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TOPICS ON THE THEORY OF HOMOGENEOUS RANDOM FIELDS

By

Ahmad Reza Soltani

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Statistics and Probability
198l

ABSTRACT
TOPICS ON THE THEORY OF HOMOGENEOUS RANDOM FIELDS
By
A.R. Soltani

A set of random variables g(t) = {gx(t), x e X, t 6 En}; where
En is the Cartesian product of E with itself n-times and X is
any given set; is called a random field, E being the set of real
numbers R or the set of integers 2. Let E denote the expected
value. It is assumed that ng(t) = 0, E|5x(t)|2 < m as elements of
a Hilbert space H of random variables 5, E|g|2 < m with the scalar
product E5fi3 E: n 6 H; and that X is a linear space. It is also
assumed that gx(t) is linear in x and continuous in t in the
Hilbert space. H. A random field g(t) is called homogeneous if

ng(t)E&(s) depends only on t-s, for all x, y e X. Let M(t) be

the closed linear span of gx(t), x 6 X, in H and M(S) = V M(t)
tES

be the closed linear span of M(t), t 6 S, in H. The dimension of
the field is defined to be the dimension of M(O). A random field is

called r-regular or regular if V n M(s: |s-t| > r)1 = M(E") or
tEE

n M(s: |s| > r) = {0} respectively. A field is called minimal
Y‘>O

if it is r-regular, for r + O.
The following topics on the theory of homogeneous random fiels

are disucssed in this thesis.

(i) Regularities, (ii) L-Markov and Markov properties, (iii)
Interpolation .

In Chapter I we assume that X is a separable Hilbert space,
and give necessary and sufficient conditions for an infinite dimensional
continuous parameter, t e R", random field to be regular. Nold-Cramér
concordance theorem is established. In particular, we prove that the
spectral measure of any regular field is absolutely continuous with
respect to the Lebesgue measure.

Chapter II deals with B(X, Y)-valued homogeneous random fields,
where B(X, Y) is the set of bounded linear operators from the Banach
space X to the Hilbert space Y. Effective conditions for a B(X, Y)-
valued homogeneous continuous or discrete parameter random field to
be regular, r-regular, minimal, L-Markov or Markov is given. This
Chapter extends our results of Chapter I on regularity, and the recent
work of Rozanov of minimality and Markov property on continuous para-
meter (t e R") Hilbert space-valued random fields to the B(X, Y)-
valued random fields. New results for discrete parameter (t E 2")
fields are also given.

In Chapter III the interpolation problem of a finite dimensional
discrete parameter homogeneous random field is discussed. A recipe
formula for the linear interpolator of the random field M(t), t 6 2",
under the assumption that the spectral density and its inverse are
square integrable is obtained.‘ This in the univariate case, extends
the recent work of Salehi and the earlier work of Rozanov, where the

boundedness of the spectral density was assumed.

To my wife Afsaneh and my son Sohrab

ii

ACKNOWLEDGEMENTS

I would like to express my sincere thanks to Professor H.
Salehi for the guidance of this thesis. The advice and encouragement
he gave are greatly appreciated.

Also, I would like to thank Professors V. Mandrekar for his
critical reading of this thesis, C. Ganser and D. Gilliland for serving
on my guidance committee.

Special thanks goes to Mrs. Clara Hanna for her excellent typing
of the manuscript.

Finally, I am grateful to the National Science Foundation and
the Department of Statistics and Probability, for financial support
during my stay at Michigan State University. I am also grateful to
the Department of Mathematics and Statistics of the Shiraz University

which gave me the opportunity of studying at Michigan State University.

TABLE OF CONTENTS

Page

INTRODUCTION .................................................. l
Chapter

I ON REGULARITY OF HOMOGENEOUS RANDOM FIELDS ........ 4

Introduction ...................................... 4

l.l. Regularity .................................. 8

1.2. The Wold-Cramér Concordance ................. 23

II 0N B(X, Y)-VALUED HOMOGENEOUS RANDOM FIELDS ....... 30

Introduction ...................................... 30

2.l. Spectral Representation and Preliminaries... 34

2.2. Regularities ................................ 38

2.3. Completely Minimal Fields ................... 56

2.4. Markov Minimal Fields ....................... 69

III A RECIPE FORMULA FOR THE LINEAR INTERPOLATOR ...... 8l

Introduction ...................................... 8l

3.l. Notations and Priliminaries ................. 82

3.2. A Recipe Formula ............................ 89

BIBLIOGRAPHY ................................................. 97

iv

INTRODUCTION

The main purpose in the thoery of homogeneous random fields
is to study and analyze the behavior of a family of random variables
{gx(t), x 6 X, t 5 En}, where X is any given set and En stands
for the Cartesian product of E with itself n times, E being
the set of real numbers R or the set of integrs Z. The questions
which are raised in this regard have drawn the attention of many
mathematicians and probabilists, and some important results in this
area are included in the work of Helson and Lowdenslager [7 ], Kotani
[l0], McKean E21], Molchan [27], [28], Kotani and Okabe [ll],

Pitt [32], [33] and Rozanov [37], [39]. prics in this theory
include extrapolation theory, interpolation theory, L-Markov and
Markov properties, regularities and prediction on finite domain.

In particular the concepts of L-Markov and Markov properties and
regularities have been investigated by several authors in recent
years, where satisfactory answers have been obtained, c.f. Kallianpur
and Mandrekar [8 J, Makagon and Heron [15 ], Molchan [28], Pitt [33],
Rozanov [39], [40], Salehi and Scheidt [44]. Each topic may be
considered for the univariate fields, multivariate fields, Hilbert
space-valued fields or Banach space-valued fields. In a general
setting with the use of the Kolmogorov isomorphism, the study of

homogeneous random fields reduces to investigate the behavior of a

family of closed subspaces H(t), t 6 En of a Hilbert space Y.
Let V H(t) denote the span closure of H(t), t e E", in Y.

tEE"

It is assumed that the family of operators U t e E", defined on

t!
V H(t) onto V H(t) by U H(s) = H(s+t) are unitary and

tEE" . tGE" t

strongly continuous. The dimension of the field is defined to be

the dimension of H(O). The problems of regularities can be formulated

as obtaining spectral charactrization for the fields with the property

that for a bounded domain sr = {t e E": |t| > r}. v n Ut H(S)*= v H(t)

tEE r téE"
or n H(Sr) = {0}, where H(S) = V H(t) and 1 stands for

r>0 tES
orthogonal complement in H(E"). Such fields are called r-regular or

regular respectively. A field is miminal if it is r-regular,for r + 0.
This thesis consists of three chapters. Each chapter starts
with an introduction which provides ancillary materials to the chapter
and contains a complete description of the topics of the chapter and
the historical background of the related problems. Here is a brief
description of the content of this thesis. In Chapter I we discuss .
finite dimensional as well as infinite dimensional regular fields.
Necessary and sufficient conditions in terms of the spectral density
of the field are obtained. Wold-Cramér concordance theorem is
established. In particular we prove that the spectral measure of a
regular field is absolutely continuous with repsect to the Lebesgue
measure. This chapter extends the work of Rozanov [37] and Pitt [32]
in the univariate case to the infinite dimensional case.
Chapter II gives a complete spectral charactrization for
a Banach space-valued field or more generally for a B(X,Y)-valued

field to be r-regular, regular, minimal, L-Markov or Markov, where

B(X, Y) is the class of bounded linear operators on a Banach space

X into a Hilbert space Y. This chapter extends the work of Rozanov [39]
on continuous parameter (t e R") Hilbert space-valued random fields

to the B(X, Y)-valued random fields. New results on discrete fields

(t e Z") in this regard are also obtained. The techniques in Rozanov's
work [39], and the existence of a square root for a nonnegative
operator-valued function from a Banach space into its dual, [13],

[25] are used in this Chapter.

Chapter III deals with the interpolation problem of a finite
dimensional homogeneous discrete parameter random field H(t), t 6 Z".
We obtain a recipe formula for the linear interpolator of the random
field H(t), t e Z". More precisely let {xk}. k = l,...,q, be an
orthonormal basis in H(O). We assume that each xk, k = l,...,q
does not belong to (txo H(t))V(H(0)\xk), where H(0)\xk = V{x2,

2 = l,...,q, 2 f k} (we call such a field a field with imperfect
interpolation, this terminalogy is being adapted from Dym and McKean
[5 J). Then we give a recipe formula for expressing the linear
projection of xk, k e Tk on V{x£: x 4 T1, 1 = l,...,q} as an
infinite series expansion, where it is assumed that all the elements
of x£(t), 2 = l,...,q are known except for the values x£(t),

t 6 T2, 2 = l,...,q; T2, 2 = l,...,q are finite domains in z".

This result constitutes an extension of the recent work of Salehi m2 1,
where similar recipe formula is obtained for univariate fields under
much stronger assumption. This problem was first studied by Rozanov

in 1960 [35].

CHAPTER I
ON REGULARITY OF HOMOGENEOUS RANDOM FIELDS

Introduction. Consider a family of real or complex-valued random

 

variables §x(t), over a probability space (9, B, P) where the
index x runs through a set X and t is a point in R"; we call
g(t) = {gx(t), x e X, t E Rn} a random field. Let E denote the
expected value. We assume that E gx(t) is zero and the correlation
function E gx(s)'§;TtT is continuous and is invarriant with respect
to simultaneous translation of s and t, for arbitary x, y e X.
In this case the random field g(t) is called homogeneous in the
wide sense.

Now let X to be a separable Hilbert space and let M(t)
be the closed linear span of the variables gx(t), x e X, considered
as elements of the Hilbert Space L2(O, B, P) of random variables
g, Elglz < m- with scalar product} E 513?. Since M(t) contains
the complete information about a at t, it is natural to call
M(t), t 6 Rn a homogenous random field. {As an example of gx(t)
we can consider an X-valued Gaussian random process 5(t) and
define gx(t) to be the inner product of 5(t) with x in X
this family is called an X-valued Gaussian family}. Following the
work [:3], [l7], [Bl], D38]. [39]. [43], [46], etc., we may assume

that gx(t), t e R", is linear in the variable x.

4

Let Ut’ t e R”, be a continuous group of unitary operators
defined by the relation Utgx(s) = gx(t + s) on the closed linear
2

(

span M(R") of all variables gx(t), x e X, t e Rn in L Q, B, P),

evidently M(t) = UtM(O).

