

2

≥ c6 r 2 ,

(2.53)

k=1

where c6 is a positive constant which may only depend on H1 , . . . , HN and N .
By using the stochastic integral representation (2.3) of X, the left hand side of (2.53)
can be written as
n

E X(u) −

ak
k=1

X(tk )

n

2

=

RN

ei u,λ

−1−

ak
k=1

43

k
(ei t ,λ

2

− 1) f (λ) dλ.

(2.54)

Hence, we only need to show
n
RN

where t0 = 0 and a0 = 1 −

ei u,λ

k
ak ei t ,λ

−

2

f (λ) dλ ≥ c6 r2 ,

(2.55)

k=0
n
k=1 ak .

We choose a function δ(·) : RN → [0, 1] in C ∞ (RN ) [the space of all infinitely differentiable functions defined on RN ] such that δ(0) = 1 and it vanishes outside the open set
t ∈ RN :

Hj
N
j=1 |tj |

ˆ
< 1 . Denote by δ the Fourier transform of δ. Then one can verify

ˆ
ˆ
that δ(·) ∈ C ∞ (RN ) as well and δ(λ) decays rapidly as |λ| → ∞.

−1
−1
Let E be the N × N diagonal matrix with H1 , · · · , HN on its diagonal and let δr (t) =

r−Q δ(r−E t) for all t ∈ RN . Then the inverse Fourier transformation and a change of
variables yield
δr (t) = (2π)−N
Since min

N
j=1 |uj

RN

ˆ
e−i t,λ δ(rE λ) dλ.

(2.56)

H
− tk | j : 0 ≤ k ≤ n ≥ r, we have δr (u − tk ) = 0 for k = 0, 1, · · · , n.
j

This and (2.56) together imply that
n

J :=

RN

ei u,λ

−

k
ak ei t ,λ

ˆ
e−i u,λ δ(rE λ) dλ

k=0
n

ak δr (u − tk )

= (2π)N δr (0) −

(2.57)

k=0

= (2π)N r−Q .

Now we split the integral in (2.57) over {λ : |λ| < 1} and {λ : |λ| ≥ 1} and denote the
44

two integrals by I1 and I2 , respectively. It follows from Lemma 2.7.1 that
n

I1 ≤

ei u,λ

k
ˆ
ak ei t ,λ |δ(rE λ)|dλ

−

|λ|<1

k=0
n

≤ c17

RN

ei u,λ

−

ak

1/2

2
k
ei t ,λ f (λ) dλ

ˆ
|λ||δ(rE λ)|dλ
|λ|<1

k=0
n

≤ c18 E X(u) −

ak

X(tk )

(2.58)

2 1/2

,

k=1

ˆ
where the last inequality follows from (2.54) and the boundedness of δ.
On the other hand, by the Cauchy-Schwarz inequality and (2.54), we have
n
2
I2

ei u,λ

≤

−

|λ|≥1

ak

2
i tk ,λ f (λ)dλ
e

k=0
n

ak X(tk )

≤ E X(u) −

2

(2.59)
r−Q

k=1

ak X(tk )

= E X(u) −

1
|λ|≥1

n

1 ˆ E 2
|δ(r λ)| dλ
|λ|≥1 f (λ)

2

r−2Q−2

k=1

f (r−E λ)

ˆ
|δ(λ)|2 dλ

1 ˆ
|δ(λ)|2 dλ.
f (λ)
|λ|≥1

ˆ
The last integral is convergent thanks to the fast decay of δ(λ). Finally, combining (2.57),
(2.58) and (2.59), we get
n

(2π)N r−Q

≤ c19 E X(u) −

ak

X(tk )

2 1/2

r−Q−1 .

k=1

Henceforth (2.53) follows, and the theorem was proved because of (2.52) and (2.53).

Proof of Theorem 2.4.2
For t ∈ RN , it is known that Xj,h =

X(t+hej )−X(t)
h

45

converges in L2 -sense, as h → 0, if

and only if
Dh,k

1
E
hk

X(t + hej ) − X(t) X(t + kej ) − X(t)

converges to a constant as h, k → 0. However,
1
C(t + hej , t + kej ) − C(t, t + kej ) − C(t + hej , t) + C(t, t)
hk
1
=
v(hej ) + v(kej ) − v (h − k)ej .
2hk

Dh,k =

(2.60)

So the first part of the theorem is proved. For the second part, it is clear that if v(t) has
second-order partial derivatives at 0 in the j-th direction then (2.14) holds [thanks to Taylor’s
theorem]. On the other hand, if (2.14) holds, then by taking h = k → 0 in (2.60) we see
that ∂v/∂tj (0) = 0. This fact, together with (2.14), implies that
v((k + h)ej ) − v(kej )
1
∂ 2v
(0) = lim lim
2
h
k→0 k h→0
∂tj
v((k + h)ej ) − v(kej ) + v(hej )
hk
k→0 h→0

= lim lim

exists. This completes the proof of Theorem 2.4.2.

Proof of Corollary 2.4.3
By Theorem 2.4.2 it suffices to show that lim Dh,k exists if and only if (2.15) holds,
h,k→0

i.e.,
N

βj γ −
i=1

1
> 2.
βi

It follows from (2.60) and (2.4) that

Dh,k =

1 − cos hej , λ − cos kej , λ + cos (h − k)ej , λ
f (λ) dλ.
hk
RN
46

(2.61)

To prove the sufficiency of (2.15), we note that for each fixed λ ∈ RN ,
1 − cos(hλj ) − cos(kλj ) + cos((h − k)λj )
= λ2
j
hk
h,k→0
lim

(2.62)

and by the mean value theorem,
1 − cos(hλj ) − cos(kλj ) + cos((h − k)λj )
≤ λ2 .
j
hk

Now we assume (2.15) holds. Then, as in the proof of Proposition 2.2.1, we have
λ2 dλ
j
λ∈RN :|λj |>1

λ2 dλj
j

∞

N
β γ
i=1 |λi | i

≤c
1

βj (γ−

λj

1
i=j βi )

< ∞.

This implies RN λ2 f (λ)dλ < ∞. By (2.61), (2.62) and the dominated convergence theorem,
j
we obtain
lim Dh,k =

h,k→0

RN

λ2 f (λ)dλ.
j

To prove the necessity of (2.15), we assume βj γ −

N 1
i=1 βi

≤ 2. Then, as in the proof

= ∞.

(2.63)

of Proposition 2.2.1, we have
λ2 dλ
j
λ∈RN :|λj |>1

N
β γ
i=1 |λi | i

We let h = k ↓ 0 and use Fatou’s lemma to (2.61) [note the integrand is non-negative] to
derive
lim inf Dh,k ≥
h=k↓0

RN

λ2 f (λ)dλ = ∞,
j

where the last equality follows from (2.63). So lim Dh,k does not exist and the proof is
h,k→0

47

finished.

Proof of Theorem 2.4.6
If v(t) has continuous second-order partial derivatives, then Theorem 2.4.2 implies that X
has mean square partial derivatives in all N directions. Let

X (t)

= X1 (t), · · · , XN (t)

T

and we show that it satisfies (2.20).
For any unit vector u in RN , we can write it as u =
uT X (t) =

N
j=1 uj Xj (t).

X(t + hu) − X(t)
=E
−
h

1
= 2 v(hu) + E
h

N
2
j=1 uj

= 1. So

2

2

N

uj Xj (t)
j=1

N

2

−

uj Xj (t)
j=1
N

2

uj Xj (t)
j=1

and

Hence

X(t + hu) − X(t)
− uT X (t)
E
h

1
= 2 v(hu) + E
h

N
j=1 uj ej

2
h

1
−
h

(2.64)

N

uj EX(t + hu)Xj (t) − EX(t)Xj (t)
j=1
N

uj vj (hu).
j=1

The last equality in (2.64) follows from (2.18).
Since v(t) is an even function with v(0) = 0 and has continuous second-order partial and
mixed partial derivatives, Taylor’s theorem implies

1
v(hu) + v(−hu) − 2v(0)
1
v(hu) = lim
= uT Ω(0)u,
2
2
2
2h
h→0
h→0 h
lim

(2.65)

where Ω(0) is an N × N matrix, with Ω(0) ij = vij (0) for i = j, and Ω(0) ii = vi (0).

48

For the second term in the last line of (2.64), note that for any i, j = 1, · · · , N and i = j,
and any l > 0, m > 0,

E

X(t + lei ) − X(t) X(t + mej ) − X(t)
l
m

1
E X(t + lei )X(t + mej ) − X(t)X(t + mej ) − X(t + lei )X(t) + X 2 (t)
lm
1
=
v(lei ) + v(−mej ) − v(lei − mej ) .
2lm
=

(2.66)

1
Let l → 0, m → 0, then the last term in (2.66) goes to 2 vij (0), where vij (0) is the second-

order mixed partial derivative of v at 0 in the i-th and j-th directions. By Theorem 2.4.4,
we have E Xj (t)

2

1
= 2 vj (0), for j = 1, · · · , N . Hence
N

2

uj Xj (t)

E
j=1

1
= uT Ω(0)u.
2

Finally for the last term in (2.64), we use Taylor’s theorem again to derive

1
lim
h→0 h

N

uj vj (hu) = uT Ω(0)u.
j=1

Combining this with (2.65) and (2.66) shows that (2.64) goes to 0, as h → 0. This completes
the proof.

Proof of Theorem 2.4.8

Under (2.15), Corollary 2.4.3 ensures that the mean square partial derivative Xj (t) exists.
In order to show that Xj (t) has a continuous version, by Kolmogorov’s continuity theorem
or general Gaussian theory [cf. Adler (1981), Adler and Taylor (2007)], it is enough to show
49

there exist constants c20 > 0 and η > 0 such that

E Xj (s) − Xj (t)

2

≤ c20 |s − t|η ,

∀ s, t ∈ [−M, M ]N .

(2.67)

Recall that

K(s, t) =
=

RN
RN

ei s,λ − 1 e−i t,λ − 1 f (λ)dλ
cos s − t, λ − cos t, λ − cos s, λ + 1 f (λ)dλ.

Thanks to (2.15), we derive

∂K(s, t)
=
∂sj
RN

− λj sin s − t, λ + λj sin s, λ f (λ) dλ

and
∂K(s, t)
=
λ2 cos s − t, λ f (λ) dλ.
j
∂sj ∂tj
RN
So

E Xj (s) − Xj (t)

2

= E Xj (s)
=2

RN

2

+ E Xj (t)

2

− 2E Xj (s)Xj (t)

λ2 1 − cos s − t, λ f (λ)dλ.
j

The rest of the proof is similar to that of Lemma 2.3.2. Denote s0 = t, s1 = (s1 , t2 , · · · , tN ),
ˆ
ˆ
s2 = (s1 , s2 , t3 , · · · , tN ), · · · , sN −1 = (s1 , · · · , sN −1 , tN ) and sN = s. Then
ˆ
ˆ
ˆ

E Xj (s) − Xj (t)

2

N

≤N

E Xj (ˆk ) − Xj (ˆk−1 )
s
s
k=1

50

2

N

= 2N
k=1

|λ|≤1

λ2 1 − cos(sk − tk )λk f (λ)dλ
j

+
|λ|>1

λ2 1 − cos(sk − tk )λk f (λ)dλ
j
1 − cos(sk − tk )λk λ2
j

N

≤ c21 |s − t|2 + c22

N
H Q+2
i=1 |λi | i

k=1 |λ|>1

(2.68)

dλ.

Now we estimate the last N integrals in (2.68). For simplicity of notation, we only consider
the case when k = j [the cases of k = j are similar]. Denote hk = sk − tk and, similar to
(2.44), (2.45) and (2.46), we derive
1 − cos(sk − tk )λk λ2
k dλ
N
Hi Q+2
|λ|>1
i=1 |λi |
∞

≤2

≤c

1 − cos(hk λk ) λ2 dλk
k

dλ1 · · · dλk−1 dλk+1 · · · dλN

N
H Q+2
i=1 |λi | i
1
∞
dλ∨
k,j0
2 dλ
+2
1 − cos(hk λk ) λk k
dλj0
N
1
H Q+2
√
0
RN −2
i=1 |λi | i
N
1
∞ 1 − cos(h λ ) λ2
k k
k dλ + c
λ2 1 − cos(hk λk ) dλk
k
k
2Hk +1
1
√
0
λk
N
1
√
N

≤ c23 |hk |2(Hk −1) log

RN −1

1
+ |hk |2 ,
|hk |

thanks to Hk > 1. Combining this with (2.68) proves (2.67).
It follows from (2.67)) that the Gaussian field Xj = {Xj (t), t ∈ RN } has a continuous
version [which will still be denoted by Xj ]. Now we define a new Gaussian random field
X = {X(t), t ∈ RN } by

X(t) = X(t1 , · · · , tj−1 , 0, tj+1 , · · · , tN )+

tj
0

Xj (t1 , · · · , tj−1 , sj , tj+1 , · · · , tN ) dsj . (2.69)

Then we can verify that X is a continuous version of X and, for every t ∈ RN , Xj (t) = Xj (t)
51

almost surely. This amounts to verify that for every t ∈ RN ,

E X(t)2 = v(t)

E X(t) − X(t)

and

2

= 0,

which can be proved by using (2.69), Theorem 2.4.4 and (2.18). Since the verification is
elementary, we omit the details. This proves Part (i) of Theorem 2.4.8.