Here we assume that there is a spectral density f(A), a
bounded linear positive operator-valued function of the variable

X 6 Rn acting on the Hilbert space X such that

E§x(s)E;TTT = I eix(s't)(f(x)x. y)dA 5. tie R"; x. y e x.
Rn

Note that (f(A)X.X) is Lebegue integrable which implies

L
f2(A) x e L2 (Rn, X, dX),where f%(X) is the square root of f(A) and

L2(Rn, X, dX) consists of all X-valued LebeSgue measurable functions
x(X) with square integrable norm Hx(X)H; the inner product between

x and y in this space is given by f (X(X), y(X))dX.
Rn

Corresponding to M(t) we will consider the unitary isomorphic

field

H(t) = e. f”2(,\)x ~

where closure is taken in L2(R , X, dX).
For 5 C R", let H(S) V H(t) be the closed linear

tGS
span of the spaces H(t), t 6 S in L

2(R", x, dX). Clearly

H(Rn) = v H(t).
teR"

We denote by “f(X)H the operator norm of f(X). In this

chapter frequently we require that Hf(X)n is integrable, i.e.,

(1.0.1) énhf(X)HdX < a.

This condition is automatically satisfied for an X-valued Gaussian

family.

l. 0.2 DEFINITION. The field H(t) is called r regular (for fixed
r>O)if
ikt

V e H(s: Isl > P)i = H(Rn).

teR"

is called minimal if it is r-regular,for r + O, and is called

regular if
n H(s: )5] > r) = {O}
r

(i stands for the orthogonal complementin H(Rn)).

We observe that "regularity" is equivalent to
v H(s: Isl > Mt = H(R").
r

Note that for any fixed r0 and [s] < r-ro,

H(t: |t| > r) : H(t: lt-sl > r0) = eHS H(t: It! > r0). Therefore

i s L
(l.0.3) Y H(t: [t| > r)l 2 San e A H(t: |t| > r0) .
This shows every r-regular field is regular and obviously every
minimal field is r-regular. But an r-regular field need not be
minimal. As Theorem l.l.l7 shows each regular field is r-regular for
some r when f has a finite rank. This problem remains open when

the rank of f is not finite.

When the dimension of the field,i.e., N = dim H(O) is one,
necessary and sufficient conditons for almmpgeneous random field
to be regular is given by Rozanov [37] for the discrete parameter

space, t e Z", and by D. Pitt for t 6 Rn [32]. The substance
of their work is that every regular field is r-regular for some r.
As we already mentioned r-regular fields are regular.

In the case of finite dimentionalhomogeneous random field,

Salehi and Scheidt [44] have obtained necessary and sufficient

conditions for r-regularity. They also considered the problem of
regularity and gave a set of sufficient conditions which amounts to
the notion of r-regularity. A. Makagon and A. Weron [15] have
the same results under slightly weaker assumptions.

The problem of minimality for infinite dimensional case has
been analyzed by Yu. A. Rozanov [39], where satisfactory answer to
this problem is obtained. Rozanov's definition of minimality is in
terms of conjugate system, and is equivalent to ours (c.f. Theorem
2.2.l4). What remains to be studied is the problem of regularity
for infinite dimensional as well as finite dimensional fields which
is the subject of this chapter.

This chapter consists of two sections. In Section l we will
give necessary and sufficient conditions for regularity (Theorem

l.l.l3). In Section 2 we give the Wold-Cramér concordance theorem

. r-
for a homogeneous random field H(t) = eIAt 2(X)X

g , where g(X)
is a positive operator-valued function which is the density of
F(X) (the spectral measure of H(t)) with respect to (w.r.t.)

some positive o-finite measure I. Our Theorem l.l.l8 shows that

the analogue of results of Rozanov ([37], p. 384) and Pitt ([32],
p. 385) for regularity of scalar-valued case remains valid for the
vector-valued case.

Before closing this discussion we point out that the concept

of one-sided regularity for stationary processes indexed by the
reals or integers was introduced in connection with the time domain
analysis of such processes. This notion played an important role

in the extrapolation theory of univariate, [4 J, [9 I, [48 I; multi-
variate [7 ], [19], [36]; infinite dimensional processes [2 3,
[l7]. [24], [38], where satisfactory analytic characterization in
term of spectral density for one-sided regularity have been obtained
(see the forthcoming article [43] for further references and in-
formation). The concept of regularity as discussed in the present
chapter is connected with the study of multiparameter stationary
processes, i.e., random fields over Rn. Its role to the problem
of minimality and interpolation is similar to the role of one-sided
regularity to the problem of extrapolation of stationary processes

with real parameter.

1.1 Regularity. In this section we discuss the problem of regularity
for a homogeneous random field, the main result being Theorem 1.1.13.
X is a seprable Hilbert space, f(x) is a spectral density. The
symbols L2(R",X,dA),Ht,H(R"), etc;are the same as introduced in the
earlier section. We discuss some of the known results as they relate

to our work.

(u

Let A be a subspace of the separable Hilbert space
L2(Rn, X, dx). Let an, n 3 1, be anorthonormal basis in A. By
A (A) a.e. X is meant the closed subspace in the Hilbert space
X generated by all values an(X), n 3 1. The subspace A (X) a.e. A
is independent of the choice of the basis an, n 3 l, (c.f. [ l).
Note that TEM;)XI (closure in X) is a subspace of X

1
a.e.A. Since 1'/2 is bounded it easily follows that

 

1/ —. -

(l.l.l) f‘(X)X = (f(-)X) (X) a.e. X.

The following lemmas (Lemma1u1.2 and1.1.3 ) are due to
Rozanov [38], and are stated here for later use; Lemma l.l.2 also

can be found in Helson's book [6 ].

l.l.2 LEMMA. Let A be a seprable Hilbert space and let 8 be a
2(R", A, dX) then the doubly invariant subspace
2(

subspace of L

L = V n eIAt B of L R", A, dX), consists of all measurable functions
tER

a(X) e L2(A, R”, dA) such that a(X) e B(X) a.e. A.

An immidiate consequence of Lemmal.l 2 fisthat E(X) = B(A)

l.l.3LEMMA. Let s c R”, then H(S)i = f“15 35. Where f-%(X) is
1,
the inverse of the restriction of f%(X) to f2

 

(X)X, and is defined

 

1
'i

L -L
from f%(X)X onto f2(X)X with the properties that f 2(X)f (X)a(X) =
'T—-—_' -L
a(X) for any a(X) e f3(i)x and f%(i)f 2(X)a(X) = a(X) for any
a(A) E f%(X)X; and BS consists of all X-valued Lebesgue measurable

functions b(X) with

ID

(i) b(X) e f3(i)x a.e. X (ii) f‘3(i)h(i) e L2(Rn, x; di) and

(iii) In e'I3t(b(i), x)dX = o for all t e s, and x e x. In
R .

addition if condition(l.[Ll) holds,then b(t), the Fourier transform

0f b(k), is a well-defined X-valued Lebsegue measurable function

for t e R". To see this note that

lfe‘lit(h(i),x)di; = l je"*t(t3(x)f’3(xlb(x). x)dxl

I A
g‘
A
fl
I
io\M
A
>2
v
U-
A
>2
V
fl h‘
N\
A

(1.1.4) X)IdA

/ “f-%(3)b(k)hhf%(k)xhdx

IA

{fhf (i) Mi2 dx} 3ifhf3m )xnzdxi3

IA

{fhf'3(x)b(h)HZdX}3{f(f(y)x, x)d,}%

{qu'3(m b( )H2 dA} W£fu H)idx}3uxh

IA

Therefore fe-1At(b(x), x)dA defines a bounded linear functional

on X, and thus there exists an X-valued Lebesgue measurable function

~

b(t) such that for each x 6 X
fe“33(h(i), x)dX = (b(t), x).
Furthermore

(1.1.5) “b(t )H < {I Hf3 )b(x)u dei3if Hf(x) )udxi3

ll

{The integrals above are taken over R", and in the future whenever
the domain of the integration is missing, it is understood that the
integral is taken over Rn}.

In this chapter we let T = {t 6 R": It! f r} and S = TC =
HER“: |t|>r.

l.l.6LEMMA. V H(S)i consists of all functions a(A) E L2(Rn, X, dA)
S

such that

Proof. For $2: $1, H(Sz) c H(Sl) and therefore H(sz)l 3 H(sl)i
which says H(Sr)i are increasing sequence of subspaces as r + w.
This permits us to consider only a countable number of H(S)is.

V H(S)l is a doubly invarriant subspace. To see this note
that forseach S and t e Rn there is a pair of S', S” such
that entH(S)i is contained in H(S')l and contains H(S”)l

therefore:

em v H(S)i = v eMt H(S)i = v H(S)l .
s s 5

Now by Lemma 1.1.2, V H(S) consists of all functions a(A) in
S
2<

L R",X, dA) such that a(A) e (V H(S)*)(A) a.e. A and the latter

-———-—fr 5
is a.e. A equal to U H253 (A) which completes the proof.
5

12

1.1.7 LEMMA. v H(S)i = H(R") if and only if

 

 

 

s
""2? 7——
U(f 2BS)(A) = f2(A)X a.e. A
s
"'55— 72 n
Proof. Suppose U (f 83) (A) = f (A)X a.e. A. Let a(A) e H(R )
s

with a(A); v H(S)‘. For any 3 by (1.0.3) we have v H(sr)i 2

s s

r

V e”t H(S)‘. Thus a(A); V eMt H(S)i for each S. This means
1: t

that je1At( a(A),c(A)))dA = 0 for any S, t 6 Rn and c(A) 6 H(S)l.

This implies that a(A); c(A) a.e A in X which is equivalent

 

to a(A) L H‘T’s‘m) a.e. A. But by Lemma 1.1.3 H(S)l = F333, therefore

 

-L
a(A) i (f 2BS)(A) a.e. A for any S. As already mentioned in the
proof of Lemma l.l.6 we may consider a countable number of H(S)i.

-L
Therefore a(A) L U (f 28$)(A) a.e. A, and hence using the assumption
S

 

we get a(A) L f%(A)X a.e. A. But a(A) é H(R") and by Lemma l.l.2

H(R") consists of all functions d(A) e L2(R", X, dA) with

d(A) E f2(A)X a.e. A; so that a(A) has to be zero a.e. A showing
v H(S)l = H(R").
S
For the necessity note that V H(S)l = H(R") implies
S
-==" =T: —.I/—_——
(g H(S) )(A)= H(R n)(A) = f2(A)X a.e. A, where the second equality

==i

holds byLema 1.1.2. Now by Lemma l.l.6 (v H(S) (A) =uHTS") ()A) a. e. A
s

-——-— -L
and by Lemma 1.1.3 HTTs 1(A) = (f 235m) a.e..A. Therefore

 

 

-L y
g(f 3 BS)(A) = f2(A)X a.e. A. This completes the proof.

l3

The following important Lemma under the condition that
f(A) is nuclear (trace class) and its nuclear norm is integrable
is due to Rozanov [39]. This Lemma is still true under our weaker
assumption (l.0.l). Since Rozanov's proof is too condenced, and is
based on the nuclearity of f(A) we will give the proof in detail
below. We point out that the technique of our proof is the same

as the one given by Rozanov.

l.l.8 Lgflflg, Suppose condition (1 0.l) is satisfied. Let G be

any closed subspace of H(S)‘. Let {ak(A)} be a complete orthonormal
system in G. Define bk(A) = f%(A)ak(A), then there exists a sub-
maximal system of bk(A), k 3 l denoted by b:(A), i = l, 2,...,MG

(MG being finite or infinite) which are a.e., A linearly independent
in X. (Maximality here means that off a set of measure zero

bk(A) e B(A), where B(A) is the linear span of b:(A), i = l,...,MG

in X). Clearly MG 5 dim X.

Proof. Let a(A) e G. Then by Lemma l.l.3, a(A) = f-%(A)b(A), where

b(A) 6 8 Furthermore by (l.l.4) (b(A), x) is integrable

{5: ISI > r}'
and B(t) is a well—defined X-valued function. Now

 

b(A) e B{s: Isl > r} implies that b(t) = 0 for |t| > r. There-
fore (b(A), x) = ——n- emmt), x)dt. Now let
(2n) It fr
. ~ n
b(z)x = 1 n e12t(b(t), x)dt, where z 6 ¢n and z . t = Z Ziti'
(2n) It fr i=1

b(z)x is an entire analytic function defining on ¢n. In fact
b(z)x is analytic in each coordinate and the analiticity follows from

Hartogs Theorem [30]. Also (b(A), x) is the boundry value

l4
of b(z)x with Re 2 = X. For [2) f 6 we have

Ib(z)x| 5 l f ethl(B(t), x)!dt

 

 

 

 

 

 

' 12w)" Itl<r
s—-‘—-—,;e'”€{I11f(A )hd INF/(A) )11 omnxh I dt.
(2") tfr
where the last inequality is by (1.1.4). Therefore with c = 1 n dt
2n) ltlfr
we have
(l.l.9) |b(z)x| f C eer{f“f(k)“dA-fflf'%(x)b(x)“2dX}%flxH for.|z) f e .
Similarly lblA), x) is the boundry value of the entire analytic
function b(z)x defined by Elz)x = If] eiZt(B(t), x)dt, Rez =
t <r
A-lblz)x) is also bounded for [2| 5 e by the same bound occuring
in (l.l.9). Also we have
_ , 1 1At ~ ' 1 ~ _
[(b(X).X)| - 1 n I e (b(t). X)l< I [(b(t). X)|dt -
(2") Itlfr (2n)n 't|<r
_ l -iXt
(l.l.lO) --—-—75 f I fe (b(A) , x )dxldt
(2n) lt|<r Rn

 

15
= C] Hop), x))dA.
n

Now by using (1.1.10) and (l.l.4) we obtain that

)(b(A), X)| 5 C I )(b(A) . x )ldA < C {Ihf( AlfldA wfflf A)“ 2dUzllel
Rn

therefore

“b(A)“ c who)11c1A-I-111="1<A)t><A11121“2

IA

(l.l.ll)

c {f“f(A) )naA fHa(A A)“2 dA11.

Now let {xk} be an orthonormal basis in X, and let {ak} be a
sequence of scalars with ak # 0 for all k and Zlaklf

We point out that for each A, bi(A) = f‘(A)ai (A A), i = l, ,N are
linearly independent in X if and only if the corresponding sequences
{(bi(A), akxk)}:=l’ i = l,...,N which can be regarded as rows of a
matrix are linearly independent. The next step is to consider the
matrix {(bi(A), akxk)}:=l’ i = l,...,N, and the Gram matrix of the
sub-matrix consisting of the first m columns of the matrix

{(b1(A), akxk)}k, i = l,...,N; k finite or infinite. Suppose

d?j(A), i, j = l,...,N are the entries of the Gram matrix. Then

 

d?j(A) =

11MB

(bi(A)a “kxk)(bj(A)’ akxk) .

k 1

We observed earlier that (bi(A), x), (ble), x) are the boundry values
31(z)x respectively. Therefore

J
d?j(A) is the boundry value of the entire analytic function d?j(z)

of entire analytic functions bi(z)x,

and

l6

m
Id?j<z)l 5 kg] lb1(2)akxkll53(2)akxkl
(1.1.12) 3 c2 e211 inf<AindAtiuf‘1<A)b.(AinzoA.iuf‘1<A1bj(A)uzi
m
X laklzllxkllz

m 2
X lakl for Ill 5 6,

= CZ ezerfflf(A)HdA

where the second inequality is by (l.l.9) and the equality is by the
fact that kaH = l for all k and f'3(A)bi(A) = ai(A), where
{ai(A)} is an orthonormal set in G with “31“ 2 K = 1.

Now since E lakl2 < m, (l.l.lZ) implies that each d?j(z),
i, j = l,...,N converges uniformly on compact subsets of ¢n to
the entire analytic function dij(z) = kg] (bi(z)akxk)(53(z)akxk)
as m +,m [30]. Define Dm(A) to be the determinant of the Gram
matrix {d?3(A)}, i. j = l,...,N, then Dm(A) are also the boundry
values of entire analytic functions Dm(z) for all m.

Clearly by uniformity in the argument given above, lim Dm(z)

m-NO
is an entire analytic function. This implies that lim Dm(A) is the
[rt-)0!)

boundry value of an entire analytic function, and therefore it either
vanishes identically or is different from zero, a.e. A. In the latter
case we agree to call the elements bi(A), i = l,...,N a.e. A linearly
independent in X. The above procedure on N permits us to construct

a sub-maximal system of bk(A), say b:i(A), i = 1, 2""’”G’ which

are a.e. A linearly independent in X. Here maximality means that
*

ki
A of a set of measure zero bi(A), bk1(A),...,bkM(A) are linearly

if bi(A) is different from b (A), i = 1. 2.....MG. then for each

l7

dependent, i.e., bi(A) 6 B(A), a.e., A. Obviously MG 5 dim X
and MG could be finite or infinite. Proof of our lemma is complete.
A X-valued function b(A) in called weakly integrable if

f(b(A), x)dA < w. A weakly integrable X-valued function b is called
S-exponential if fe'ikt(b(A), x)d = 0 for t e 5. When
5 = (t 6 R": (t) > r) we use the phrase r-exponential instead of
S-exponential. Rozanov [39] and Pitt [32] have adopted this
definition which reduces to the classical definition of exponential
functions of type r on R by the help of the Paley-Wiener theorem,
Dym and McKean [5], and exponential functions of type T = Sc on
Rn by the help of an n-dimensional version of the Paly-Niener theorem,
Stein [45]. An operator-valued function ¢(A): X + X is called
r-exponential if for each x e X, the X-valued function o(A)x is

r-exponential.

The following Theorem gives necessary and sufficient conditions
for regularity. Characterization for r-regularity could be deduced
from Rozanov's work [39]. The extension of this criterion to the

Banach space is given by our Theorem 2.2.14.

l.l.13 THEOREM. Ahoniogeneous random field over a separable Hilbert
space X, with spectral density f(A) satisfying the condition
(l.0.l) is regular if and only if there exists a family of r-exponential

operator-valued functions ¢r(A), r + m, such that

l
(1) Each ¢r(A) in a Hilbert-Schmidt operator and or(A)X C ffi(A)X

a.e. A.

l8

 

 

(ii) f"1(A)o (A)x : f'%(A)o (A)x a.e. A with r < r2 and
r1 r‘2 1

 

 

u f'1(A)or(x)x = f1(A>x a.e. A
Y‘

-L
(iii) For each r, f 2(A)or(A) is a Hilbert-Schmidt operator
a.e. A and (Hf-%(A)or(A)H§ dA < w. Here H “2 stands for the

Hilbert-Schmidt norm.

Note: The condition(l.0.l)<ui f(A) is used only for the necessity

part of the Theorem.

Proof. Sufficiency: Suppose there exists a family of r-exponential
Hilbert-Schmidt operator-valued functions ¢r(A) satisfying (i),
(ii) and (iii). For each x 6 X define ¢:(A) = ¢r(A)x. The fact
that ¢r is r-exponential satisfying (i) and (iii) implies

¢:(A) belongs to BS where Sr = (t E R": |t| 3 r). Thus
r
orX c BS , where orX and B5 are well defined classes of X-valued
r r

function and the inclusion is pointwise. Now orX : BS implies
r

2

(

f'%¢rx : f‘ng , where the inclusion is in L Rn,XodX)- This implies
r

 

 

 

that (f'gorX)(A) s «"23S )(A) a.e. A. Since f‘%(A)¢r(A) is
l”

 

 

 

bounded linear operator a.e. A we have (f'gorx)(A) = f-%(A)¢r(A)X

 

 

*-

-y -/ ‘
a.e. A, therefore f ’(A)¢r(A)X c (f 1285 )(A) a.e. A. This means
r

g; ‘:$7"'
that g f ’(A)¢r(A)X G g (f 285 )(A) a.e. A. Using (ii) we get
.___.——-——- r r

;;T“"‘ '737""
2(A)X :Ag (f 2 BS)(A) a.e. A, and since the converse of this inclusion

 

 

 

 

 

 

 

. . L -
is always true we obtain f2(A)X = U (f ng)(A) a.e. A. Thus by
S

l9

Lemmal.l.7 we have H(R") = V H(S)i which says the process is regular.
5
Necessity: {H(Sr)*: Sr = (t: It) > r)} is an increasing
se uence w.r.t. r, i.e., H(S )i : H(S )i for s :3 s .
q r1 r2 r1 r2

Let {ar k(A)}k be an orthonormal basis in H(Sr)i. Consider the
sequence or k(A) = f%(A)ar,k(A), k = 1,... .
By Lemma l.l.8 for each H(Sr)i there exists a maximal system

to: 1.(A)} i = 1,...,Mr (Mr being finite or infinite) which a.e. A

are linearly independent in X, and such that if Br(A) denotes

the linear span of b: 1.(A), i = l,...,Mr, then each or k belongs

i can be approximated by

21

to Br(A) a.e. A. Each a(A) 6 H(S )

2(R” R”, x, dA),

ar,k(A) in L , X, dA), i.e., k=l akar’k(A) + a(A) in L

therefore there exists a subsequence of { g Gk ar k(A)}N which

k=l _L
converges to a(A) pointwise in X. But 9 ak a k(A) 6 f 2(A)B (A)

k=l ”’ r
-A
a.e. A. Thus a(A) E f 2(A)Br(A) a.e. A. This clearly shows that

 

 

 

 

 

(1.1.14) h(§:)1(A) : f‘1(A)sr(A) a e. A.
(l-l l5) f-%(A)Br(A) : f-%(A)Br (A) a.e. A.
r 2

Now based on b* (A), i = l,...,Mr; we define the operator

r,i
* a
valued function ¢r(A)xk = ”k br,k(A), where {xki( in an orthonormal
basis in X and u > 0 for k = l,...,N with 2 u < m;
k r k=1 k

and “k = O for superplus x; s.