It remains to prove Part (ii) of Theorem 2.4.8. By applying Part (i) to j = 1, we obtain
(1)

a continuous version X (1) of X such that ∂ X (t) is continuous. Then we apply Part (i) to
∂t1
X (1) with j = 2 and obtain a version X (2) of X (1) defined by

X (2) (t) = X (1) (t1 , 0, t3 , · · · , tN ) +

tj
0

X (1) 2 (t1 , s2 , t3 , · · · , tN ) ds2 .

(2.70)

(2)
(2)
Then ∂ X (t) and ∂ X (t) are almost surely continuous. Repeating this “updating” pro∂t1
∂t2

cedure for j = 3, · · · , N , we obtain a continuous version X (N ) of X such that all first-order
partial derivatives of X (N ) are continuous almost surely. Hence the sample function of
X (N ) is almost surely differentiable in the sense of (2.23). The proof of Theorem 2.4.8 is
complete.

Proof of Proposition 2.6.1
By stationarity, we’ll have

E X(x, t) − X(y, s)
= E X(x, t)

2

2

+ E X(y, s)

2

− 2E X(x, t)X(y, s)

= 2K(0, 0) − 2K(x − y, t − s),
52

as well as

2K(0, 0) − 2K(x, t)
= 2σ 2 −

2σ 2
(1 + a|t|2α )βd/2

exp −

(1 + a|t|2α )βd/2 − exp −
= 2σ 2

c|x|2γ
(1 + a|t|2α )βγ

(2.71)

c|x|2γ
(1 + a|t|2α )βγ

(1 + a|t|2α )βd/2

.

By using Taylor expansion, we can write (2.71) as
c|x|2γ
c|x|2γ
−o
(1 + a|t|2α )βγ
(1 + a|t|2α )βγ
2σ 2
(1 + a|t|2α )βd/2
c|x|2γ
c|x|2γ
βd
+ o(|t|2α ) − o
a|t|2α +
2
(1 + a|t|2α )βγ
(1 + a|t|2α )βγ
= 2σ 2
.
(1 + a|t|2α )βd/2
1 + βd a|t|2α + o(|t|2α ) − 1 +
2

Hence we can find positive constants c24 ≤ c25 such that

c24 |x|2γ + |t|2α ≤ 2K(0, 0) − 2K(x, t) ≤ c25 |x|2γ + |t|2α

(2.72)

for all (x, t) ∈ Rd+1 with |x| and |t| small. Replace x and t in (2.72) by x − y and t − s
respectively; (2.33) follows.

To prove (2.34), we make use of the fact that for any Gaussian random vector (U, V )
with mean 0,
ρ2 − (σU − σV )2 (σU + σV )2 − ρ2
U,V
U,V
,
Var(U |V ) =
2
4σV
2
2
where ρ2 = E (U −V )2 , σU = E(U 2 ) and σV = E(V 2 ). Let U = X(x, t) and V = X(y, s),
U,V

53

we derive

Var X(x, t)|X(y, s) =

K(0, 0) − K(x − y, t − s) K(0, 0) + K(x − y, t − s)
K(0, 0)

≥ c24 |x − y|2γ + |t − s|2α .
This proves (2.34).

Proof of Proposition 2.6.2
Eq. (2.35) follows from Proposition 2.6.1 and Theorem 2.5.1. Then let’s prove (2.36),
where 0 < α ≤ γ ≤ 1. By Proposition 2.6.1 and Theorem 2.5.1, we get
k

dimGrX([0, 1]d+1 )

=

min

1≤k≤d+1

d+1

Hk
1
+ d + 1 − k + (1 − H k )p;
,
Hj
Hj
j=1
j=1

where H 1 = α, H 2 = · · · = H d+1 = γ. Denote
k

S(k) =

Hk
+ d + 1 − k + (1 − H k )p.
Hj
j=1

γ
We have S(1) = d + 1 + (1 − α)p, S(k) = d + α + (1 − γ)p
d+1 1
j=1 H
j

S, for 2 ≤ k ≤ d + 1. Also

1
d
1
= α + γ . We can verify directly that if p < α , then

d+1

S(1) < S <

1
,
Hj
j=1

1
1
which yields dimGrX([0, 1]d+1 ) = S(1). The verifications for the cases α ≤ d < α + d and
1
p ≥ α + d are similar. We omit the details.

54

Proof of Proposition 2.6.4
d
1
By Proposition 2.6.1 and Theorem 2.5.2 we find that when α + γ < p, for every x ∈ Rp ,
1
d
X−1 (x) = ∅ a.s. Also, when α + γ > p, for any x ∈ Rp , with positive probability

k

dim X−1 (x) =

min

1≤k≤d+1

Hk
+ d + 1 − k − Hk p .
Hj
j=1

If 0 < α ≤ γ < 1, we have H 1 = α, H 2 = · · · = H d+1 = γ. Denote
k

T (k) =

Hk
+ d + 1 − k − H k p,
Hj
j=1

γ
then T (1) = d + 1 − αp, T (k) = d + α − γp

T , for 2 ≤ k ≤ d + 1. Since T (1) < T , if and

1
only if p < α , (2.38) follows.

If 0 < γ ≤ α < 1, then H 1 = · · · = H d = γ and H d+1 = α. It follows that T (d + 1) =
dα
γ

+ 1 − αp and T (k) = d + 1 − γp

˜
˜
T , for 1 ≤ k ≤ d. Since T < T (d + 1), if and only if

d
p < γ , we obtain (2.39). The proof is complete.

55

Chapter 3
Criteria for equivalence and
asymptotically optimal predictions

3.1

Introduction

Optimal linear prediction has been widely used in spatial statistics and geostatistics, where
it is known as kriging. In kriging, to guarantee good linear predictors based on an estimated
Gaussian probability measure, it is of great value to be able to distinguish between two orthogonal probability measures and to determine when one can tell which measure is correct
and which is not. Many authors have created various criteria for the equivalence and orthogonality of two Gaussian measures corresponding to one-dimensional Gaussian processes
or Gaussian random fields. The references include Gihman and Skorohod (1974), Ibragimov and Rozanov (1978), Parzen (1963), Chatterji and Mandrekar (1978), Kallianpur and
Oodaira (1963), Yadrenko (1983), Stein (1999b) and so on. In fact, Parzen (1963) developed
an approach for equivalence of two Gaussian measures by using two concepts: the notion of
probability spectral density function and the notion of a reproducing kernel Hilbert space of a
56

time series. Chatterji and Mandrekar (1978) also used the method of RKHS to find sufficient
and necessary conditions for the equivalence of two Gaussian measures in a general setting.
It is worth noting that the approach which uses RKHS has no constrains like stationarity or
isotropy on the underlying process, and the results are applicable to random fields. Ibragimov and Rozanov (1978) obtained the conditions for equivalence of two Gaussian measures
involving the entropy of distributions, and developed the conditions for stationary processes
by associating a Hilbert space spanned by analytic functions. Moreover, given two equivalent
Gaussian processes, Kallianpur and Oodaira (1973) defined the notion of a non-anticipative
representation of one of the processes with respect to the other, for one dimensional case.
Later, Yadrenko (1983) extended Ibragimov and Rozanov’s results to stationary and isotropic
random fields. Du (2009) reviewed of the basic results for the equivalence and orthogonality of two Gaussian measures, and provided a detailed re-proof of Theorem 4 in Yadrenko
(1983), page 156, under the setting of stationary and isotropic random fields. However, in
the literature, there are few explicit results available for the equivalence of two Gaussian
measures in a non-stationary random field, especially for anisotropic cases.
In this chapter, we extend Ibragimov and Rozanov’s method to study intrinsically stationary random fields. We determine the relationships among three corresponding Hilbert
spaces: the random variable space generated by the random field, the reproducing kernel
Hilbert space corresponding to the covariance kernel and the complex function space spanned
by the analytic functions of the form λ → ei λ,t − 1, t ∈ D. Criteria for equivalence and
orthogonality of intrinsically stationary Gaussian random fields are given in terms of their
probability spectral density functions and the structures of their reproducing kernel Hilbert
spaces. The results we have obtained are different from those for stationary processes [see
Ibragimov and Rozanov (1978)]. Moreover, given the equivalence of two random fields, we
57

obtain a representation of one of the random fields with respect to the other. The advantage
is that we can use the equivalent representation instead of the original one whenever the
representation is simpler with respect to some prediction questions.
As we know, in practice, the true probability distribution of our Gaussian model is always
unknown and must be estimated from the gathered data. To this end, it is of great value to
investigate the effect of using a fixed but incorrect probability distribution, especially, when
more sample data can be obtained by sampling the spatial or temporal domain increasingly
densely (fix-domain asymptotics). Actually, the asymptotic optimality of linear predictions
of intrinsically stationary Gaussian models and the convergence rates are established in
this chapter. Moreover, the asymptotic efficient prediction of non-stationary, anisotropic
space-time models with a misspecified probability distribution is studied. The main results
show that under the equivalence of two Gaussian measures, the prediction based on the
incorrect distribution is asymptotically optimal and efficient relative to the prediction under
the correct distribution, as the points of observations become increasingly dense in the study
domain. Our results extend those of Stein (1988, 1990, 1999a, 1999b) which were concerned
with isotropic and stationary Gaussian random fields.
The rest of this chapter is organized as follows. Section 2 studies the relationships among
the three Hilbert spaces we have constructed. In Section 3 we obtain criteria for equivalence
and orthogonality of two Gaussian measures in the intrinsically stationary random fields.
We study the asymptotic optimality of linear predictions in Section 4 and the convergence
rates of the predictors are established in Section 5. In Section 6, we show the proofs of the
main results in this chapter.
In the spatial statistics contexts, one would feel more comfortable using the space model
as X = {X(t), t ∈ Rd } and space-time model as X = {X(x, t), (x, t) ∈ Rd × R}, where d
58

denotes the dimension for the space variable.
In this chapter, we study the asymptotic and prediction properties of the random field
X = {X(t), t ∈ Rd }.

3.2

Three corresponding Hilbert spaces and equivalence

Let X = {X(t), t ∈ Rd } be a real-valued, centered intrinsically stationary Gaussian random
field (i.e. Gaussian random field with stationary increments) with X(0) = 0. We assume
that X has continuous covariance function K(s, t) = E[X(s)X(t)]. As in Chapter 2, K(s, t)
can be represented as

K(s, t) =

Rd

ei s,λ − 1 e−i t,λ − 1 F (dλ),

(3.1)

where F (dλ) is a nonnegative symmetric measure on Rd \ {0} satisfying
|λ|2
F (dλ) < ∞.
2
Rd 1 + |λ|

(3.2)

Moreover, X has the following stochastic integral representation:

X(t) =

Rd

ei t,λ − 1 Φ(dλ),

where Φ(dλ) is a centered complex-valued Gaussian random measure which satisfies

E Φ(A)Φ(B) = F (A ∩ B) and Φ(−A) = Φ(A)
59

(3.3)

for all Borel sets A, B ⊆ Rd with finite F -measure.

Let D be a bounded region in Rd . Without loss of generality, we assume 0 ∈ D. Let
L0 be the linear hull of the complex exponential functions λ → ei λ,t − 1, t ∈ D and take
D
LF (D) to be the closure of L0 under the inner product
D

ϕ1 , ϕ2 F

ϕ1 , ϕ2 L (D) =
ϕ1 (λ)ϕ2 (λ)F (dλ),
F
Rd

where ϕ1 , ϕ2 ∈ L0 . Let HF (D) be the closed linear hull of the random variables X(t),
D
t ∈ D with respect to the inner product

ei λ,s − 1 e−i λ,t − 1 F (dλ).
X(s), X(t) H (D) = K(s, t) =
F
d
R
On HF (D), there exist mean and covariance operators, which we also call m and K, such
that for η1 , η2 ∈ HF (D), E(η1 ) = m(η1 ) and Cov(η1 , η2 ) = K(η1 , η2 ). We will freely switch
between the functions m and K and the operators m and K in the rest of this chapter, the
meaning being apparent from context.

We denote by RK (D) the reproducing kernel Hilbert space (RKHS, for short) of the
random field X(t) with reproducing kernel K(s, t), s, t ∈ D. That is, for every real function
f ∈ RK (D), we have
f, K(·, t) R (D) = f (t),
K

∀ t ∈ D.

In fact, the Hilbert space RK (D) is the closure of the subspace spanned by real functions
K(·, t), t ∈ D, with respect to the inner product ·, · R (D) . Note that RK (D) is separable
K
because of the continuity of K(s, t).
60

Define a mapping ρ from HF (D) onto the RKHS RK (D) such that, for every t ∈ D,

ρ (X(t)) = K(·, t).

(3.4)

It can be proved that ρ is a linear, isometric, one to one mapping. First, we obtain the
following lemma, which gives a representation for random variables in HF (D) with respect
to the analytic functions in LF (D).

Lemma 3.2.1. Each random variable η ∈ HF (D) can be represented as

η(ϕ) =

Rd

ϕ(λ)Φ(dλ)

(3.5)

for some ϕ(λ) ∈ LF (D). For every function ϕ(λ) ∈ LF (D), (3.5) is well defined and
η ∈ HF (D). The mapping φ : η → ϕ is linear, isometric and one to one.

Regarding (3.4), for every η ∈ HF (D) and t ∈ D,

ρ(η)(t) = E (ηX(t)) .

(3.6)

The function t → ρ(η)(t) in (3.6) belongs to RK (D).
Since X(t) = Rd (ei λ,t − 1 Φ(dλ), and η = η(ϕ) = Rd ϕ(λ)Φ(dλ) by Lemma 3.2.1,
(3.6) can be rewritten as

ρ(η)(t) =

Rd

ϕ(λ) e−i λ,t − 1 F (dλ).