20

We extend each ¢r(A) to a Hilbert-Schmidt operator-valued

r
is a Hilbert-Schmidt operator a.e. A with {Hf’%(A)¢r(A)ug dA < w.

function on X such that ¢r(A)A: f%(A)X a.e. A and f'%(A)¢ (A)

Furthermore each ¢r(A) is r-exponential, i.e., fe'1At(¢(A)x,y)dA = O
for xoy E X, t 6 Sr (see p. ll of [39] and Chapter II p. 49

for extension to the Banach space).

By the way that ¢r(A) is constructed we have f'%(A)Br(A) =

 

 

 

 

 

 

f'%(A)¢r(A)X a.e. A. Thusby O.l.15) f'g(A)or (A)X : f'%(A)or (A)X.
_____ 1 __g 2
Furthermore (f'l/ZBS (A) = H(Sr)*(A) : f-%(A)Br(A) = f-%(A)¢r(A)X

r
a.e. A. The first equality is by Lemma l.l.3 and the inclusion is

by (l.l.l4). Thus

 

U (f'%B )(A) c U f-%(A)¢r(A)X a.e. A.
Y‘ Y‘ Y‘

 

 

 

S

But by Lemma H .7, regularity implies that

 

 

U (f-ZBS )(A) = f2(A)X a.e. A.
r

r

 

 

-L
2(A)¢ (A)X a.e. A and the proof

1/
Therefore we get f2(A)X = U f r
r

is complete.

l.l.lG COROLLARY. Let f(A), the spectral density of a regular
homogeneous random field satisfy the condition (l.0.l). Then f(A)
has constant rank a.e. A. This constant value is called the rank

of f.

2l

Proof. From the construction of or(A) in the proof of Theorem l.l.l3

 

we see that rank Of ¢r(A)X is constant a.e, A = Mr. By l.l.l3
A -L L
(i), ¢r(A)X c f2(A)X a.e. A. But f 2(A) is 1-1 on f2(A)X
L .9
onto f2(A)X. Thus rank of f 2(A)¢r(A)X = Mr a.e. A. The result

follows by l.l.l3 (ii).
As a Corollary to our Theorem l.l.l3 we get the following

interesting result.

l.l.l7lHEOREM. Let H(t), t e R”, be a regular homogeneousrandom
field with spectral density f(A) satisfying (l.0.l). If f is of

finite rank then the process is r-regular for some r > 0.

Proof. The construction procedure-for~ ¢r(A) shows that each
¢r(A)X has a constant dimension Mr a.e. A. Also by Corollary l.l.l6,

f%(A)X has a constant dimension a.e. A. Further more

 

 

-y -/
f 2(X)¢r (A)X E f a(A)¢r (A)X for r1 < r2. Therefore by Theorem l.l l3
1 2 _ :7"“
(ii), there exists r0 such that f %(A)¢r (A)X = f2(A)X a.e. A.
O
and this is equivalent to say that the process is ro-regular,

 

see [39] page 12. This also follows from our Theorem 2.2.l4.

This completes the proof.

We remark that for the univariate case the class 85, which
was introduced in Lemmal.l.3 coincides with the class of all functions
¢ of exponential type T = SC with f lgdg-dA < m. For additional
information on functions of exponentialntype see [5 J and [45].

In the case that t e Z", the elements of BS are polynomials of

the form 2 ake1k°e with In f do < m, where Tn is
kET T

 

the n-dimensional torus and, do in the Lebesques measure on T".

The following Theorem which is an immidiate consequence of our The-
orem l.l.l7 extends the results of Rozanov [37] and Pitt [32]

to q-variate processes over R" and, completes the work of Salehi
and Scheidt [44] and Makagon and Weron [15] in interpolation
theory.

l.l.l8. THEOREM. Let g(t) be a q-variate homogeneous random field
with t E Rn and spectral distribution F(A). Then 5(t) is
regular if and only if F(A) is absolutly continuous w.r.t. The
Lebesgue measure and there exists a nonzero r-exponential matrix-

valued function o(A) such that

(i) rang o(A) = rang F'(A) a e. A, and

(if) f¢*F'-1¢ dA EXlStS and f 0-

Proof. The fact that for a regular field, F(A) is absolutely
continuous will be proved in section 2. When the dimension of the
field in finite, condition (l.0.l) automatically holds. From the
proof of Theorem l.l.l7 follows that in the finite dimensional case,
the spectral set of conditions for regularity given in Theorem
l.l.l3 reduces to the present set of conditions (i) - (ii), and the

proof is complete.

23

2. The Wold-Cramér concordance. Suppose the homogeneousrandom
field 5(t) has a spectral distribution F(A) which is absolutly
continuous w.r.t. a o-finit positive measure t, i e., there
exists a weakly integrable bounded linear positive operator-valued

function g(A) with
_ iA(t-s)
ng(t)€y(5) " f e (g(A)X:.Y)dTa X,y e X’

The bounded operator-valued function g(A) is called the density

of F(A) w.r.t. t, i.e., g(A) = ngll .

Thefnmogeneous random field M(t) can be represented

within a unitary isomorphism as

(1.2.1) H(t) = elitgf(A)x

where the closure is taken in L2(R", X, dT).

Similar to Lemmalflu3 H(S); admits the following representation.

l.2.2l£MMA. H(S)i = gJ2 8% where g'%(A) is the inverse of the

 

restriction of 9%(A) to 92(A)X and 83 consists of all
X- valued measurable function b(A) with (i) b(A) 6 9%(A)X a.e. T
(ii) g'1(A)o(A) e L2( R", x, dt), (iii) fe1kt(b(A),X)dT = o,
for all x E X and t E S.
The proof is similar to the proof of Lemma l.l.3, and there-

fore it is only sketched.

Proof. Take a(A) 6 H(S)‘, then f e'iAt(a(A), 9%(X)X)dT = 0
for all t e S or f e'lxt(g%(A)a(A), X)dT = O, t 6 S. Let

b(A) = g%(A)a(A), then (i) and (iii) are obvious, and

24

-L 2

9 2(AMA) = a(A) 6 L n

(R , X, dT) which implies (ii). The proof

of the converse follows by reversing the order of discussion given
above.
Let F = Fa + FS and T = T + TS be the Cramér-Lebesgue

decomposition w.r.t. A. Clearly F(A) = f ——3-dA + FS(A). Now
A

F is absolutly continusous w.r.t. T, therefore

- 9f. . 15...; fl
F(A) - A dA dT i dT dA dA + A d1 dTS. Therefore
d‘l' dF
915...}. - _9. e 515 _ . . .
i dA dA dA i dA dA g d1 dTS FS(A) wh1ch 1mpl1es
dFa dta
(1.2.3) (TX—=- g()\) 8-5:— a e A

Because g(A) is a.e. A a bounded operator-valued function, (l.2.3)
defines géé- as a bounded operator~valued function a.e. A.

We denote ggé- by the usual notation f(A). In summary this

discussion shows that f(A) is a weakly integrable positive operator-

valued function. With this preparation we state the following

lemma.

l.2.4L_EflllA;. Let H€(t) be ahomogeneous random field

over a separable Hilbert space X, with spectral distribution

F(A) admitting the spectral representation (l.2.l). Let Hn(t)
be the random field corresponding to the absolutly continuous part
of F(A). Then H€(S)i is isomorphic to Hn(S)*, where S is

the complement of a bounded set T.

25

iAt dFa
Proof. Let Hn(t) = e 2(A)X , where f = dA—' as above. Then
i = —% f i = -% 9 f 9
Hn(S) f BS and H€(S) g 85 where BS and BS were

described in Lemmas l.l.3 and l.2.3. We establish a correspondence

between Hn(5)l and H€(S)l as follows. Take b(A) in 3% , then

[ e"*t(b(A), X)dT = o for all x e x and t 6.3. This implies that

(b(A), X)dT is absolutly continuous w. r. td The Lebesgue measure
with density [(b(A), x)dtl/dA = (b(A), x) d——- for all x E X.
Therefore b(A) can be taken to be zero on the singular part of
t and fel*t(b(A),) x) =fe ‘*t(b(A), x)dt. Now let
d(A)= —b(A). Then d(A) is a X-valued Lebesgue measurable

function dsatisfying (i) d(A) E f%(A)X a.e.A by (1.2.3),

dTa
(ii) [€1At(d(A) x)d A fe'lxt(fi b(A), x)dA

fe “*t<b(A) x)dgie-1At(b(A),x)dt =

for all x E X, t E S and

2dr
(iii) qu'1(A)d(A)u2dA = 119 11A)ub(A)2dTa dA

wiug'1)b<A)n at = 119‘11A1b1A)uzdt

f
S.
to g'g(A)b(A) in H€(S)*. Computations in (iii) shows that this
map is norm preserving. Furthermore for any d(A) in 8: define
dta
b(A) through b a——-= d. The calculation above shows that this

therefore d(A) 6 B Let f-%(A)d(A) in Hn(S)i correspond

map is also onto. Hence the proof is complete.

l.2.5 DEFFINITION. A random field H(t), t e R", is said to be singular

 

ianwQ=HmM.

26

1.2.6 COROLLARY. Let H€(t) be arnmogeneous random field

as in Lemmal.2.4. Then Hé(t) is singular if and only if B; s O
for all S.

f - -% f . i
Proof. Suppose BS ; 0, then f 85 a O wh1ch says Hn(S) 2 0.

Therefore byLemmal.2.4 Hg(Sf’= 0 which is equivalent to say that

H£(t) is singular. Now assuming the process is singular, then

-1’
H€(S)l a o and byLemmal.2.4 Hn(s)* a 0 or f 2 a; s o and

L -L
since for any b(A) 6 B; we have b(A) = f2(A)f 2(X)b(X) we

get 8: a 0 and this completes the proof.

1 2.7LEMMA. Let H€(t), t e R", be ahomogeneous random

field with spectral distribution F(A) satisfying (l.2.l). Let
Fa(X) and FS(X) denote the absolutely continuous part and the
singular part of F(A) w.r.t. the Lebesgue measure. Then

€x(t) = nx(t) e cx(t). x e x, t e R”; Hn(Rn) 1 HC(Rn), where n

and c have spectral measures Fa and FS respectively. Moreover
H (S) = Hn(S) e HC(S) for any S the complement of a bounded

5

set T C R", and H€(S)l

H:(S).

Proof. We note that dr dra + drs, where r and r are

a s
absolutely continuous part and singular part of the measure I

w.r.t. the Lebesgue measure. Therefore

 

(1.2.8) g(X)dr = g(A)d~ra + g(A)drS.
. 17—— .
Now let Hn(t) = e1At1 C92(X)X and Hc(t) = e1At1Ag%(A)X, where

_ A 1
A is the support of Ts and closure is taken in L2(Rn, X, d1).

27

Hence gx(t) = nx(t) e ;x(t), x E X, t e R“. The subspaces Hn(Rn)
and HC(Rn) are perpendicular to each other in L2(Rn, X, dT),
because

. / . L
f(e‘AtI cg’zmx, e‘*51 cg:

A A (x)y)dr = xe‘*(t‘s’(g%<x)x, ge(,)y)1A1 cdr =

We also note that by (l.2.3) Hn(t) admits the spectral representation
Hn(t) = eithEq13;' with closure in L2(Rn, X, dA). Therefore
by Lemnal.2.4 +155)i = Hn(S)J‘. Furthermore Hn(S) = Hnm") e Hn(5)*c
H€(Rn) e 115(3)i = H€(S) this along with gx(t) = nx(t) + gx(t)
show that HC(S) : ”5(5) which completes the proof.

The well-known Wold decomposition theorem for the random
field H€(t), t e R", states that g(t) can be uniquely decomposed

into €x(t) = n (t) e 5x(t), x e x, t e R”; Hn(Rn) L H€(Rn), where

x
Hn(t) is a regular and HC(t) is a singular field, and that they
are subordinated to H€(t). The following result establishes the
so-called Wold-Concordance Theorem which extends the work of Rozanov
[37], Pitt.[32], Makagon and Neron E15], Salehi and Scheidt [44]

to the infinite dimensional case.

l.2.9 THEOREM. (Wold-Cramer Concordance) let Hg(t), t E Rn be
a homogeneous random field with spectral representation

given by (l.2.l). Let Fa(A) and FS(A) denote absolutely
continuous part and singular part of F(A) w r t. the Lebesgue
measure respectively. Suppose the density of Fa(A) satisfies
the conditions (i), (ii) and (iii) of Theorem l.l.7. Then

a(t), t 6 R", can be uniquely decomposed in the form gx(t) =

28

nx(t) 9 tx(t), x E X, t 6 R”, where the field Hn(t) is regular,

H,(t) is singular and Hn(Rn) L HC(Rn). Moreover for any set
5

S the complement of a bounded set T, H€(S) = Hn(S) e HC(S)

and Fn(X) = Fa(X) and F; = FS.

PTOOT- By Lenmal.2.7 Ex(t) can be written as gx(t) =

n n n ,
nx(t) 9 cx(t). x 6 X, t e R . Hn(R )i HC(R ). Hn(t), H;(t) have

spectral measures Fa and FS respectively and Hn(t)’ Hc(t)

are subordinated to H€(t). Since the density of Fa satisfies

the conditions (i), (ii) and (iii) of the Theorem l.l l3, by Theorem
lJ.7 thxt) is regular Furthermore byLemmal.2.6 Ht(t) is singular,
therefore the proof is complete by the uniqueness of the Wold

decomposition.

l.2.lO THEOREM. The spectral measure of a regular homogeneous random
field H (t), t e R", with spectral representation given by (l.2.l)

is absolutely continuous with respect to the Lebesgue measure.

Proof. Let Hn(t) and Hc(t) be the components of H (t) in

E

the decomposition of H€(t) 'h1Lemmal.2.7. Now H (t) is regular

5

which implies v 115(5)l = H€(Rn). But by Lemma 1.2.7 Haw)l = Hn(3)i.
5

Therefore V Hn(S)i = HE(R"). Also by the construction of Hn(t)

S n n n L n
we have Hn(R ) : H€(R ). Hence Hn(R ) 2 V Hn(S) = H€(R ),
S
which implies Hn(Rn) = H€(Rn), Hn(Rn) is regular and HC(Rn) s 0.
Thus ”c(t) 5 0 which by Lemma l.2.7implies F the singular part

5
of the spectral measure F is zero and this completes the proof.

29

l.2.ll COROLLARY. Suppose F the spectral measure of a regular
homogeneous random field is of trace class (this is satisfied for

the finite dimensional case). Then F is absolutely continuous

w.r.t. Lebesgue measure.

Proof. Spectral representation (1.2.1) is automathically valid in

this case. Now apply Theorem l.2.l0.

CHAPTER II
ON B(X, Y)-VALUED HOMOGENEOUS RANDOM FIELDS

INTRODUCTION. Let X be a complex-Banach space and (a, B, P)

 

be some probability space. Let g be a random variable of second
order over (9, B, P) taking values in the Banach space X, i.e.,

2

* , * * *
g: n + x, x (g(w)) e L (o, B, P) for all x e x , where x

is the space of bounded conjugate linear functionls on X, namely
X* = {x*: x* = i", x' 6 X'}, where X' stands for the space of
bounded linear functionals on X. Let us define the linear operator
T on x* into L2(Q, B, P) by Tx* = x*( (b)). Then by closed
graph theorem T belongs to B(X*, Y), where Y = L2(n, B, P) and
B(X*, Y) denotes the space of bounded linear operators from X*
to Y. This association induces a B(X*, L2(n, 8, P))-valued process
for each X-valued process. Therefore in general, we may consider
B(X, Y)-valued stochastic processes, where X and Y are arbitary
complex Banach and Hilbert spaces respectively.

Let at, t E Rn = the n-dimensional euclidean space, be a
B(X, Y)-valued stochastic process, i.e., for each t e R”, at e B(X, Y).
it is called homogeneous if (gtx, asy)Y depends only on t-s for
all x,y e X; t, s e R"; it is called continuous if (gtx, gsy)Y
is continuous in t and s, where ( , )Y stands for the inner

Product in the Hilbert space Y. In this work we assume that the

30

31

the processes are always continuous. Corresponding to define

gt

a closed subspace in Y spanned by the elements x, x e X. This

5
t
closed subspace is denoted by M(t). Since the strong closure of

the family {gtAz A e B(X, X)} is the same as B(X, M(t))(c.f.[22j]
p. 9, [Bl] p. 335), as far as terminology is concerned we will make

no distinction between and M(t), t e R", and we call them

at
B(X, Y)-valued homogeneous random fields.

For any S c R", define M(S) = V M(t), the span closure
tES

in Y of the subspaces M(t), t 6 S. Clearly M(R") = V n M(t).
tER

2.0.l DEFINITION. A B(X, Y)-valued homogeneous random field
at, t E R", is called r-regular (r > 0) if

v n ut M(t: [t] > r)i = M(Rn),
tER

where Ut is the continuous unitary shift operator defined on

M(R") onto M(Rn) by U x for all x E X, and t, s e Rn

tgsx = gt+s
(L stands for orthogonal complement in M(Rn)).

2.0.2 DEFINITION. A B(w, Y)-valued homogeneous random field
”t’ t E R", is called an r-conjugate field to the random field

at, t e R" if

Mn(O) e145“: It] > r)*,

dim Mn(0) 5 dim M (0),

E

where N is a complex Banach space and r > 0.

32

2.0.3 DEFINITION. A B(X, Y)-valued homogeneous random field

 

S t E R", is called minimal if it is r-regular, as r + O.

t’
NOTE. The notion of minimality was first introduced by Rozanov [39]
in terms of the conjugate system. Indeed he called a random field
minimal, if for each r, r + 0, the field has an r-conjugate field.
Clearly this concept of minimality implies ours. We will prove

that under the assumption (2.2.l) these two concepts of minimality

are equivalent.

2.0.4 DEFINITION. A B(X, Y)-valued homogeneous random field

 

at, t e R", is called regular if
n M(t: |t| > r) = {0}.
r>0
This concept of regularity is equivalent to
v M(t: It] > r)l = M(Rn).
r>O
This chapter consists of four sections. Section I consists
of spectral representation of a B(X, Y)-valued homogeneous random
field and some ancillary results for later use. In Section 2
necessary and sufficient conditions in terms of the spectral density
for r-regularity, minimality or regularity is given respectively.
It will be shown that every r-regular field has an r-conjugate field
(Theorem 2.2.l4) (evidently a random field with an r-conjugate field
is r-regular). In Theorem 2.2.l4, we will also give necessary and
tsufficient conditions for a homogeneous random field to admit an
?‘-conjugate field. This result is the extension of the work of

Rozanov [39] to the B(X, Y)-valued random fields (Rozanov obtained

33

necessary and sufficient conditons for a Hilbert space valued homog-
eneous continuous parameter, t E R", random field to admit an
r-conjugate field (c.f. [39], 1976)). As a corollary to our work

we derive necessary and sufficient conditions for minimality of a
discrete parmaeter, t e Z", random fields which independently has
been obtained by Makagon [141. We will also show that the conjugate
field is a Hilbert space-valued field acting on Y with certain
spectral representation. The work of regularity problem extends

our result in Chapter 1 to B(X, Y)-valued fields.

In Section 3 the concept of complete minimality for the Banach
space case will be discussed, and sufficient conditions for a minimal
field to be completely minimal will be given. The key to this result
is Theorem 2.3.8, which says that under certain conditions on spectral
density, each r-regular fields admits a Hilbert space-valued spectral
representation with a nuclear (trace class) density. The notion of
complete minimality was introduced by Rozanov in 1976 [39] for Hilbert
space-valued continuous parameter homogeneous random fields, and plays
important role in analyzing Markov property. The role of conjugate
field, which is an extension of biorthagonality (c.f. Masani [18],
Nadkarni [29]) in Markov property was first observed by Kallianpur
and Mandrekar [8]. We also introduce such a notion for discrete parameter
random fields. Similar results as the continuous case are obtained for
the discrete case. In particular we prove that every finite dimensional
<1iscrete parameter homogeneous minimal random field satisfies certain
geometrical property (c.f. Theorem 2.3.18).