Hence, there is a linear, isometric, one to one mapping θ from the Hilbert space LF (D) onto
61

the RKHS RK (D) such that, for every ϕ ∈ LF (D),

θ(ϕ)(·) =

Rd

ϕ(λ) e−i λ,· − 1 F (dλ).

(3.7)

Some remarks about the intrinsically stationary random field follow.
Remark 3.2.2

(1) Not like the case of stationary random fields, the covariance function K(s, t) for an
intrinsically stationary random field can not be represented as the Fourier transform
of the spectral measure F .

(2) The Hilbert space LF (D) for the intrinsically stationary random field is different from
that for stationary case; the latter contains real constants as members.

Let P0 be the Gaussian probability measure on the σ-algebra U(D) generated by X(t) for
t ∈ D, with the second-order structure (0, K0 ) and the spectral measure F0 (dλ). Let P1 be
the Gaussian probability measure on U(D) for a random field X1 (t), with the second-order
structure (0, K1 ), which has the form

X1 (t) = X(t) − m1 (t),

where m1 (t) = E1 (X(t)). Denote the spectral measure of X1 (t) as F1 (dλ), then we have the
following lemma, which is similar to (1.27) of Ibragimov and Rozanov (1978), page 70.

Lemma 3.2.3. Suppose the Gaussian measures P0 and P1 are equivalent on the σ-algebra
62

U(D), then1
ϕ F0

ϕ F1 ,

ϕ ∈ L0 .
D

(3.8)

Lemma 3.2.3 implies that when the Gaussian measures P0 and P1 are equivalent on U(D),
we have LF0 (D) = LF1 (D), HF0 (D) = HF1 (D) and RK0 (D) = RK1 (D).

3.3

Some conditions for equivalence of two Gaussian
measures

Many authors have created various criteria for the equivalence and orthogonality of two
Gaussian measures. Parzen (1963) studied the equivalence and orthogonality of two Gaussian measures using the tools of RKHS RK (D) and determined the corresponding RandonNikodym derivative under two different cases: sure signal case ( with two Gaussian measures
having the same covariance function) and stochastic signal case (with two Gaussian measures having the same mean function), respectively. Chatterji and Mandrekar (1978) also
used the method of RKHS to find sufficient and necessary conditions for the equivalence
of two Gaussian measures in a general setting. It is worth noting that the approach which
uses RKHS has no constraints like stationarity or isotropy on the underlying process, and
the results are applicable to random fields. Kallianpur and Oodaira (1963) gave necessary
and sufficient conditions for equivalence of two Gaussian measures by defining an operator
between the corresponding reproducing kernels (covariance functions), and obtained a nonanticipative representation of one Gaussian process by another. Sottinen and Tudor (2006)
applied Kallianpur and Oodaira (1963)’s idea to investigate the equivalence in law of mul1

ϕ F0

ϕ F1 means 0 < c1 ≤ ϕ F0 / ϕ F1 ≤ c2 < ∞, where c1 and c2 are constants.
63

tiparameter Gaussian processes, i.e. Gaussian random fields, with a Brownian sheet and a
fractional Brownian sheet. They surveyed multiparameter analogous of Hitsuda, Girsonov
and Shepp representations. On the other hand, Ibragimov and Rozanov (1978) investigated
equivalence of stationary processes by using analytic tools, namely, the Hilbert space LF (D).
In this section, we apply Ibragimov and Rozanov’s method to study the criteria for the equivalence of two Gaussian measures in an intrinsically stationary random field and compare our
results with the existing ones.
Let P0 and P1 be Gaussian measures on the σ-algebra U(D) generated by all random variables of the intrinsically stationary random field X(t), t ∈ D. The second-order structures
of P0 and P1 are (0, K0 ) and (m1 , K1 ), respectively. It is known that Gaussian measures
have the following property. See page 77 and 78 of Ibragimov and Rozanov (1978) for more
details and proofs.
Lemma 3.3.1. The Gaussian measures P0 and P1 are equivalent if and only if there exists
a Gaussian measure P such that the pairs P0 and P , and P1 and P are equivalent; for the
equivalent measures P0 and P1 , the density P1 (dω)/P0 (dω) is such that
P (dω) P (dω)
P1 (dω)
= 1
.
P0 (dω)
P (dω) P0 (dω)

Based on Lemma 3.3.1, in this section, we may consider two cases. In case one, the covariance function K0 coincides with K1 , such that ϕ1 , ϕ2 F0 = ϕ1 , ϕ2 F1 , for all ϕ1 , ϕ2 ∈ L0 .
D
From Lemma 1 of Bonami and Estrade (2003), we know that K0 = K1 implies F0 = F1 .
We write F0 = F1 = F in this case. In case two, the mean function m1 (t) ≡ 0, but the
covariance functions K0 and K1 are different. We obtain necessary and sufficient conditions
for P0 and P1 to be equivalent under the two cases, respectively. Let us first consider the
64

case where the two measures differ only in the mean functions.

3.3.1

Case I: Same covariance function

By Lemma 3.2.1, each random variable η ∈ HF (D) can be expressed as η(ϕ) = Rd ϕ(λ)Φ(dλ),
for some ϕ(λ) ∈ LF (D). Denote m1 (ϕ) as the mean of η(ϕ) under the second-order structure (m1 , K1 ). We will then have the following extension of Theorem 3 of Ibragimov and
Rozanov (1978), page 78, where the case of stationary processes is considered.
Theorem 3.3.2. Suppose K0 = K1 , the Gaussian measures P0 and P1 are equivalent on
U(D) if and only if, the mean value m1 (ϕ) is a linear continuous functional on the Hilbert
space LF (D):
m1 (ϕ) = ϕ, ψ F ,

ϕ ∈ L0 ,
D

(3.9)

for some ψ(λ) ∈ LF (D).
As a consequence, we obtain a more explicit necessary and sufficient condition for the
equivalence of two measures which differ only in the mean functions.
Theorem 3.3.3. Suppose K0 = K1 , the Gaussian measures P0 and P1 are equivalent on
U(D) if and only if the mean function m1 (t), t ∈ D, permits a representation as

m1 (t) =

Rd

e−i λ,t − 1 ϕ(λ)F (dλ),

(3.10)

for some ϕ(λ) ∈ LF (D). And in the latter case, the Randon-Nikodym derivative p(ω) =
P1 (dω)/P0 (dω) on the σ-algebra U(D) can be expressed as

p(ω) = exp

Rd

ϕ(λ)Φ(dλ) −

65

1
ϕ 2
F
2

.

(3.11)

Corollary 3.3.4. Under the conditions of Theorem 3.3.3, the Gaussian measures P0 and
P1 are equivalent on U(D) if and only if, the mean function m1 (t), t ∈ D, is in the RKHS
RK (D).

Proof

The proof of Corollary 3.3.4 follows directly from Theorem 3.3.3 and (3.7).

We need to mention that Corollary 3.3.4 is consistent with the results obtained by Parzen
(1963) and Chatterji and Mandrekar (1978), where the tools of RKHS are applied to study
the equivalence and orthogonality of two Gaussian measures. This criterion obtained by
using the method of RKHS is general, has no constraints like stationarity or isotropy on the
underlying process, and the result can be applied to any multi-dimensional case.
We now assume P0 and P1 have spectral densities f0 and f1 . We have known that
K0 = K1 implies f0 = f1 .
Corollary 3.3.5. If K0 = K1 has a bounded density function f (λ), then P0 and P1 are
equivalent on U(D) if and only if, m1 (t), t ∈ D can be extended to all t ∈ Rd and there
exists a square-integrable function ψ on Rd such that

m1 (t) =

Rd

e−i t,λ − 1 ψ(λ)dλ

(3.12)

and
|ψ(λ)|2
dλ < ∞.
Rd f (λ)

(3.13)

Gaussian random fields whose spectral densities are described by a power law model
provide a simple and flexible class of models for inferences. This class includes fractional
Brownian fields as a special case. Ibragimov and Rozanov (1978) obtained necessary and
sufficient conditions for the equivalence of two Gaussian measures with power law densities,
66

under the setting of stationary processes [see Theorems 10 in Chapter III of Ibragimov
and Rozanov (1978)]. Michael Stein stated in a SAMSI workshop in 2010 that Ibragimov
and Rozanov (1978)’s results for stationary random fields might be extendable to certain
nonstationary processes. In the following, we give a necessary condition for the equivalence
under the setting of intrinsically stationary random fields. For the sufficient condition, we
restrict to the one-dimensional case.
Corollary 3.3.6. If f (λ) is bounded and satisfies

f (λ) ≤

K
,
(1 + |λ|2 )n

(3.14)

for some constants K > 0 and n ≥ 1, then a necessary condition for Gaussian measures P0
and P1 to be equivalent on U(D) is that m1 (t) must have partial derivatives in each variable
up to the order n − d+1 . Equivalently, ∀j = 1, 2, · · · , d
2
∂k
k
m (t) =
− iλj e−i t,λ ψ(λ)dλ,
k 1
∂tj
Rd
for all k = 1, 2, · · · , n − d+1 .
2
To obtain a sufficient condition for P0 and P1 to be equivalent, we restrict ourselves to
d = 1.
Corollary 3.3.7. Suppose f (λ) is bounded and the mean function m1 (t) is differentiable on
D = [0, τ ] and m1 (t) can be extended to be a mean-square integrable function on R. If the
Fourier transform ψ(λ) of m1 (t) satisfies ψ ∈ L1 (R) and
|ψ(λ)|2
dλ < ∞,
2
R |λ| f (λ)
67

then P0 and P1 are equivalent on the σ-algebra U([0, τ ]).

3.3.2

Case II: Same mean function

In this subsection, we consider the case where the two Gaussian measures differ only in the
covariance functions. Assume m1 (t) ≡ 0, for all t ∈ D. In analogy to L0 defined before, let
D
i λ,s − 1 e−i µ,t − 1 of s, t ∈ D and λ, µ ∈ Rd .
L0
D×D be the linear hull of the functions e

Take LF ×F (D × D) to be the closure of L0
D×D under the inner product

ϕ1 , ϕ2 F ×F =

Rd ×Rd

ϕ1 (λ, µ) ϕ2 (λ, µ)F (dλ)F (dµ),

where ϕ1 , ϕ2 ∈ L0
D×D . Let HF ×F (D × D) be the closed linear hull of functions X(s)X(t) −
K(s, t), s, t ∈ D, under the second-order structure (0, K). Let us consider the linear space
everywhere dense in HF ×F (D × D) of all variables represented in the symmetric form

ckj X(tk )X(tj ) − K(tk , tj )

η=

(3.15)

k,j

with symmetric real coefficients ckj = cjk , k, j = 1, 2, · · · .

We recall the general formula for products of Gaussians [see Ibragimov and Rozanov
(1978), page 16]:

E X(t1 )X(t2 )X(t3 )X(t4 )
=K(t1 , t2 )K(t3 , t4 ) + K(t1 , t3 )K(t2 , t4 ) + K(t1 , t4 )K(t2 , t3 ).
68

For any variables η1 , η2 of the given type in (3.15), such that

ckj X(tk )X(tj ) − K(tk , tj )

η1 =
k,j

and
ckj X(tk )X(tj ) − K(tk , tj ) ,

η2 =
k,j

we derive that

E(η1 η2 )
ckj cmn K(tk , tm )K(tj , tn ) +

=
k,j m,n

ckj cmn K(tk , tn )K(tj , tm )
k,j m,n

(3.16)

ckj cmn K(tk , tm )K(tj , tn ).

=2
k,j m,n

Let us define a new random measure Ψ(dλ, dµ) as

Ψ(A × B) = Φ(A)Φ(B) − F (A ∩ B),

for all Borel sets A, B ⊆ Rd with finite F -measure, where F (A ∩ B) = E Φ(A)Φ(B) . [see
Section 2.2]
We can see that each variable of the type given in (3.15) can be expressed as

η(ϕ) =

Rd ×Rd

ϕ(λ, µ)Ψ(dλ, dµ),

(3.17)

where
−i µ,tj
ckj ei λ,tk − 1 e
−1 .

ϕ(λ, µ) =
k,j

69

(3.18)

For more details on multiple stochastic integrals, see Major (1981), where systematic accounts on multiple integrals of Gaussian measures are given.
From (3.16), we obtain

E(η1 η2 ) = 2

Rd ×Rd

ϕ (λ, µ)ϕ (λ, µ)F (dλ)F (dµ) = 2 ϕ , ϕ F ×F ,

(3.19)

where
−i µ,tj
ck,j ei λ,tk − 1 e
−1

ϕ (λ, µ) =
k,j

and
−i µ,tj
ck,j ei λ,tk − 1 e
−1 .

ϕ (λ, µ) =
k,j

It is seen from (3.17)–(3.19) that the convergent sequence {ηn ∈ HF ×F (D × D)} is
associated with a sequence of functions {ϕn ∈ LF ×F (D × D)}, and the double stochastic
integral in (3.17) can be extended using L2 -convergence to all ϕ ∈ HF ×F (D ×D). Moreover,
any variable η ∈ HF ×F (D × D) as the limit of a sequence ηn of the type given in (3.15)
can be represented by (3.17), where the function ϕ(λ, µ) ∈ LF ×F (D × D) is the limit of the
corresponding functions ϕn of the type given in (3.18). Given any function ϕ(λ, µ), (3.17)
defines a certain variable η ∈ HF ×F (D × D). So we have finished the proof of the following
Lemma.

Lemma 3.3.8. Each random variable η ∈ HF ×F (D × D) can be represented as

η(ϕ) =

Rd ×Rd

ϕ(λ, µ)Ψ(dλ, dµ),

70

(3.20)

for some ϕ(λ, µ) ∈ LF ×F (D × D). Especially, for any s, t ∈ D,

X(s)X(t) − K(s, t) =

Rd ×Rd

ei λ,s − 1 e−i µ,t − 1 Ψ(dλ, dµ).

(3.21)

For every function ϕ(λ, µ) ∈ LF ×F (D × D), (3.20) is well defined and η ∈ HF ×F (D × D).
Similar to the definition of LF ×F (D × D), let us take LF0 ×F1 (D × D) to be the closure
of L0
D×D with respect to the inner product

ϕ1 , ϕ2 F0 ×F1 =

Rd ×Rd

ϕ1 (λ, µ) ϕ2 (λ, µ)F0 (dλ)F1 (dµ),

where ϕ1 , ϕ2 ∈ L0
D×D . For random variables η(ϕ), η(ψ) ∈ HF (D), denote

b(ϕ, ψ) = K0 η(ϕ), η(ψ) − K1 η(ϕ), η(ψ) ,

where K0 , K1 are covariance operators, which are defined in Section 2. The following
theorem gives a criterion for the equivalence of two Gaussian measures with the same mean
function, which is an extension of Theorem 5 of Ibragimov and Rozanov (1978), page 84,
where stationary processes are considered.
Theorem 3.3.9. Gaussian measures P0 and P1 with 0 mean values are equivalent on U(D)
if and only if, b(ϕ, ψ) being a functional on the class of functions ϕ(λ)ψ(µ) ∈ L0
D×D , can be
extended to a linear continuous functional on LF0 ×F1 (D × D).
Proof

The proof is similar to that of Theorem 5 of Ibragimov and Rozanov (1978), page

84. It should also be based on the entropy of Gaussian distribution and the definition of the
LF0 ×F1 (D × D). We omit the proof, and leave it to interested readers.
71

As a consequence, we obtain a more explicit necessary and sufficient condition for the
equivalence of two Gaussian measures which differ only in the covariance functions.

Theorem 3.3.10. Gaussian measures P0 and P1 with 0 mean values are equivalent on U(D)
if and only if, the difference of the two covariance functions b(s, t) = K0 (s, t) − K1 (s, t) can
be expressed as

b(s, t) =

Rd ×Rd

e−i λ,s − 1 ei µ,t − 1 ϕ(λ, µ)F0 (dλ)F1 (dµ)

(3.22)

for all s, t ∈ D, where ϕ(λ, µ) ∈ LF0 ×F1 (D × D). Moreover, the Randon-Nikodym derivative
p(ω) = P1 (dω)/P0 (dω) on the σ-algebra U(D) can be represented as

p(ω) = C exp −

1
2

Rd ×Rd

ϕ(λ, µ)Ψ(dλ, dµ) ,

(3.23)

where C is a normalizing multiplier, and the definition of the double integral in (3.23) is the
same as (3.20).

Let RK0 ×K1 (D × D) be the reproducing kernel Hilbert space corresponding to the kernel
K0 × K1 , which is a function of four variables (s, s1 , t, t1 ) defined by

K0 × K1 (s, s1 , t, t1 ) = K0 (s, t)K1 (s1 , t1 ).

Similar to Corollary 3.3.4, we have the following result, which is consistent with the results in
Parzen (1963) and Chatterji and Mandrekar (1978), where the tools of RKHS are applied to
study the equivalence and orthogonality of two Gaussian measures. It is worth noting that
the criterion obtained by using the method of RKHS is general, which has no constraints
72

like stationarity or isotropy on the underlying process, and the result are applicable to any
multi-dimensional case.

Corollary 3.3.11. Under the conditions of Theorem 3.3.10, the Gaussian measures P0 and
P1 are equivalent on U(D), if and only if b(s, t) = K0 (s, t) − K1 (s, t) is in the RKHS
RK0 ×K1 (D × D).

Proof

The proof is similar to that of Corollary 3.3.4, and it follows directly from Theorem

3.3.10.
We now assume P0 and P1 have spectral densities f0 and f1 , respectively.

Theorem 3.3.12. Gaussian measures P0 and P1 with 0 mean are equivalent on U(D) if
and only if, b(s, t) can be represented as

b(s, t) =

Rd ×Rd

e−i λ,s − 1 ei µ,t − 1 g(λ, µ)dλ dµ

(3.24)

for all s, t ∈ Rd (i.e. b(s, t) is extendable to be a function on Rd × Rd ) and g(λ, µ) satisfies
|g(λ, µ)|2
dλ dµ < ∞.
Rd ×Rd f0 (λ)f1 (µ)

Remark 3.3.13 If f0 (λ) ≤

K
,
(1+|λ|2 )n

Rd ×Rd

for |λ| large, then g(λ, µ) satisfies

1 + |λ|2

n

1 + |µ|2
73

n

|g(λ, µ)|2 < ∞.

(3.25)

This implies that for any k, m = 0, 1, · · · , n − d+1 ,
2

Rd ×Rd



|λ|k |µ|m |g(λ, µ)|dλ dµ
2

|λ|k

≤

Rd

·

dλ ·

1 + |λ|n

Rd ×Rd

|µ|m
1 + |µ|n

Rd

2

1/2
dµ

(1 + |λ|n )2 (1 + |µ|n )2 |g(λ, µ)|2 dλ dµ

1/2

< ∞.

Therefore, the function b(s, t) has all partial derivatives in each variable up to the order
n − d+1 : e.g. ∀ k, m = 0, 1, · · · , n − d+1 ,
2
2
∂ k+m
b(s, t) =
∂sk ∂tm
j

Rd ×Rd

(−iλj )k (−iµ )m e−i( λ,s − µ,t ) g(λ, µ)dλ dµ

for all j, = 1, 2, · · · , d.
When d = 1 and the processes are stationary, Ibragimov and Rozanov (1978) gave a
necessary and sufficient condition for P0 and P1 to be equivalent on U([0, τ ]) in terms of
the (2n)th-order derivative [see Theorem 13 of Ibragimov and Rozanov (1978), page 99]. It
seems to be an open problem whether analogous results still hold for intrinsically stationary
Gaussian random fields (i.e. Gaussian random fields with stationary increments).
In the following, we prove a sufficient condition for the equivalence of the Gaussian
measure P0 and P1 on U([0, τ ]) when d = 1.

Corollary 3.3.14. Assume d = 1,

∂ 2 b(s,t)
∂s∂t

∂ 2 b(s, t)
=
∂s∂t

is the Fourier transform of the form

e−i(λs−µt) ψ(λ, µ)dλdµ
R×R

74

for some ψ(λ, µ) ∈ L1 (R2 ), which satisfies
|ψ(λ, µ)|2
dλdµ.
2 2
R×R λ µ f0 (λ)f1 (µ)

(3.26)

Then P0 and P1 are equivalent on the σ-algebra U([0, τ ]).

For our conjecture, the following is another sufficient condition for P0 and P1 to be
equivalent on the σ-algebra U([0, τ ]), which extends Theorem 17 in Ibragimov and Rozanov
(1978), page 104.
Conjecture 3.3.15 We assume d = 1 and the spectral densities f0 and f1 satisfy the
condition
f0 (λ)

f1 (λ)

(1 + λ2 )−n .

If
f0 (λ) − f1 (λ)
R

2
f0 (λ)

2

dλ < ∞,

then P0 and P1 are equivalent on the σ-algebra U([0, τ ]).
We have listed several criteria for two Gaussian measures to be equivalent in the intrinsically stationary random fields on the above. Then, with these conditions for the equivalence
of two Gaussian measures, you may ask “What if the two measures are equivalent?” Here
is an answer. Theorem 3.2 of Sottinen and Tudor (2006) states that every mean square
continuous Gaussian random field {X(t), P1 } which is equivalent to a given Gaussian random field {X(t), P0 } admits a non-anticipative representation with respect to {X(t), P0 }.
Now, we work to derive an explicit representation under the equivalence of two intrinsically
stationary random fields.

75

Theorem 3.3.16. Suppose Gaussian measures P0 and P1 are equivalent, then {X(t), P1 }
has a representation x(t) with respect to {X(t), P0 }, such that

x(t) = X(t) +

Rd [−∞,λ]

b(µ, λ)Φ(dµ) ei λ,t − 1 dλ,

where b is a square integrable Volterra kernel.

We can find more consequences of the equivalence of two Gaussian measures from the
next section.

3.4

Asymptotic optimality of linear predictions

In practice, the true probability distribution of our Gaussian model is always unknown and
must be estimated from the gathered data. To this end, it is of great value to investigate the
effect of using a fixed but incorrect probability distribution, especially, when more sample
data can be obtained by sampling the spatial or temporal domain increasingly densely. This
section studies the effect of misspecifying the mean and covariance function of a random
field on optimal linear predictions of the random field.
Suppose P0 and P1 are two equivalent Gaussian measures. Write HF0 (D) as H0 (D) for
short in this section. Let h1 , h2 , · · · be a complete system of linearly independent elements
from H0 (D), and take ψ1 , ψ2 , · · · to be the Gram-Schmidt orthogonalization of h1 , h2 , · · ·
under (0, K0 ), such that

K0 (ψj , ψk ) =





1, j = k,
and K1 (ψj , ψk ) =


 0, j = k,





2
σk , j = k,


 0,
76

j = k.

Of course, the closed linear hull of ψ1 , ψ2 , · · · under the inner product defined by (0, K0 )
is H0 (D). Let ψ ∈ H0 (D), then the best linear predictor of ψ given ψ1 , · · · , ψn under
ˆ
(0, K0 ) is ψn = kn Ψn , where Ψn = (ψ1 , · · · , ψn ) and kn = (K0 (ψ, ψ1 ), · · · , K0 (ψ, ψn )) .
ˆ
Let e0 (ψ, n) = ψ − ψn be the prediction error under (0, K0 ). Similarly, define e1 (ψ, n) to be
the error of the best linear prediction with respect to (m1 , K1 ). In the following, we suppose
(m1 , K1 ) to be the presumed second-order structure when in fact (0, K0 ) is the actual secondorder structure. We will then consider the behavior of the best linear predictor as n → ∞.
Conventionally, we assume 0/0 = 0 throughout this section.
First, any ψ ∈ H0 (D) can be written as ψ =
K0 (ψ, ψi ). So

∞ 2
i=1 ci

∞
i=1 ci ψi ,

where ci = ψ, ψi K0 =

< ∞. We can then write
∞

e0 (ψ, n) =

ci ψi .
i=n+1

Define
µj = E1 ψj ,

for j = 1, 2, · · ·

and
bjk = K1 (ψj , ψk ) − K0 (ψj , ψk ),

for j, k = 1, 2, · · · .

The following results of asymptotic theory are from Stein (1988, 1990, 1999a and 1999b),
which hold for any Gaussian random field, including both stationary and intrinsically stationary ones.
Theorem 3.4.1. Suppose P0 and P1 are two equivalent Gaussian measures. As n → ∞,
E1 e0 (ψ, n)2 − E0 e0 (ψ, n)2
= Λn ↓ 0
E0 e0 (ψ, n)2
ψ∈H0 (D)
sup

77

and
E1 e0 (ψ, n)2 − E0 e0 (ψ, n)2
= λn ↑ 0,
E0 e0 (ψ, n)2
ψ∈H0 (D)
inf

where Λn and λn are, respectively, the largest and smallest eigenvalues of the infinite matrix
(bjk + µj µk )∞
jk=n+1 .
Switching the roles of (0, K0 ) and (m1 , K1 ), we can define the corresponding largest and
smallest eigenvalues as Λn and λn , respectively, such that
E0 e1 (ψ, n)2 − E1 e1 (ψ, n)2
= Λn ↓ 0
E1 e1 (ψ, n)2
ψ∈H0 (D)
sup

and
E0 e1 (ψ, n)2 − E1 e1 (ψ, n)2
= λn ↑ 0.
E1 e1 (ψ, n)2
ψ∈H0 (D)
inf

The above theorem comes from Stein (1990), page 855. Moreover, using some elementary
results, we obtain the following results, see Stein (1999b), page 130.

Corollary 3.4.2. Suppose P0 and P1 are two equivalent Gaussian measures. Then
E1 e0 (ψ, n)2 − E0 e0 (ψ, n)2
= 0,
n→∞ ψ∈H (D)
E0 e0 (ψ, n)2
0
lim

sup

E0 e1 (ψ, n)2 − E0 e0 (ψ, n)2
=0
n→∞ ψ∈H (D)
E0 e0 (ψ, n)2
0
lim

sup

and
E0 e1 (ψ, n) − e0 (ψ, n)
lim
sup
n→∞ ψ∈H (D)
E0 e0 (ψ, n)2
0

Switching the roles of (0, K0 ) and (m1 , K1 ), then
78

2

= 0.

E0 e1 (ψ, n)2 − E1 e1 (ψ, n)2
= 0,
n→∞ ψ∈H (D)
E1 e1 (ψ, n)2
0
lim

sup

E1 e0 (ψ, n)2 − E1 e1 (ψ, n)2
=0
n→∞ ψ∈H (D)
E1 e1 (ψ, n)2
0
lim

sup

and
E1 e0 (ψ, n) − e1 (ψ, n)
lim
sup
n→∞ ψ∈H (D)
E1 e1 (ψ, n)2

2

= 0.