Section 4 discusses the L-Markov and Markov properties for the

Banach space case. L-Markov property for discrete fields was first

34

introduced by Rozanov (c.f. [37] 1967). Recent works in this topic
are [1 ],[282h [10], [28], [40]. Necessary and sufficient conditions
for a completely minimal field to be L-Markov or Markov are given.

This also is an extension to the work in [39] to the B(X, Y)-

valued homogeneous fields. Similar results for the discrete case

are obtained. In summary this chapter extends Rozanov's work [39]

and the work in Chapter I to B(X, Y)-va1ued continuous parameter
random fields with new results on discrete parameter random fields.

The techniques that we use are similar to the ones employed by

Rozanov [39]. The existence of a square root for a B+(X, X*)-

valued function (c.f. Miamee-Salehi [25] and Masani [20] is crucial in
carrying out our work. Throughout this chapter the measurablity of any X* -
valued function b is understood to be in weak* sense, i.e., for

each x 6 X, b(A)x is a complex-valued measurable function. For

such functions the integrals are taken in the sense of Pettis.

2.1 Spectral Representation and Preliminaries. Let gt, t e R",
be a B(X, Y)-valued homogeneous random field. It is known that

there exists a unique B+(X, X*)-va1ued measure F such that

(etx. 55y) = h" ei*(t’s) (F(dA)x)y . x. y e X.

where B+(X, X*) stands for bounded linear positive operators on
X to X* E J. F(A) is called the spectral measure of the process
at. When the derivative of F with respect to the Lebesgue measure
exists we say that the process has a density. In this chapter we

assume that the process has a density f(A), a unique measurable

gt
+ * , ,
3 (X, X )-va1ued funct1on w1th

(2.1.1) (gtx, 55y) = In e f(X)x)y dX x,y E X.
R

We also assume that the Banach space X is separable.

An interesting factorization result which was proved by
A.G. Miamee and H. Salehi [25] is the following: for a separable
Banach space X, any weakly integrable B+(X, X*)-valued function
f(A) has a square root. More precisely there exists a separable
Hilbert space K (dim K g dim X) and a measurable B(X, K)-valued

function Q(X) such that
(2.1.2) f(X) = Q (X)Q(X) a.e. X,

'k 'k
where Q (X): K + X is the adjoint of Q(X) defined by

(2.1.3) (0 (A)Q(X)x)y = (Q(A)x. Ohm. x. y e x

EQIE: As we mentioned, the factorization result given above under
the assumption that the Banach space X is separable was proved by
A.G. Miamee and H. Salehi. This assumption was later relaxed to
the separability of F(Rn)X by A. Makagon [13], [141, where F

is the spectral measure of the field. Since such a factorization
is essential in our work, we assume that F(Rn)X is separable.

The justification for the separability assumption and the study

of the Banach space-valued fields is discussed in [25] p. 548.

We will continue with developing the spectral representation of a
B(X, Y)-va1ued homogeneous random field.

From (2.1.2) and (2.1.3) we have

*

(2.1.4) (f(A)X)y = (Q (A)Q(A)X)y = (Q(A)x', Q(A)y), x. y 6 X

36

Let f(X) be the density of the process it. Then (2.1.1) and

(2.1.4) give the following:

(atx. asy) = (n el*(t's) (Q(X)x, Q(X)y)KdX

(2.1.5)

The above identity defines the so called Kolmogorov isomorphism
map between the time and spectral domains. To be more precise let
H(t) be the span closure of eiAt Q(X)x, x e X in L2(K) = L2(Rn, K, dX),
where L2(K) consists of all measureable K-valued functions x(X)
with square integrable norm Hx( )H, with the inner product defined
by (x(X), y(X)) = é" (x(X), y(X))KdX. Then by (2.1.5) M(t) c Y

is isomorphic to H(t) c L2(K). Evidently from (2.1.5) we have

(2.1.6) H(t) = e”t Q(X)X ,
where closure is taken in L2(K). Define H(S) = V H(t), the span

tES
closure in L2(K) of H(t), t e S s R”. Evidently H(S) and M(S)

are isomorphic. Therefore we may consider H(t) instead of M(t).
Definitions 2.0.1, 2.0.2, 2.0.3, and 2.0.4 can be defined for

H(t), t 6 Rn in a similar way.

NOTATIONS. Let A be a subset of L2(K), then A is separable and

._ I
has an orthonormal basis. By X(X), we mean the span closure of the
elements of the orthonormal basis of A in X (see Section 1.1).
It is clear that ilX) is defined, a.e., X and is independent of

the choice of the orthormal basis. Let Q(X) be the square root

37

of f(X). Then Q(XSX is a well defined closed subspace of K

a.e. X, and since Q(X) is bounded and linear we have Q(X)X =

 

(Q X)(X) a.e. X (c.f. Section 1.1). We also introduce the following

notations which are used heavily through this Chapter.

(2.1.7) K(X) = W X x and x*(>.) = Q(X)(Q X x a.e. X.
The following elementary lemma is essential.

2.1.8 LEMMA.
(a) For any a(A) e GTXTX' we have <0<X>x. a(X))K = (0*(X)a(x))(x)

*‘1 *
(b) There exists a linear operator Q (X): X (X) + K(X)

 

with the following properties:

(i) 0* (x)o*(x)a(x)

a(X) for any a(X) E K(X)

*‘1

(ii) o*<x)o (x)b<x)

b(X) for all b(X) e x*(X)

Proof. (a) Let a(X) e K(X), then there exists a sequence
{Q(X)yn, yn e X} such that Q(X)yn tends to a(X) in K norm.

This obviously implies

(Q(X)X. a(X)) lim (Q(X)X. Q(X)yn)

n

1im (0*(X>o(x>x)<yn) by (2.1.3)
n

11m <o*<x>o(x)yn)(x).

n

 

But Q(X)yn tends to a(X) implies that Q*(X)Q(X)yn tends to
Q*(X)a(X) in X* norm, because 0*(X) is bounded. Now strong

*
convergence in X implies weak convergence, therefore

 

1im (Q (X)Q(X)y )(x) = (Q (X)a(X))(x) and the proof is

finished.

(b) All we need is to show that Q*(X): K(X) + X*(X) is
one-to-one. Let 0*(X)a(X) = Q*(X)b(X) for a(X), b(X) e K(X).
This implies that for any x 6 X, 0*(X)a(X)x = Q*(X)b(X)x. This
by part (a) is equivalent to (a(X), Q(X)x) = (b(X), Q(X)x) for
all x e X, or (a(X)-b(X), Q(X)x) = 0 for all x e X, which implies
a(X)-b(X) L K(X). But a(X)-b(X) E K(X). Therefore for a(X) = b(X)

which completes the proof.

2.2 Regularities. The main purpose of this Section is to obtain
necessary and sufficent conditions for a B(X, Y)-valued homogeneous
field H(t), t 6 R", to be r-regular, minimal or regular. We will
show in this section that every r-regular field has an r-conjugate
field. The main results are Theorems 2.2.14, 2.2.19 and 2.2.22.
Theorems 2.2.14(b) and 2.2.22 extend the work of Rozanov (c.f. [39]
p. 12) and the work in Chapter 1 to B(X, Y)-va1ued homogeneous
random fields respectively. Throughout this Section we assume that
the random field H(t), t e R", admits the spectral representation
(2.1.6) and its spectral density f(X) satisfies the following

condition
(2.2.1) é” Hf(X)HdX < m .
We start with the following lemma.

2.2.2 LEMMA. For any X*-valued measurable function b(X) satisfying

39

-1

(i) b(X) e X*(X), a.e., X and (ii) I “0* b(X)“2dX < a,

Rn
the function b(t), t e Rn defined by b(t)x = In e-ixtb(X)de is
an X*-va1ued measurable function and R

.~ . *“ 2 9
(2.2.3) “b(t)“ 5 (fhf(X)th-IHQ (A)b(X)n dA) .

When the domain of integration is not specified, it is understood that

the integration is taken over Rn.

 

-i>\t

Proof. All we need is to show that (e b(X)de defines a bounded

 

linear functional on X. The linearity of fe'1ktb(X)de is obvious
once we have shown that f|b(X)xldX < m. By using (i), (ii) and
2.1.8(a) we have

* *‘1 *‘1
B(Xlk = (Q (X)Q (X)b(X))(X) = (Q(X)x.Q (X)b(X)). Therefore

. *-l
lfe"*tETXT§qug f](Q(X)x, Q (X)b(X))|dX

[110(X)XIH£Q* (X)b(X)lldX
1

IA

(l~llo(X>xllZ)‘=-(lllo*' (Albmllzdoi.

IA

But “Q(X)xu2 = (Q(X)x.Q(X)X) = (0*(X)Q(X)X)(X) = (f(X)X)(X) s “f(X)thH2.
Thus

in: — — *‘1 .2 i,
(2.2 4) lfe b(X)de| 5 Ilb(X)xldX§ (fo(X)HdX-Ih0 (X)b(X)h ) “x“
This shows that eiAt b(X)de defines a bounded linear functional

on X satisfying (2.2.3). The proof is complete.

40

2.2.5 DEFINITION. A x*-valued measurable function b(X) is called
weakly integrable if f|b(X)x|dX < a for any x e X, and is called

S-exponential if in addition
-iXt . _
fe b(X)de - o for all x e x and t e s,

where S is a subset of Rn (see the discussion on S-exponential
functions following Lemma 1.1.8).

Similar to Chapter 1, let BS denote the class of all S-ex-
ponential functions b(X) with

l

(i) b(X) e X*(X), a.e. X, and (ii) [no*- (X)b(X)H2dX < e.

With this notation the following important lemma characterizes the
orthogonal complement of H(S) in H(R") which will be denoted by
H(S)‘.