0

Taking the observations as ψ1 , ψ2 , · · · , which form a basis of the Hilbert space H0 (D) is
convenient mathematically, but in fact excludes some common and interesting applications
in the asymptotics. In real life, we care more about the prediction for an unknown value
X(t), t ∈ D, based on the observations X(t1 ), · · · , X(tn ), where t1 , · · · , tn ∈ D but different
ˆ
from t. Let us take Xi (n) for i = 0, 1 to denote the best linear predictor of X(t), using
ˆ
(mi , Ki ) as the second-order structure, where m0 ≡ 0. Define ei (n) = X(t) − Xi (n), the
error of the corresponding prediction. We obtain the following results which are directly
related to Corollary 3.4.2.
Corollary 3.4.3. Suppose P0 and P1 are two equivalent Gaussian measures. Let t ∈ D
and {ti }∞ be a sequence in D not containing t but having t as its limit point, such that
i=1
E0 e0 (n)2 > 0. Then
lim

E1 e0 (n)2 − E0 e0 (n)2
= 0,
n→∞
E0 e0 (n)2

(3.27)

E0 e1 (n)2 − E0 e0 (n)2
=0
n→∞
E0 e0 (n)2

(3.28)

lim

and
E0 e1 (n) − e0 (n)
lim
n→∞
E0 e0 (n)2
79

2

= 0.

(3.29)

The corollary above follows directly from Theorem 10 [Stein (1999b), page 132], and
switching the roles of (0, K0 ) and (m1 , K1 ) is also feasible. Note that, the assumption
of E0 e0 (n)2 → 0 as n → 0 in Theorem 10 [Stein (1999b), page 132] is guaranteed by
mean-square continuity of X(t), t ∈ D, under P0 . We obtain Equation (3.28), saying there
ˆ
is an asymptotically efficient predictor X1 (n) under the presumed second-order structure
(m1 , K1 ) when, in fact, (0, K0 ) is the correct second-order structure, as long as P0 and
P1 are equivalent. Moreover, the predictions obtained under those two second-order structures are asymptotically close to each other [see (3.29)], and the discrepancy between the
presumed mean-squared prediction error and the actual mean-squared prediction error is
asymptotically 0 [see (3.27)].

3.5

Explicit bounds with equal covariance functions

In this section, we want to obtain the bounds on Λn and λn of Theorem 3.4.1 for intrinsically stationary Gaussian random fields. The bounds can be obtained by approximating an
element of a Hilbert space by an element of a finite-dimensional subspace. This problem has
been considered as it applies to optimal design for estimating the regression coefficients of
a stochastic process. The references include Sacks and Ylvisaker (1966, 1968, 1970), Wahba
(1971, 1974) and Eubank, Smith and Smith (1981). Stein (1990) obtained results on these
bounds for less smooth mean functions than those considered in previous work for stationary, second-order random fields. We extend Stein (1990)’s method to investigate the bounds
for intrinsically stationary random fields. Actually, the general case appears to be rather
difficult, however, it simplifies considerably under equal covariance functions, like the case I
80

in Section 3.3.1. From Stein (1990), page 857, we have
∞

µ2
j

Λn =

and λn = 0.

j=n+1

Moreover, Λn has an upper bound as follows:

Λn ≤ E0 (ν − νn )2 ,

(3.30)

for some νn ∈ Hn (D) (the subspace of HF (D) generated by ψ1 , · · · , ψn ), where ν is the
Randon-Nikodym derivative of P1 with respect to P0 . In the following, we derive bounds
on Λn under certain conditions on F , by using the characteristics of the associated function
space LF (D). Let us start from the one-dimensional process.

3.5.1

One-dimensional Processes

Let D = [0, τ ], for τ > 0. Suppose F (dλ) = f (λ)dλ, and f (λ) satisfies R (1 ∧ |λ|2 )f (λ)dλ <
∞ and for a positive integer m,

f (λ)

(1 + λ2 )−m .

(3.31)

Theorem 3.5.1. Under the condition (3.31), all elements of the function space LF (D) can
be expressed as
τ

ϕ(λ) = P (iλ) + (1 + iλ)m−1

(eiλt − 1)c(t)dt,

(3.32)

0

with
m−1

ck (iλ)k ,

P (iλ) =
k=1

81

(3.33)

where ck ’s are real and c(t) is a square-integrable real function on D = [0, τ ].

Remark 3.5.2
(1) If the condition (3.31) changes with m as a positive non-integer number, then Theorem
3.5.1 still holds, by replacing m with its integer part m in (3.32) and (3.33).
(2) The conclusion of Theorem 3.5.1 still holds well under the following weaker condition
than (3.31):
f (λ) ≤ βλ−2m ,

as |λ| → ∞,

where β > 0, m > 0 are constant. We can derive this statement by applying Lemma
2.7.1 to the proof of Theorem 3.5.1.
(3) The analytic function space LF (D) for an intrinsically stationary random field is different from that of a stationary random field. All elements in LF (D) of a stationary
random field under condition (3.31) are given as
m−1

ck (iλ)k + (1 + iλ)m
k=0

τ

eiλt c(t)dt,

(3.34)

0

where ck ’s and c(t) are the same in (3.32) [see Stein (1990)]. As we can see from (3.34)
the Hilbert space LF (D) for the stationary case contains real constants as members.
In fact, if the spectral density f (λ) satisfies (3.31), the Gaussian process X(t) has (m − 1)th
mean-square derivative. Besides, Theorem 2.4.8 in Chapter 2 shows that the sample functions
are differentiable up to (m − 1) orders. Without loss of generality, we will take τ = 1 in the
following. Let Hn,p be the subspace generated by X (j) (tk ) for j = 0, · · · , p with p ≤ m − 1
and 0 = t0 < · · · < tn = 1. Let Ln,p be the subspace of LF (D) isomorphic to Hn,p , and let
82

Pn,p be the operator that projects elements of LF (D) onto Ln,p , so that

inf

ϕn ∈Ln,p

ϕ − ϕn 2 = ϕ − Pn,p ϕ 2
F
F

(3.35)

for all ϕ ∈ LF (D). From Theorem 3.4.1 and (3.30), for any ϕn ∈ Ln,p , ϕ − ϕn 2 is an
F
uniform bound for Λn . Define ∆k = tk −tk−1 , for k = 1, · · · , n. Assume f (λ) satisfies (3.31).
From Theorem 3.3.3, we know that a necessary and sufficient condition for the equivalence
of Gaussian measures P0 and P1 with the same covariance function is
1

m1 (t) =

e−iλt − 1 ϕ(λ)f (λ)dλ,

(3.36)

0

for some ϕ(λ) ∈ LF (D), which can be written as (3.32). In the rest of this subsection, we
assume m1 (t) can be represented as (3.36). Then we obtain the following upper bounds,
which extend Theorem 4.1 and Theorem 4.2 of Stein (1990), page 859.

Proposition 3.5.3. Suppose there exists

≤ m, such that c(t) given in (3.32) has an

absolutely continuous ( − 1)th derivative and c( ) (t) is square-integrable on [0, 1]. Let h(t) =
c(t)e−t , then

ϕ − Pn,m−1 ϕ

2
F

≤ c ( − 1)!

−2

n

∆k 2
2

k=1

tk

h( ) (t)2 dt,

(3.37)

tk−1

where c is a positive constant. Moreover, if ∆k = 1/n for all k, then the upper bound in
(3.37) can be written as
1

−2
c n−2 2−2 ( − 1)!
0

83

h( ) (t)2 dt.

Remark 3.5.4 If we specify (3.31) as that there exist two positive constants α and β, such
that
α(1 + λ2 )−m ≤ f (λ) ≤ β(1 + λ2 )−m ,

(3.38)

then the constant c in (3.37) can be expressed as 4πβe2 .

Proposition 3.5.5. Suppose ϕ(λ) given in (3.36) is of the form
1

ϕ(λ) =

c(t) eiλt − 1 dt,

0

where |c(t)| is uniformly bounded by C on [0, 1]. Let ρ = n max{∆k , 1 ≤ k ≤ n}. Then for
m > 1,
16βnC 2
4ρ(m − 1) 2 −2m+1
2m
ϕ − Pn,0 ϕ 2 ≤
2ρ(m − 1)
max 1,
n
.
F
2n − 1
m!

3.5.2

Two-dimensional random fields

We now give an analogue to Theorem 3.5.1 for two-dimensional random fields. The extension
to high dimensions is not difficult. For convenience, let us introduce the separable random
field first.
Suppose D = [0, τ ] × [0, τ ] for τ > 0, the spectral density satisfies that for λ = (λ1 , λ2 ) ∈
R2 and positive integers m1 , m2 ,

f (λ)

1 + |λ1 |2

−m1

1 + |λ2 |2

−m2
.

(3.39)

Theorem 3.5.6. Under the condition given by (3.39), all elements of the function space
84

LF (D) can be expressed as

ϕ(λ) = P (iλ) + Q(λ) + (1 + iλ1 )m1 −1 (1 + iλ2 )m2 −1

(eiλ t − 1)c(t)dt,

(3.40)

D

for
m1 −1 m2 −1

j

ajk ij+k λ1 λk
2

P (iλ) =
j=0

(3.41)

k=0

and
m2 −1

Q(λ) =

ak (iλ2 )k (1 + iλ1 )m1 −1

k=0
m1 −1

+

τ
0

(eiλ1 t1 − 1)b1 (t1 )dt1
(3.42)

ak (iλ1 )j (1 + iλ2 )m2 −1
˜

j=0

τ
0

(eiλ2 t2 − 1)b2 (t2 )dt2 ,

where a00 = 0, ajk ’s, ak ’s and ak ’s are real, b1 (t1 ), b2 (t2 ) are square-integrable real functions
˜
on [0, τ ] and c(t) is a square-integrable real function on D = [0, τ ] × [0, τ ].

Remark 3.5.7
(1) If the condition (3.39) changes with m1 and m2 as positive non-integer numbers, then
Theorem 3.5.6 still holds, by replacing m1 (m2 ) with its integer part [m1 ] ([m1 ]) in
(3.40), (3.41) and (3.42).
(2) The conclusion of Theorem 3.5.6 still holds under the following weaker condition than
(3.39):
−2m1 −2m2
λ2
,

f (λ) ≤ βλ1

as |λ1 |, |λ2 | → ∞,

where β > 0 is constant. We can derive this statement by applying Lemma 2.7.1 to
the proof of Theorem 3.5.6.
85

For the two-dimensional case, we can have similar results to Propositions 3.5.3 and 3.5.5.
We leave it to interested readers.

3.6

Proofs

Proof of Lemma 3.2.1
Let us prove the representation of η ∈ HF (D) first. By (2.3), it is obvious that ∀ t ∈ D,
η = X(t) satisfies (3.5), with ϕ(λ) = ei λ,t − 1 ∈ LF (D). Then for any positive integer m,
the linear combination of X(t), such as η =

m
k=1 ck X(tk ),

where ck ’s are real and tk ’s ∈ D,

can be written as
m

ck

Rd

k=1

ei λ,tk − 1 Φ(dλ)

m

=

Denote ϕ(λ) =
that for any η1 =

m
k=1 ck

Rd k=1

ei λ,tk − 1 ∈ LF (D), then (3.5) is satisfied. We can also verify

m
k=1 ck X(tk )
m

η1 , η2 =

∈ HF (D), η2 =

ck d

Rd

n
=1 d

X(t ) ∈ HF (D),

n

k=1 =1
m

=

ck ei λ,tk − 1 Φ(dλ).

Rd

ei λ,tk − 1 ei λ,t − 1 F (dλ)
n

ck ei λ,tk − 1
k=1

d ei λ,t − 1 F (dλ)

(3.43)

=1

= ϕ1 , ϕ 2 F ,

where ϕ1 (λ) =

m
k=1 ck

ei λ,tk − 1 ∈ LF (D), ϕ2 (λ) =
86

n
=1 d

ei λ,t − 1 ∈ LF (D). It

follows at once that if there is a sequence {ηn } ∈ HF (D) of the form
mn

ηn =

ckn X(tkn ),
k=1

with a limit point η ∈ HF (D), i.e.,

E|ηn − η|2 → 0,

as n → ∞,

there should exist a corresponding sequence {ϕn } ∈ LF (D), which can be written as
mn

ckn ei λ,tkn − 1 ,

ϕn =
k=1

such that
ϕn − ϕ 2 =
F

Rd

|ϕn (λ) − ϕ(λ)|2 F (dλ) → 0,

as n → ∞,

where ϕ ∈ LF (D). In fact, ϕ depends only on η instead of the sequence {ηn } ∈ LF (D),
such as η = Rd ϕ(λ)Φ(dλ). So (3.5) holds for any η ∈ HF (D).
By (3.43) and a similar limiting argument, we see that for any function ϕ ∈ LF (D), (3.5)
is well defined, which then yields η ∈ HF (D).

Proof of Lemma 3.2.3
It is clear that if ϕ F0 = 0 and ϕ F1 = 0 for a function ϕ(λ) ∈ L0 , the measures P0
D
and P1 are orthogonal, since the corresponding random variable η(ϕ) ∈ HF (D) from (3.5)
satisfies
P0 {η(ϕ) = 0} = 1 and P1 {η(ϕ) = 0} = 0.
87

Furthermore, if there exists a sequence {ϕn (λ)} ∈ L0 , such that
D

ϕn F0 = 1 and σn = ϕn F1 → 0,

as n → ∞,

then for m1 (ϕn ) = E1 (η(ϕn )), we show that as n → ∞,

P0 {|η(ϕn ) − m1 (ϕn )| <

√

P1 {|η(ϕn ) − m1 (ϕn )| <

σn } =

√

√
|x−m1 (ϕn )|< σn

σn } =

2
1
√ e−x /2 dx → 0,
2π

2
1
√ e−x /2 dx → 1.
√
|x|<(1/ σn ) 2π

Similar relations hold true if

ϕn F1 = 1 and ϕn F0 → 0,

as n → ∞.

Hence, the desired result follows.

Proof of Theorem 3.3.2
First of all, suppose P0 and P1 are two equivalent Gaussian measures. We first prove
that the linear functional m1 (·) is bounded. Let {ϕn (λ)} be a sequence in L0 , such that
D
σn = ϕn F1

ϕn F0 = 1. Suppose m1 (ϕn ) → ∞ as n → ∞, then

∞

P0 η(ϕn ) >

m1 (ϕn ) = √

2
1
√ e−x /2 dx → 0,
m1 (ϕn ) 2π

∞

P1 η(ϕn ) >

m1 (ϕn ) =

−m1 (ϕn )+

√

m1 (ϕn )

√

2
2
1
e−x /2σn dx → 1,
2π σn

which imply a contradiction to the equivalence of P0 and P1 . So m1 (ϕ) is a linear bounded
88

functional on the Hilbert space LF (D), which is equivalent to saying that the linear functional
m1 (ϕ) is continuous on LF (D). Hence, ∃ ψ(λ) ∈ LF (D), such that m1 (ϕ) = ϕ, ψ F .
To prove the converse, suppose the mean value m1 (ϕ) is a continuous linear functional on
the Hilbert space LF (D), then there exists a unique ψ ∈ LF (D) such that m1 (ϕ) = ϕ, ψ F ,
for all ψ ∈ LF (D). Let {ϕk } ∈ L0 be a complete orthonormal system in LF (D). It is known
D
that the entropy distance between P0 and P1 on the σ-algebra Un generated by the variables
η(ϕk ), k = 1, · · · , n, is
n

n

rn =

m1 (ϕk

)2

k=1

ϕk , ψ 2 ;
F

=
k=1

see (2.9) of Ibragimov and Rozanov (1978), page 76. So
∞

ϕk , ψ 2 = ψ 2 < ∞.
F
F

lim rn =

n→∞

k=1

The equivalence of the Gaussian measures P0 and P1 follows now from Lemma 3 of Ibragimov
and Rozanov (1978), page 77.

Proof of Theorem 3.3.3
Since the system of functions ϕ(λ) = ei λ,t − 1, t ∈ D, is complete in LF (D), a necessary
and sufficient condition for P0 and P1 to be equivalent is (3.10), which follows from Theorem
3.3.2.
Now suppose P0 , P1 are two equivalent Gaussian measures, and the Randon-Nikodym
derivative is p(ω) = P1 (dω)/P0 (dω) on U(D). Choose a complete orthonormal system ϕ1 (λ),
ϕ2 (λ), · · · ∈ L0 . First, consider the density pn (ω) = P1 (dω)/P0 (dω) on the σ-algebra
D
Un , each of which is generated by the variables η(ϕk ), k = 1, · · · , n. Actually, pn (ω) =
89

E (p(ω)|Un ). By the martingale convergence theorem,

p(ω) = lim pn (ω).
n→∞

n
k=1 ak ϕk (λ).

Let ak = m1 (ϕk ), k = 1, 2, · · · , and ψn (λ) =

By Theorem 3.3.2, there exists

ϕ(λ) ∈ LF (D) such that
ak = ϕk , ϕ F ,

∀ k ≥ 1.

So
n

ψn (λ) =

ϕk , ϕ F ϕk (λ).
k=1

According to formula (2.2) of Ibragimov and Rozanov (1978), page 75, we have
n

pn (ω) = exp
k=1
n

= exp

1
ak η(ϕk ) −
2
ak

k=1

= exp

Rd

Rd

n

a2
k
k=1

ϕk (λ)Φ(dλ) −

ψn (λ)Φ(dλ) −

1
ψn 2
F
2

1
ψn 2
F
2

.

Moreover,
n

lim ψn (λ) = lim

n→∞

n→∞

lim

n→∞

ϕk , ϕ F ϕk (λ) = ϕ(λ),
k=1

ψn 2 = ϕ 2 .
F

F

Therefore, (3.11) holds.

Proof of Corollary 3.3.5
On one hand, if P0 and P1 are equivalent on U(D), then by Theorem 3.3.3, there exists
ϕ ∈ LF (D) such that (3.10) holds. We define ψ(λ) = ϕ(λ)f (λ), then
90

Rd

|ψ(λ)|2 dλ =

Rd

|ϕ(λ)|2 f 2 (λ)dλ ≤ c

Rd

|ϕ(λ)|2 f (λ)dλ < ∞,

where c is a positive constant. So ψ ∈ L2 (Rd ), and (3.10) can be rewritten as (3.12).
Moreover, ψ satisfies (3.13).

On the other hand, suppose there exists ψ ∈ L2 (Rd ), such that (3.12) and (3.13) hold.
We take ϕ(λ) = ψ(λ)/f (λ), then (3.13) implies

Rd

|ψ(λ)|2
dλ < ∞.
Rd f (λ)

|ϕ(λ)|2 f (λ)dλ =

Let ϕ be the projection of ϕ into LF (D), then (3.12) implies that

m1 (t) =

Rd

e−i t,λ − 1 ϕ(λ)f (λ)dλ.

Hence, P0 and P1 are equivalent by Theorem 3.3.3.

Proof of Corollary 3.3.6

Suppose P0 and P1 are equivalent, then there exists ψ ∈ L2 (Rd ) such that (3.12) and
(3.13) hold. By (3.13) and (3.14), we have

Rd

1 + |λ|2

n

|ψ(λ)|2 dλ < ∞.

91

This and H¨lder’s inequality imply that ∀ k ≤ n − d+1 ,
o
2

Rd

≤

|λ|k |ψ(λ)|dλ
|λ|2k

Rd

1 + |λ|n

1/2
2

dλ
2

Rd

1 + |λ|n |ψ(λ)|2 dλ

1/2

< ∞.

Hence the conclusion follows from the dominated convergence theorem.

Proof of Corollary 3.3.7
Since we can write
e−iλs ψ(λ)dλ

m1 (t) =

∀ t ∈ R.

R

Fubini’s theorem gives
t

m1 (t) =

0

t

m1 (s)ds =
t

=

0

e−iλs ψ(λ)dλds
R

e−iλs ds ψ(λ)dλ

0

R

e−iλs − 1

=−
R

ψ(λ)
dλ.
iλ

Since
ψ(λ) 2 1
|ψ(λ)|2
dλ =
dλ < ∞,
2
f (λ)
R iλ
R λ f (λ)
the conclusion follows from Corollary 3.3.5

Proof of Theorem 3.3.10
Since the functions ϕ(λ, µ) = ei λ,s − 1 e−i µ,t − 1 , s, t ∈ D form a complete system
in LF0 ×F1 (D × D), (3.22) is equivalent to the condition given in Theorem 3.3.9.
Now, assume P0 , P1 are equivalent and choose a sequence t1 , t2 , · · · everywhere dense
92

in D. Then {X(tk )} forms a complete system in HF (D). Let us first consider the density
pn (ω) = P1 (dω)/P0 (dω) on the σ-algebra Un , generated by the variables X(tk ), k = 1, · · · , n.
Analogous to (3.11) of Ibragimov and Rozanov (1978), page 89, we have

1
log pn − E log pn = −
2

where (ckj ) = K1 (tk , tj )

−1

n

ckj X(tk )X(tj ) − K(tk , tj ) ,
k,j=1

− K0 (tk , tj )

−1

, difference between the two matrix inverses.

Let
n

ckj X(tk )X(tj ) − K(tk , tj ) .

ηn =
k,j=1

By Lemma 3.3.8, we can also write

ηn =

Rd ×Rd

ϕn (λ, µ)Ψ(dλ, dµ),

where
n

ϕn (λ, µ) =

−i µ,tj
ckj ei λ,tk − 1 e
−1 .

k,j=1

Since
K0 (tk , tj ) (ckj ) K1 (tk , tj ) = b(s, t),
for s, t = t1 , · · · , tn , it can be verified that

Rd ×Rd

e−i λ,s − 1 ei µ,t − 1 ϕn (λ, µ)F0 (dλ)F1 (dµ) = b(s, t),
93

which is rewritten as

s, t ∈ Tn = {t1 , · · · , tn },

ϕn , ϕ0 F0 ×F1 = b(s, t),

where
ϕ0 (λ, µ) = ei λ,s − 1 e−i µ,t − 1 .
For any m ≤ n, ϕm (λ, µ) coincides with the projection of ϕn (λ, µ) ∈ LF0 ×F1 (Tn × Tn ) onto
the subspace LF0 ×F1 (Tm × Tm ), so that

ϕn − ϕm 2 ×F = ϕn 2 ×F − ϕm 2 ×F ≥ 0.
F0 1
F0 1
F0 1

So ϕn 2 ×F is nondecreasing. Since
F0 1

j,k b(tj , tk )

2

< ∞, which follows from the equiva-

lence of P0 and P1 [see (2.20) of Ibragimov and Rozanov (1978), page 81], then limn→∞ ϕn 2 ×F
F0 1
exists. So
lim ϕn (λ, µ) = ϕ(λ, µ) ∈ LF0 ×F1 (D × D).

n→∞

Hence, by the bounded convergence theorem,

ηn → η =

Rd ×Rd

ϕ(λ, µ)Ψ(dλ, dµ),

as n → ∞.

It follows from (1.33) of Ibragimov and Rozanov (1978), page 73 that the limit of E log pn
exists for equivalent Gaussian measures, so denote the limit as log C. Also, analogous to the
proof of Theorem 3.3.3, we have p(ω) = limn→∞ pn (ω). Hence

log p(ω) = log C −

1
2

Rd ×Rd

94

ϕ(λ, µ)Ψ(dλ, dµ),

and the desired result follows.

Proof of Theorem 3.3.12
On one hand, suppose P0 and P1 are equivalent. Then by Theorem 3.3.10, b(s, t) can be
written as: ∀ s, t ∈ D,

b(s, t) =

Rd ×Rd

e−i λ,s − 1 ei µ,t − 1 ϕ(λ, µ)f0 (λ)f1 (dµ)dλdµ

for some ϕ ∈ LF0 ×F1 (D × D).
Define
g(λ, µ) = ϕ(λ, µ)f0 (λ)f1 (µ),
then
|g(λ, µ)|2
dλ dµ
Rd ×Rd f0 (λ)f1 (µ)
=

Rd ×Rd

|ϕ(λ, µ)|2 f0 (λ)f1 (µ)dλ dµ < ∞.

i.e. g satisfies (3.25).
On the other hand, suppose that (3.24) and (3.25) hold. We take

ϕ(λ, µ) =

g(λ, µ)
.
f0 (λ)f1 (µ)

By (3.25), we have

Rd ×Rd

=

|ϕ(λ, µ)|2 F0 (dλ)F1 (dµ)

|g(λ, µ)|2
dλ dµ < ∞.
Rd ×Rd f0 (λ)f1 (µ)
95

i.e. ϕ ∈ LF0 ×F1 (D × D) (or we may take the projection of ϕ onto LF0 ×F1 (D × D)).
Note that we can rewrite b(s, t) as

b(s, t) =

e−i λ,s − 1 ei µ,t − 1 ϕ(λ, µ)f0 (λ)f1 (µ)dλ dµ.

Rd ×Rd

That is, (3.22) holds. It follows from Theorem 3.3.10 that P0 and P1 are equivalent on the
σ-algebra U(D).

Proof of Corollary 3.3.14
We write
s

b(s, t) =
0

t

∂ 2 b(u, v)
du dv
∂u∂v
0

s

t

e−i(λu−µv) ψ(λ, µ) dλ dµ du dv

=
0

0

R×R

e−iλs − 1 e−iµt − 1

=
R×R

ψ(λ, µ)
dλ dµ.
λµ

By (3.26), the function
g(λ, µ) =

ψ(λ, µ)
λµ

satisfies (3.25) in Theorem 3.3.12; the conclusion follows.

Proof of Theorem 3.3.16
By Theorem 3.2 of Sottinen and Tudor (2006), we know that under the equivalence of
P0 and P1 , {X(t), P1 } has a non-anticipative representation x(t) with respect to {X(t), P0 }
and x(t) ∈ HF (D). According to Lemma 3.2.1, there is an isometric isomorphism ζ from
HF (D) to LF (D), such that ζX(t) = ei ·,t − 1, for t ∈ D. By the sample path continuity
96

of X(t), it follows from (4.3) of Kallianpur and Oodaira (1973) that ζx(t) can be written as

ζx(t) = (I + B)ζX(t) = (I + B) ei ·,t − 1 ,

where I is an identity operator and B is a Volterra operator on LF (D). Therefore, we can
write
(I + B) ei ·,t − 1 = ei ·,t − 1 +

Rd

b(·, λ) ei λ,t − 1 dλ,

where b is the Volterra kernel corresponding to the Volterra operator B. We thus obtain the
representation

x(t) =

Rd

ei λ,t − 1 Φ(dλ) +

= X(t) +

Rd [−∞,λ]

Rd Rd

b(µ, λ) ei λ,t − 1 dλΦ(dµ)

b(µ, λ)Φ(dµ) ei λ,t − 1 dλ.