*-1

2.2.6 LEMMA. H(S)l = Q 85

Proof. Let a(X) e H(S)‘. Then a(X) L H(S) which is equivalent to
f(a(X),eiAt Q(X)x)K dX = 0 for all t E S and x e X. But by lemma
2.1.8(a) (an). eiAt Q(X)X)K = e'm (an). Q(X)x)K = e“'”(o*<i)a(m(xl.
(Here we note that by Lemma 1.1.2 H(R") consists of all K-valued
functions a(X) which take values in K(X), a.e., X. Therefore
a(X) é H(S)‘ satisfies the requirement of Lemma 2.1.8 which already
used in the above identity). Therefore fe'ikt (0*(X)a(X))(x)dX = 0
for all t e s and x e x. This implies that b(X) = 0*(X)a(X)

is an S-exponential function. b(X) obviously takes values in

X*(X) a.e. X. But Q*- (X)b(X) = a(X) which belongs to

L2(K) i.e. [no*' (X)b(X)H2dX < e. Thus b(X) e BS which finishes

41

one part of the proof, i.e., any function a(X) e H(S)l is of the
form a(X) = 0*-1(X)b(X) with b(X) 6 BS' The proof of the remaining
follows by just reversing the argument given above.

The following lemma plays an important role in developing the
main results of this section. As it was mentioned earlier in Chapter
I, under the assumption that f(X) is nuclear, namely of trace class,
and its nuclear norm is integrable the lemma below was proved by
Rozanov [39] for the Hilbert space-valued homogeneous random fields.
In Chapter I we relaxed the nuclearity of f and the integrability
of its nuclear norm to the integrability of “f(X)“ (c.f. Lemma 1.1.8).
Although with the help of Lemmas 2.2.2 and 2.2.6 the proof for the
Banach space case can be carried out similar to the one given in Lemma
1.1.8, nevertheless we give the proof in detail for the convenience of the

readers. In summary we extend below Lemma 1.1.8 to the B(X, Y)-valued

homogeneous random fields under the assumption (2.2.1).

2.2.7 LEMMA, Let G be any closed subspace of H(s: Isl > e)l

and let {ak(X)} be a complete orthonormal system in G. Also let
bk(X) = Q*(X)ak(X). Then there exists a sub-maximal system of bk(X)'s
denoted by bki(X), i = l, 2,...,MG (MG being finite or infinite)
which are a.e. X linearly independent in X* (maximality means that
off a set of measure zero bk(X) e B(X) for all k, where B(X) is
the linear span of bki(X), i = l, 2,...,MG in X*). Clearly

* 'k
M 5 dim X (X) f dim X , a.e. X.

G
*-1
Proof. Let a(X) e G. Then by Lemma 2.2.6 a(X) = Q (X)b(X), where

b(X) e 3 Furthermore by the proof of Lemma 2.2.2 b(X)x

{5: (SI > €}°

42

~ '1:
is integrable and b(t) is a well defined X -va1ued function.

Now b(X) 6 3(5, implies that B(t) = 0 for |t| > e.

 

 

is! > 6} .
Therefore b(X) x = n e1At b(t)xdt. Now let
(2n) Itlge
. n
b(z)x = 1 n e1Zt b(t)xdt, where z e ¢n and zt = Z ziti'
(2n) Itlfé i=1

b(z)x is an entire analytic function defined on ¢n. In fact b(z)x
is analytic in each coordinate and the analiticity follows from
Hartogs Theorem [30]. Finally b(X)x is the boundry value of
b(z)x with Re z = X.

For |z| 5 r we have

 

lb(z)xl < ‘ I leml IE<t>xldt

 

- (2n)n lt|<€
5 (21)" It! 6 9’ Ztl 'b(t)x'dt
11’ <
5 ‘ n eer [b(t)x|dt
(2n) It|<€

*-l
‘ n eff (fo(X)HdX-IHQ <i)b<X>n2dxiiuxn I dt.
(2n) Itlfé

 

IA

where the last ineguality is by (2.2.3). Therefore with
l

n dt we have
(2n) Itle

 

1

(2-2-8) lb(Z)X| E C eer(fo(X)HdX°fHQ*- (X)b(X)H2dX)%HXH f0r Ill 5 r

Similarly b(X)x is the boundry value of the entire analytic function

E(z)x defined by B(z)x = If! e1Zt'b(t),x dt, Rez = X. |E(z)xl
t 56

is also bounded for |z| 5 r by the same bound occurring in (2.2.8).

43

Also we have

 

[b(X)xl = | 1 f eiAt B(t)xdtl 5 ——l——- f ib(t)x|dt
t

 

(2.2.9)

 

 

C In [b(X)xldX.
R

Now by using (2.2.9) and (2.2.4) we obtain that

1

[b(X)xl g C é" [b(X)xldX s C (fo(X)HdX-IHQ*- (X)b(X)H2)%an .

Therefore

*‘1 L
“b(X)“ c (fo(X)H X 1H0 (X)b(X1H212

IA

(2.2.10)

I
6

c (fo(X)HdX:fHa(X)H2dX)

Now let {xk} be a dense linear set in X with kau = l for all
k, and let {ak} be a sequence of scalars with bk f 0 for all k,
k

2 lakl2 < m. Each bi(X) = 0*(X)ai(X) takes values in X*. For
k

each X and each x e X, bi(X)x is uniquely determined by the sequence

on

k=l
bounded linear functional on X and each bounded linear functional

{bi(X)okxk} In fact for each X, bi(X)x is a conjugate of a

is uniquely determined by its values on a dense linear subset.

Also for each X, bi(X), i = 1,. .,N are linearly independent

44

akxk:k=],
i = l,...,N which can be regarded as rows of a matrix are

'k
in X if and only if the corresponding sequences {bi(X)

linearly independent. To see this let us assume that bi(X), i = l,...,N
*

are linearly independent in X and there exists Bi’ i = l,...,N

such that

+...+ 8N{bN(X)akxk} = 0 for all k.

s {b (X)a x } + B {b (X)a x }
llkkk22kkk k

This implies that 8] b](X)okxk + 82 b2(X)akxk +...+ Ban(X)akxk = 0

for all k or
{B1 b](X) + 82b2(X) +...+ BNbN(X)}akxk = 0 for all k. But

81b](X) + 82b2(X) +...+ BNbN(X) is a conjugate of a bounded linear
functional and by the argument above it has to be the zero element
in x*, i.e. 81b](X) + 62b2(X) +...+ erN(i) = 0. But bi(X),

i = l,...,N are linearly independent therefore Bi’ i = l,...,N
have to be zero.- The proof of the other part is stright forward.
The next step is to consider the matrix {bi(X)ak xk}k, i = l,...,N
(k being finite or infinite) and look at the Gram matrix of the
sub-matrix consisting of the first m columns of the matrix
{bi(X)okxk}k, i = l,...,N; k finite or infinite. Suppose

m

dij(X), i, j = l,...,N are the entires of the Garm matrix. Then

m
d?j(I) = kg] (bi(X)akxk)(531X)E;§;).

we observed earlier that bi(X)x and blelx are the boundry values
of entire analytic functions b.(z)x and bfi(z)x respectively.

1
Therefore d?j(X) is the boundry value of the entire analytic function

45

d?j(z) and
m I’ll
1d,J( )1 5 kg] l(b,(z)akxk)lb ( kku
m *-1 1..- *‘1 1/
(2.2.11) C2 ezer fuf(X Hunk)1 (IHQ (X)b.(X)H2dI)°(IHQ (X)bj(X)n2dX)2
laklzllxkllz

e2 m
C2 er fo(X )HdX k2] lakl2 for |z| f r,

where the second inequality is by (2.2.8) and the equality is by the
-l
*

fact that flxku = l for all k and Q (X)bi(X) = ai(X), where

{ai(X)} is an orthonormal set in G with Hai “L2 =1.

fl( )
Now since 2 lakl2 m, (2.2.11) implies that each d?j(z z), i, j = l,...,N
converges uniformly on compact subsets of ¢n to the entire analytic
function dij(z) = kél (bi(z)akxk)(bi(z)akxk) as m + m [30].
Define Dm(X) to be the determinant of the Gram matrix {d?j(X)}
i, j = l, 2,...,N, then Dm(X) are also the boundry Values of entire
analytic functions Dm(z) for all m. Clearly by uniformity of the

argument given above, 1im Dm(z) is an entire analytic functions.
m-HD

This implies that lim Dm(X) is the boundry value of an entire analytic
function, and there$3:e it either vanishes identically or is different
from zero a.e. X. In the latter case we agree to call the elements
bi(X), i = l,...,N a.e. X linearly independent in X*. The

above procedure on N permits us to construct a sub-maximal system

of bk(X), say bki(X), i = 1, 2""’MG’ which are a.e. X linearly

*
independent in X . Here maximality means that if bj(X) is different

from bk1(X), i = l, 2,...,MG, then for each X off a set of measure

46

zero. bj(X), bk](X), bk2(X),...,bkM(X) are llnearly dependent, i.e.,
* *

bj(X) 6 B(X) a.e. X. Obviously MG 5 dim X (X) f dim X and MG

could be finite or infienite. The proof of our lemma is complete.

We recall from the introduction that a homogeneous field

* ' *

Hr(t) = e1At Hr(0) is called an r-conjugate field of the field
H(t) if (i) H:(0)c:H(s: |s| sr)*, (ii) H:(Rn)=H(Rn) and
dim H:(0) 5 dim H(O) 5 dim K.

The following lemma is an immediate consequence of lemma 1.1.2.

2.2.12 LEMMA. Let A be any subspece of H(R”), then

lXt

 

 

V e A = H(R") if and only if A (X) = K(X) a.e. X
t
Proof: By Lemma 1.1.2 V eMt A = H(R") if and only if XXX) =
t
H(R")(X) a.e. X. But by the same lemma H(R") = V e”t Q(X)X implies

t
that H(Rn)(X) = K(X) a.e. X. and proof is finished.

 

 

*
2.2.13 COROLLARY. Let Hr(t) be an r-conjugate field of H(t), then

 

 

il- ' s
Proof: Use Lemma 2.2.12 and the fact that H(R") = Hr(Rn) = V elit

t
The following theorem gives necessary and sufficient conditions

H:(t).

for a Banach space valued homogeneous random field H(t), t 6 R".

to be r-regular. It also shows that the Defenitions 2.0.1 and 2.0.2
are equivalent, in particular, every r-regular field has an r-conjugate
field. Part (D) of the theorem given below extends the work of

Rozanov (c.f. [39] p. 12) to the B(X, Y)-va1ued homogeneous fields.

47

2.2.14 THEOREM. Let H(t), t e R", be a B(X, Y)-valued homogeneous

random field with spectral density f(X) satisfying (2.2.l). Then

(a) the field H(t), t e R", is r-regular if and only if there exists
an r-exponential B(K,X*)-va1ued function ¢r(X) such that

(i) cpr(X)KCX*(X) a.e.X (c.f. (2.1.7) for X*(X)),

 

(ii) Q (X)¢ (X) K = K(X) a.e.X (c.f. (2.1.7) for K(X)),

-l -l
(111) 9r(X) = [0* (X)¢r(X)]*EQ* (X)¢ (X)] is a nuclear function
-1
*

in K i.e. Q (X)¢(X) is a.e. X a Hilbert-Schmidt operator

on K to K and In “0* (X)¢r(X)H§dX < m, where H “2
R

stands for the Hilbert-Schmidt operator norm.