Proof of Theorem 3.5.1
On one hand, we show that the function satisfying (3.32) belongs to LF (D). For any
s ∈ [0, τ ], by the bounded convergence theorem, iλ eiλs is the limit of
eiλ(s+h) − 1 − eiλs − 1
,
h
as h → 0, under the inner product defined on LF (D); and for k = 2, · · · , m − 1, (iλ)k eiλs
being limits of the form
lim

h→0

(iλ)k−1

eiλ(s+h) − eiλs
h

belong to LF (D). Take s = 0, we can show that each polynomial P (iλ) =
97

m−1
k
k=1 ck (iλ)

∈

τ
LF (D), where ck ’s are real. We can show that (1+iλ)m−1 0 (eiλt −1)cs (t)dt is also contained

in LF (D), where
cs (t) =


 t
 e , 0 ≤ t ≤ s,

 0, s < t ≤ τ.

In fact,
τ

(1 + iλ)m−1

s

(eiλt − 1)cs (t)dt = (1 + iλ)m−1
0

0

s

e(1+iλ)t dt −

et dt

0

= (1 + iλ)m−2 (eiλs − 1)es + iλ(1 − es )

is contained in LF (D).

It can also be seen that the linear hull of “step” functions cs (t), for s ∈ [0, τ ] is everywhere
dense in the space L2 ([0, τ ]), which is the space of all square-integrable functions c(t), 0 ≤
t ≤ τ [see Ibragimov and Rozanov (1978), page 30]. Moreover, for any ϕj (λ), j = 1, 2 of the
form
τ

(1 + iλ)m−1
0

(eiλt − 1)cj (t)dt,

where cj (t) is a linear combination of step functions cs (t), we derive from (3.31) that

ϕ1 (λ) − ϕ2 (λ) 2 =
F
≤c

∞

|ϕ1 (λ) − ϕ2 (λ)|2 f (λ)dλ

−∞
∞

1
2
−∞ 1 + λ
τ

≤c
0

τ
0

2

(eiλt − 1)(c1 (t) − c2 (t))dt dλ

|c1 (t) − c2 (t)|2 dt,

where c is a positive constant.

On the other hand, we can get eiλs − 1 by means of iterated integration from (1 +
98

iλ)m−2 e(1+iλ)s − es and
t

m−1
k
k=1 ck (iλ) ,

since

(1 + iλ)m−2 e(1+iλ)s − es ds

0

= (1 + iλ)m−3 e(1+iλ)t − 1 − (1 + iλ)(et − 1)
= (1 + iλ)m−3 e(1+iλ)t − et + (1 + iλ)m−3 iλ)(1 − et ).
τ
So it is proved that the closed linear hull of functions (1 + iλ)m−1 0 (eiλt − 1)cs (t)dt and

(iλ)k , 1 ≤ k ≤ m − 1 forms the space LF (D). Hence the theorem is proved.

Proof of Proposition 3.5.3

First of all, we need to prove the following fact: For l = 0, 1, · · · , m − 2,
tk
tl

(1 + iλ)m−1

e(1+iλ)t − et dt,

k = 1, · · · , n,

(3.44)

0

belong to Ln,m−1 . To verify (3.44), let us first check, for l = 0, 1, · · · , m − 2,
tk

(1 + iλ)m−1

tl e(1+iλ)t dt

0

=(1 + iλ)m−2 tl e(1+iλ)tk − l
k

tk
tl−1 e(1+iλ)t dt
0

=(1 + iλ)m−2 tl e(1+iλ)tk − l (1 + iλ)m−3 tl−1 e(1+iλ)tk
k
k
tk

− (1 + iλ)m−3 (l − 1)

tl−2 e(1+iλ)t dt

0
l

=e(1+iλ)tk

l−j

(1 + iλ)m−j−2 tk (−1)j l!/(l − j)! + (−1)l+1 l!(1 + iλ)m−l−2 .
j=0

99

(3.45)

As a consequence, we show that
tk
tl

(1 + iλ)m−1

e(1+iλ)t − et dt

0
l
l−j

(1 + iλ)m−j−2 tk (−1)j l!/(l − j)! + (−1)l+1 l!(1 + iλ)m−l−2

=e(1+iλ)tk

j=0
tk

− (1 + iλ)m−1

tl et dt

(3.46)

0
l

=etk

eiλtk

−1

l−j

(1 + iλ)m−j−2 tk (−1)j l!/(l − j)! + (−1)l+1 l!(1 + iλ)m−l−2
j=0

l

+ etk

tk

l−j

(1 + iλ)m−j−2 tk (−1)j l!/(l − j)! − (1 + iλ)m−1
j=0

tl et dt.

0

It can be seen that the first term on the second equality of (3.46) is in Ln,m−1 . In order to
get the result of (3.44), we just need to check
l
l−j
tk (−1)j

etk

l!/(l − j)! + (−1)l+1 l!

tk

=

tl et dt,

0

j=0

which is the same as (3.45), with λ = 0. Hence, (3.44) is proved.

Secondly, based on the fact in (3.44), we know that for any constant bjk , j = 0, · · · , m−2
and k = 1, · · · , n
n m−2

(1 + iλ)m−1

bjk
k=1 j=0

tk
0

tj e(1+iλ)t − et dt ∈ Ln,m−1 .

100

m−1
k
k=1 ck (iλ)

And obviously, P (iλ) =

is in Ln,m−1 . Then by (3.32), (3.35) and (3.38)

ϕ − Pn,m−1 ϕ 2

F

∞

1

f (λ) (1 + iλ)m−1

≤
−∞

e(1+iλ)t − et h(t)dt

0
n m−2

− (1 + iλ)m−1

tk
tj

bjk

∞

1

1
2
−∞ (1 + λ )
1

−

≤ 4βπ

−

= 4βπ

2

bjk tj et I{t≤t } dt
k

k=1 j=0

bjk tj et I{t≤t }
k

e2t h(t) −

dt

2

n m−2

k=1 tk−1

dλ

2

n m−2

et h(t) −
tk

k

k=1 j=0

k=1 j=0

0
n

bjk tj e(1+iλ)t I{t≤t } dt

n m−2

et h(t)

dt dλ

n m−2

e(1+iλ)t h(t) −

0

0
1

− et

0

k=1 j=0

≤β

2

e(1+iλ)t

bjl tj

dt.

l=k j=0

Referring to Stein (1990), page 861, we get the desired result.

Proof of Proposition 3.5.5
For m > 1, with any constant ak , k = 0, · · · , n, we have

ϕ − Pn,0 ϕ

2
F

∞

≤β

1

(1 + λ2 )−m

−∞

2

n

c(t)(eiλt

0

− 1)dt −

ak

(eiλtk

− 1) dλ.

(3.47)

k=0

Similarly to page 862 of Stein (1990), the right side of (3.47) can be obtained by polynomial
interpolating quadratures [Krylov (1962), Chapter 6]. Denote

bk =

tm−1
t0

c(t)

u(t)dt
,
(t − tk )u (tk )

101

where k = 0, · · · , m − 1 and u(t) = (t − t0 ) · · · (t − tm−1 ). From Equation (6.1.9) of Krylov
(1962), page 81, we know that for any real function h(t), whose mth derivative is bounded
by M,
tm−1

m−1

c(t)h(t)dt −

t0

bk h(tk ) ≤
k=0

≤

M tm−1
|c(t)h(t)| dt
m! t0
MC
(t
− t0 )m+1 .
m! m−1

Applying this bound to the real and imaginary parts of eiλt − 1 separately, we will have
tm−1

m−1

c(t)

eiλt

bk eiλtk − 1

− 1 dt −

t0

k=0

≤

2C m
|λ| (tm−1 − t0 )m+1 .
m!

The rest of the proof is the same as on page 863 of Stein (1990)

Proof of Theorem 3.5.6

Similarly to the proof of Theorem 3.5.1, we first show that any function which satisfies
(3.40) belongs to LF (D). For any s ∈ D, by the bounded convergence theorem, iλ1 eiλ s and
iλ2 eiλ s are the limits of
1
h

ei(λ s+λ1 h) − 1 − (eiλ s − 1)

and

1
h

ei(λ s+λ2 h) − 1 − (eiλ s − 1) ,

respectively, as h → 0, under the inner product defined on LF (D). For k = 0, · · · , m1 − 1,
j
j = 0, · · · , m2 − 1 and k = j = 0 does not hold, ij+k λ1 λk eiλ s , being the limits of the form
2

1 j+k−1 j−1 k i(λ s+λ h)
1 − eiλ s
i
λ1 λ2 e
h→0 h
lim

102

or
1 j+k−1 j k−1 i(λ s+λ h)
2 − eiλ s
i
λ1 λ2
e
h
h→0
lim

m1 −1
j=0

belong to LF (D). So each polynomial P (iλ) =

m2 −1
j+k λj λk
1 2
k=0 ajk i

∈ LF (D),

where a00 = 0, ajk ’s are real. From the proof of Theorem 6.1, we know that Q(λ) ∈ LF (D).
We can show that

(1 + iλ1 )m1 −1 (1 + iλ2 )m2 −1

(eiλ t − 1)cs (t)dt
D

is also contained in LF (D), where for t = (t1 , t2 ) ∈ D and s = (s1 , s2 ) ∈ D,

cs (t) =


 t1 t2
 e e , 0 ≤ t1 ≤ s1 and 0 ≤ t2 ≤ s2 ,



0,

otherwise.

In fact, for m1 , m2 > 1,
(1 + iλ1 )m1 −1 (1 + iλ2 )m2 −1

(eiλ t − 1)cs (t)dt
D

= (1 + iλ1 )m1 −1 (1 + iλ2 )m2 −1
s1

−
0

0

0

s1
e(1+iλ1 )t1 dt1

0

s2
e(1+iλ2 )t2 dt2

s2
et1 et2 dt1 dt2

= (1 + iλ1 )m1 −2 (1 + iλ2 )m2 −2 e(1+iλ1 )s1 − 1 e(1+iλ2 )s2 − 1
− (1 + iλ1 )(1 + iλ2 )(es1 − 1)(es2 − 1)
= (1 + iλ1 )m1 −2 (1 + iλ2 )m2 −2 es1 +s2 (eiλ s − 1) − es1 (eiλ1 s1 − 1) − es2 (eiλ2 s2 − 1)
− (iλ1 + iλ2 + i2 λ1 λ2 )(es1 es2 − es1 − es2 + 1)
is contained in LF (D). It can also be seen that the linear hull of “step” functions cs (t),

103

for s ∈ D is everywhere dense in the space L2 ([0, τ ] × [0, τ ]), generated by square-integrable
functions c(t), t ∈ [0, τ ] × [0, τ ]. Moreover, for any ϕi (λ), i = 1, 2 of the form

(1 + iλ1 )m1 −1 (1 + iλ2 )m2 −1

D

(eiλ t − 1)ci (t)dt,

where ci (t) is a linear combination of step functions cs (t),
ϕ1 (λ) − ϕ2 (λ) 2 =
F

∞

∞

|ϕ1 (λ) − ϕ2 (λ)|2 f (λ)dλ

−∞ −∞
∞
∞

≤β

1
2 )(1 + λ2 )
−∞ −∞ (1 + λ1
2

≤c
D

2

D

(eiλ t − 1)(c1 (t) − c2 (t))dt dλ1 dλ2

|c1 (t) − c2 (t)|2 dt,

where c is a positive constant.
On the other hand, we can get eiλ s −1 by means of iterated integration from Q(λ), P (iλ)
and (1 + iλ1 )m1 −2 (1 + iλ2 )m2 −2 es1 +s2 (eiλ s − 1). So it is proved that the closed linear hull
of functions expressed as (3.40) forms the space LF (D). Hence the theorem is proved.

104

Chapter 4
Conclusion and future work
In this dissertation, we propose a family of anisotropic space-time intrinsically stationary
Gaussian model. We study the smoothness and fractal properties of the model, all in terms
of the parameters of the models explicitly, and obtain the criteria for two Gaussian measures
to be equivalent in intrinsically stationary random fields. We derive upper and lower bounds
for the prediction errors of the model, and investigate its asymptotically optimal predictions.
This work is of importance in studying the statistical properties of non-stationary Gaussian
random fields.
There are some open problems for future work. First, how to estimate the parameter (now
a vector (H1 , · · · .HN )). Guo, Lim and Meerschaert (2009) develop the local Whittle method
to simultaneously estimate the Hurst index H = (H1 , H2 ) of self-similarity, based on the
asymptotic properties of the spectral density of a stationary and anisotropic random field near
the origin. They prove the consistency of the local Whittle estimators of the long memory
parameters and obtain the asymptotic distribution of the local Whittle estimators. The main
goal here is to construct consistent estimators for (H1 , · · · .HN ) that are applicable in various
space-time modeling and to study their asymptotic normality. It is a great challenge for the
105

realization, since multiple smoothness parameters have to be estimated simultaneously in
our model which has non-stationary and anisotropic properties. In order to estimate all
the parameters of an anisotropic random field model, one has to work with a multivariate
random field X as defined by (2.25).
Second, Gaussian random fields whose spectral densities are described by a power law
model provide a simple and flexible class of models for inferences. Because most of these
random fields are nonstationary, the extensive results available on equivalence of Gaussian
measures for stationary models [see Theorems 10 and 13 in Chapter III of Ibragimov and
Rozanov (1978)] do not apply to them. Basically, the result of Theorem 13 states that
for mean 0 stationary Gaussian processes on the interval [0, T] with two possible spectral
densities f0 and f1 , if f0 (w)(1+w2 )n is bounded away from 0 and ∞ and b is the difference of
the two covariance functions viewed as a function on [0, T ]2 , then the measures are equivalent
if and only if
T
0

T
0

∂ 2n b(s, t)
∂sn ∂tn

2

ds dt < ∞.

It seems to be an open problem whether analogous results still hold for Gaussian random
fields with stationary increments.

106

BIBLIOGRAPHY

107

BIBLIOGRAPHY

[1] Adler, R. J. (1981), The Geometry of Random Fields. Wiley, New York.
[2] Adler, R. J. and Taylor, J. E. (2007), Random Fields and Geometry. Springer, New
York.
[3] Anderes, E. B. and Stein, M. L. (2008), Estimating deformations of isotropic Gaussian
random fields on the plane. Ann. Statist. 36, 719–741.
[4] Aronszajn, N. (1950), Theory of reproducing kernels. Trans. Am. Math. Soc., 68,
337–404.
[5] Banerjee, S. and Gelfand, A. E. (2003), On smoothness properties of spatial processes.
J. Multi. Anal. 84, 85–100.
[6] Banerjee, S., Gelfand, A. E. and Sirmans, C. F. (2003), Directional rates of change
under spatial process models. J. Amer. Statistical Assoc. 98, 946–954.
[7] Berg, C. and Forst, G. (1975), Potential Theory on Locally Compact Abelian Groups.
Springer-Verlag, New York-Heidelberg.
[8] Bierm´, H., Lacaux, C. and Xiao, Y. (2009), Hitting probabilities and the Hausdordorf
e
dimension of the inverse images of anisotropic Gaussian random fields. Bull. London
Math. Soc. 41, 253–273.
[9] Bonami, A. and Estrade, A. (2003), Anisotropic analysis of some Gaussian models.
J. Fourier Anal. Appl. 9, 215–236.
108

[10] Calder, C. A. and Cressie, N. (2007), Some topics in convolution-based spatial modeling. In: Proceedings of the 56th Session of the International Statistics Institute.
Lisbon, Portugal.
[11] Chan, G. and Wood, A. T. A. (2000), Increment-based estimators of fractal dimension
for two-dimensional surface data. Statist. Sinica 10, 343–376.
[12] Chan, G. and Wood, A. T. A. (2004), Estimation of fractal dimension for a class of
non-Gaussian stationary processes and fields. Ann. Statist. 32, 1222–1260.
[13] Chatterji, S. D. and Mandrekar, V. S. (1978), Equivalence and singularity of Gaussian measures and applications. Probabilistic Analysis and Related Topics (Ed. A.T.
Barucha Ried), 1, AP, 169–197.
[14] Constantine, A. G. and Hall, P. (1994), Characterizing surface smoothness via estimation of effective fractal dimension. J. Roy. Statist. Soc. Ser. B 56, 97–113.
[15] Cram´r, H. and Leadbetter, M. R. (1967), Stationary and Related Stochastic Proe
cesses. John Wiley & Sons, Inc., New York.
[16] Cressie, N. (1993), Statistics for Spatial Data (rev. ed.). Wiley, New York.
[17] Cressie, N. and Huang, H.-C. (1999), Classes of nonseparable, spatiotemporal stationary covariance functions. J. Amer. Statist. Assoc. 94, 1330–1340.
[18] Davies, S. and Hall, P. (1999), Fractal analysis of surface roughness by using spatial
data (with discussion). J. Roy. Statist. Soc. Ser. B 61, 3–37.
[19] de Iaco, S., Meyers, D. E. and Posa, D. (2001), Space-Time analysis using a general
product-sum model. Statist. Probab. Letters 52, 21–28.
[20] de Iaco, S., Myers, D. E. and Posa, D. (2002), Nonseparable space-time covariance
models: some parametric families. Math. Geology 34, 23–42.
[21] de Iaco, S., Myers, D. E. and Posa, D. (2003), The linear coregionalization model and
the product-sum space-time variogram. Math. Geology 35, 25–38.
[22] Du, J (2009), Asymptotic and computational methods in spatial statistics. Ph.D.
thesis, Michigan State University.
109

[23] Eubank, R. L., Smith, P.L. and Smith, P.W. (1981), Uniqueness and eventual uniqueness of optimal designs in some time series models. Ann. Statist. 9, 486–493..
[24] Falconer, K. J. (1990), Fractal Geometry – Mathematical Foundations and Applications. Wiley & Sons, New York.
[25] Fuentes, M. (2002), Spectral methods for nonstationary spatial processes. Biometrika
89, 197–210.
[26] Fuentes, M. (2005), A formal test for nonstationarity of spatial stochastic processes.
J. Multi. Anal. 96, 30–54.
[27] Gihman, I. I. and Skorohod, A. V. (1974), The Theory of Stochastic Processes, vol.
1. Springer-Verlag, Berlin.
[28] Gneiting, T. (2002), Nonseparable, stationary covariance functions for space-time
data. J. Amer. Statist. Assoc. 97, 590–600.
[29] Gneiting, T., Kleiber, W. and Schlather, M. (2009), Mat´rn cross-covariance functions
e
for multivariate random fields. Preprint.
[30] Guo, H., Lim, C. and Meerschaert, M. M. (2009), Local whittle estimator for
anisotropic random fields. J. Multi. Anal. 100, 993–1028.
[31] Hall, P. and Wood, A. T. A. (1993), On the performance of box-counting estimators
of fractal dimension. Biometrika 80, 246–252.
[32] Higdon, D. (2002), Space and space-time modeling using process convolutions. In:
Quantitative Methods for Current Environmental Issues (Anderson, C., Barnett, V.,
Chatwin, P.C., El-Shaarawi, A.H., editors), pp. 37–56. Springer-Verlag, New York.
[33] Higdon, D., Swall, J., and Kern, J. (1999), Nonstationary spatial modeling. In:
Bayesian Statistics, (J. M. Bernardo et al, editors), Vol 6, pp. 761–768, Oxford University Press, Oxford, U.K.
[34] Ibragimov, I. A. and Rozanov, Y. A. (1978), Gaussian Random Processes. Springer,
Verlag.
[35] Jones, R. H. and Zhang, Y. (1997), Models for continuous stationary space-time
processes. In: Modelling Longitudinal and Spatially Correlated Data, (T. G. Gregoire,
110

D. R. Brillinger, P. J. Diggle, E. Russek-Cohen, W. G. Warren, and R. D. Wolfinger,
editors), Lecture Notes in Statist. No. 122, pp. 289–298, Springer, New York.
[36] Kahane, J.-P. (1985), Some Random Series of Functions. 2nd edition, Cambridge
University Press, Cambridge.
[37] Kallianpur, G. and Oodaira, H. (1963), The equivalence and singularity of Gaussian
measures. Proc. Sympos. Time Series Analysis. Wiley, New York, 279–291.
[38] Kallianpur, G. and Oodaira, H. (1973), Non-anticipative representations of equivalent
Gaussian processes. Ann. Probab. 1, 104–122.
[39] Kent, J. T. and Wood, A. T. A. (1997), Estimating the fractal dimension of a locally
self-similar Gaussian process by using increments. J. Roy. Statist. Soc. Ser. B 59,
679–699.
[40] Kolovos, A., Christakos, G., Hristopulos, D. T. and Serre, M. L. (2004), Methods for
generating non-separable spatiotemporal covariance models with potential environmental applications. Adv. Water Resour. 27, 815–830.
[41] Krylov, V. I. (1962), Approximate Calculation of Integrals. Macmillan, New York.
[42] Kyriakidis, P. C. and Journel, A. G. (1999), Geostatistical space-time models: a
review. Math. Geology 31, 651–684.
[43] Li, Y. and Xiao, Y. (2010), Multivariate operator-self-similar random fields. Stoch.
Process. Appl., to appear.
[44] Luan, N. and Xiao, Y. (2010), Spectral conditions for strong local nondeterminism
and exact Hausdorff measure of ranges of Gaussian random fields. Submitted.
[45] Ma, C. (2003a), Families of spatio-temporal stationary covariance models. J. Statist.
Plan. Infer. 116, 489–501.
[46] Ma, C. (2003b), Spatio-temporal stationary covariance models. J. Multivariate Anal.
86, 97–107.
[47] Ma, C. (2004), Spatial autoregression and related spatio-temporal models. J. Multivariate Anal. 88, 152–162.
111

[48] Ma, C. (2005a), Spatio-temporal variograms and covariance models. Adv. in Appl.
Probab. 37, 706–725.
[49] Ma, C. (2005b), A class of stationary random fields with a simple correlation structure.
J. Multivariate Anal. 94, 313–327.
[50] Ma, C. (2007), Stationary random fields in space and time with rational spectral
densities. IEEE Trans. Inform. Th. 53, 1019–1029.
[51] Ma, C. (2008), Recent developments on the construction of spatio-temporal covariance
models. Stoch. Environ. Res. Risk Assess. 22 suppl. 1, 39–47.
[52] Major, P. (1981), Multiple Wiener-Itˆ Integrals. Lecture Notes in Math. 849,
o
Springer-Verlag, Berlin.
[53] Meerschaert, M. M., Wang, W. and Xiao, Y. (2010), Fernique-type inequalities and
moduli of continuity of anisotropic Gaussian random fields. Submitted.
[54] Paciorek, C. J. and Schervish, M. J. (2006), Spatial modelling using a new class of
nonstationary covariance functions. Environmetrics 17, 483–506.
[55] Parzen, E. (1963), Probability density functionals and reproducing kernel Hilbert
spaces. Proc. Sympos. Time Series Analysis. John Wiley & Sons, Inc., New York,
155–169.
[56] Potthoff, J. (2010), Sample properties of random fields III: differentiability. Comm.
Stoch. Anal. 4, 335–353.
[57] Rogers, C. A. (1970), Hausdorff Measures, Cambridge University Press.
[58] Sacks, J. and Ylvisaker, D. (1966), Designs for regression problems with correlated
errors. Ann. Math. Statist 37, 66–89.
[59] Sacks, J. and Ylvisaker, D. (1968), Designs for regression problems with correlated
errors: Many parameters. Ann. Math. Statist 39, 49–69.
[60] Sacks, J. and Ylvisaker, D. (1970), Designs for regression problems with correlated
errors. III. Ann. Math. Statist 41, 2057–2074.
112

[61] Schmidt, A. and O’Hagan, A. (2003), Bayesian inference for nonstationary spatial
covariance structure via spatial deformation. J. Roy. Statist. Soc. Ser. B 65, 745–
758.
[62] Sottinen, T. and Tudor, C. A. (2006), On the equivalence of multiparameter Gaussian
processes. J. Theoret. Probab. 19, 461–485.
[63] Stein, M. L. (1988), Asymptotically efficient prediction of a random field with a
misspecified covariance function. Ann. Statist. 16, 55–63.
[64] Stein, M. L. (1990), Uniform asymptotic optimality of linear predictions of a random
field using an incorrect second-order structure. Ann. Statist. 18, 850–872.
[65] Stein, M. L. (1999a), Predicting random fields with increasing dense observations.
Ann. Appl. Probab. 9, 242–273.
[66] Stein, M. L. (1999b), Interpolation of Spatial Data: Some Theory for Kriging.
Springer, New York.
[67] Stein, M. L. (2005), Space-time covariance functions. J. Amer. Statist. Assoc. 100,
310–321.
[68] Wahba, G. (1971), On the regression problem of Sacks and Ylvisaker. Ann. Math.
Statist. 42, 1035–1053.
[69] Wahba, G. (1974), Regression design for some equivalence classes of kernels. Ann.
Statist. 2, 925–934.
[70] Xiao, Y. (2007), Strong local nondeterminism of Gaussian random fields and its applications. In: Asymptotic Theory in Probability and Statistics with Applications, (T.-L.
Lai, Q.-M. Shao and L. Qian, editors), pp. 136–176, Higher Education Press, Beijing.
[71] Xiao, Y. (2009), Sample path properties of anisotropic Gaussian random fields. In:
A Minicourse on Stochastic Partial Differential Equations, (D. Khoshnevisan and F.
Rassoul-Agha, editors), Lecture Notes in Math. 1962, pp. 145–212, Springer, New
York.
[72] Xiao, Y. (2011), Properties of strong local nondeterminism and local times of stable
random fields. In: Seminar on Stochastic Analysis, Random Fields and Applications
VI, pp. 279–310. Progr. Probab., 63, Birkh¨user, Basel.
a
113

[73] Xue, Y. and Xiao, Y. (2011a), Fractal and smoothness properties of anisotropic Gaussian models. Frontiers Math. China, to appear.
[74] Xue, Y. and Xiao, Y. (2011b), Criteria for equivalence and asymptotically optimal
predictions of intrinsically stationary random fields. To be submitted.
[75] Yadrenko, M. I. (1983), Spectral Theory of Random Fields. Optimization Software,
New York.
[76] Yaglom, A. M. (1957), Some classes of random fields in n-dimensional space, related
to stationary random processes. Th. Probab. Appl. 2, 273–320.
[77] Yaglom, A. M. (1987), Correlation Theory of Stationary and Related Random Functions, Vol. 1. Springer-Verlag, New York.
[78] Zhu, Z. and Stein, M. L. (2002), Parameter estimation for fractional Brownian surfaces. Statist. Sinica 12, 863–883.

114