(b) Each ¢r(X) satisfying (i), (ii) and (iii) defines a certain

r-conjugate field for H(t), namely

 

(2.2.15) H:(t) = eiAt 0* (X)¢ (X)K, t E R",

with spectral density gr(X). Furthermore corresponding to each
r-conjugate field there exists an r-exponential B(K, X*)-valued
function ¢r(X) satisfying conditions (i), (ii), (iii) given above,
and each r-conjugate field admits the spectral representation (2.2.15)

with the help of the corresponding r-exponential function.

NOTE. We point out that the conjugate field is a K-valued field with

density gr(X) defined on K to K.

Proof. Suppose the field H(t), t 6 Rn is r-regular, i.e.
e”t H(s: Isl > r)1 = H(Rn).
Put Hr(0) = H(s: |s| > r). Let {ak(X)} be an orthonormal

V
t

basis in Hr(0)l. Also let bk(X) = Q*(X)ak(X). By Lemma 2.2.7

48

there exists a sub-maximal system bki(X), i = 1, 2”"’Mr (Mr is

finite or infinite) such that bki(X), i = l, 2,...,Mr are a.e. X
linearly independent in X*. Take a(X) E Hr(0)i, then there exists
a sequence of the form 2 akak(X) which converges to a(X) in
L2(K), and therefore it has a subsequence which converges to a(X)
a.e. X. But by Lemma 2.2.7 each bk(X) e B(X) a.e. X, where B(X)
is the linear span of bki(X), i = 1, 2""’Mr’ and therefore

- -T
ak(X) 6 0* (X)B(X) a.e. X. Thus we obtain that a(X) e 0* (X)B(X)

 

 

a.e. X. This implies that Hr(D)L(X) c Q* (X)B(X) a.e. X. But
-1
Q* (X)B(X) is the linear span of aki(X), i = l, 2,...,Mr therefore

 

-l _____.

*
Q (X)B(X) : Hr10)*(X) a.e. X. Therefore

*-1
Q (X)B(X), a.e., X.

 

 

 

(2.2.15) W10)

r

But H(t), t e R", is r-regular, i.e., V eIAt Hr(0)l = H(R"), and
. t
this by Lemma 2.2.12 is equivalent to

 

(2.2.16) il‘Tdfim = K(X) a.e. X.

r

Now from (2.2.15) and (2.2.16) we obtain that

 

*-1
(2.2.17) 0 (X)B(X) = K(X) a.e. X.

 

Also note that X*(X) = 0*(X)K(X) which implies dim X*(X) 5 dim K(X).
But dim B(X) = Mr and (2.2.17) implies dim K(X) f dim B(X) = Mr'
Also according to Lemma 2.2.7 we have Mr 5 dim X*(X), therefore by

putting all these together we obtain

 

* *
dim X (X) f dim K(X) 5 Mr 5 dim X (X) .a.e. X.

49

 

*

This shows that the dim X (X) = dim K(X) = Mr’ a.e., X, which says

*
f(X) = Q (X)Q(X) has a.e.X constant rank.

*

The B(K,X )-valued function ¢r(X) can be constructed in
the followlng way, deflne cpr(X)yk = ukbk(X), k = 1, 2""’Mr’ where
{yk} is an orthogonal basis in K and uk's are scalars subject

to E ufi < m and pk = 0 for superplus yk with k > Mr‘ Since
k

)Hmfl( )ywz =znub<<X1M

IA

czfi<mflnmwmqom%n

C fo(X) )ndX) “k

(The inequality is by (2.2.10)), or(X) can be extended continuously
to an operator over the whole K. We denote this extension also by

m (X). g (X) satisfies the required properties, i.e.

 

r (i) o,(X)K c x*(X) . (ii) 0*- (x)d,(X1K = K(X) a.e.X.
(ii) je‘m (q3r(X)y)x dX = o for (t) > r, y e K, x e x and
(iv) Q(X) = LQ*-](X)¢r(X)];LQ*-](X)wr(X)] is nuclear a.e.X
with f trace g(X) )dX = fl( X)yk, yk)dX < e.
For (i) note that
-l -l

é" E “0* (x)d,(X)ykn§dX (n E ”0* (i)ukbku2dX = (n g uEuak(X)ude

(2.2.18)

Efiénmgufim=gufi.m

-1 -1
which implies X HQ*( (X)¢( Myku m a.e. X, i.e., Q* (X)¢ (X)
k

can be extended to a Hilbert-Schmidt operator on K _to K dentoed
- -l
*
by wr(X). Obviously ur(X)yk = Q (X)¢r(X)yk e K(X) which implies

50

'k *
b(X)K : K(X), a.e. X. But since 0 (X) 6 B(K, X ), a.e. X, we may

* ~
consider the composition of Q (X) and pr(X) denoted by ¢r(X), i.e.,

$r(R)y==Q*(X)ur(X)y a.e. X

~ *
clearly ¢r(X) is a B(K, X )-valued function satisfying (i) and
-l
* ~
Q (X)¢r(X) is, a.e. X, Hilber-Schmidt operator-valued function

on K to K. But $r(X)yk = ¢r(X)yk for all k, a.e., X, and since

N

or, or are bounded linear operators we obtain that $r(X)y = ¢r(X)y

 

for any y E K a.e. X which gives (1). (iv) follows from (2.2.18).
-1

For (ii) note 0* (X)o (X) K = 0* (X)B(X) (this is by the way

 

that ¢r(X) was construZted). From this and (2.2.17) we obtain that
Q*-T(X)¢r(X) K = K(X), a.e., X.
For (iii) note that
Ie"kt(w,<X)y)x d> - I -1At(Q*(A)Q*-](A)mr(X)y)x dX
= fe'i*t(Q*-](X)¢,(X)y. Q(X)x)KdX
= f(Q*-1(X)¢r(X)y, eiAtQ(X)x)KdX .

*-1

Therefore (iii) is equivalent to show that Q (X)¢r(X)y E Hr(0)l

* *
H(t: |t| > r)‘. In fact we will show that Q (X)¢r(X) 6 Hr(0)’

where H*(0) = V{a (X), k = l, 2,...,M } in L2(K). Let {y }
r k r n k
be an orthonormal basis in K, for y e K we have y ~ 2 akyk, i.e.,
n k=1
Hy- Z okykn + 0 as n + m, where 6k, k = l, 2,... are scalars.
k=1
Also let {ak(X)} be an orthonormal basis in H:(0) : Hr(0)l, then

-1 n 2 *-l n *-1
HQ (X)vr(\)y kg] ukakak(X)H 2 K) = “Q (X)vr(X)y - k=1 ukakQ (X)
bk(X)H2L2(K)
*-1 *-1 n 2
= NO (X)vr(X)y - 0 kg] akukbk(X)HL2(K)
*-l *-l n 2
= “Q (X)¢r(X)y - Q (X) kg] ok¢r(X)kaL2(K)
*-l *-l n 2
= “Q (X)¢r(X)y - Q (X)mr(X) kg] akkaL2(K)
X-1 n 2
= “Q (X)¢r(X)(y - kg] akyk)H 2( )
. *" " 2
= I “Q (X)vr(X)(y - kg] dkyk)HK dX
" 2 *" 2
5 My - Z akkaK é" HQ (X)cp,.(X)HK dX

n ‘-l 2

My - z akykufi é" “0* (X)¢r(X)“2 dX

IA

n
2
5 c “y - Z akkaK by (2.2.18).
*-1
Therefore Q (X)¢r(X)y can be approximated by the members of
*
H:(0) in L2(K), and since Hr(0) is a closed subspace, therefore
*-1
o (uwuheh
*‘T 2
Q (X)¢r(X)K C H (0), where closure is taken in L (K). But
*‘1 *‘T

ak(X) = Q (X)cpr(X)u;1yk which implies that H:(O) C Q (X)¢r(X)K

(0). This gives (iii) and also shows that

 

“III-”5*

 

52

2( iXt H*(O) =

as closed subspaces of L r

-l
1%
Q (X)¢r(X)K, t E R", which is (2.2.15). The above spectral
-1
* *

*
representation says that Hr(t) has a density g(X) = [Q (X)@r(X)1

K). Therefore H:(t) = e

 

iXt
e

(X)mr(X)l with the desired nuclear property. Finally since

(om) = Vfak(X), k =1, 2,...,Mr} in K we have M:
*'1 *‘1 *‘1
Q (X)B(X) .e. X. But Q (X)B(X) = Q (X)¢ (X)K = K(X) a.e. X.

r
Therefore H (0)(X) = K(X)a.e. X. Now Lemma 2.2.l2 implies that

I {—1
3I- O
3‘.

(0)(X) =

 

 

 

a
i=§
r

' * 'k
V e”t Hr(0) = H(R"). Also note, dim Hr(0) = Mr 5 dim K(X). There-
t

*
fore Hr(t) is an r-conjugate field of H(t). The proof of one part
is now complete.
For the proof of the other part, suppose there exists an

'k
r-exponential B(K, X )-valued function ¢r(X) satisfying 2.2.l4

(i), (ii) and (iii). Define H:(t) = eiAtQ*-I(X)¢r(X)K, then
H:(t) is an r-conjugate field of H(t) i.e. (a) H:(0) : H(s: |s| > r)i
(b) tan H:(t) = H(R") and (c) dim H30) 511(0).
*-1 .
For (a) note that f(Q (X)or(X)y, e1AtQ(X)x)dX
. * *-l
= le'mm (X10 (X)QrUb’HXNX
= fe'ikt(¢r(X)y)(x)dX = D for It] > r, X E X, Y €1<
*-l .

Therefore Q (X)¢ (X)y I e1AtQ(X)x in L2(K) for It! > r, x E X, y E K

I"
This implies that H:(0)<: H(s: (s( > r)‘.

(b) follows from 2.2.14 (ii) and Lemma 2.2.12. (c) also follows

form 2.2.l4 (ii). Now note that (b) also says that

V n e1xt H(s: Isl > r)i = H(R") which is equivalent to say that
tER

53

the process H(t) is r-regular (this argument also shows that every
field admitting an r-conjugate field is r-regular). The proof of

part (a) is now complete.

(b) We already in the sufficiency part of the part (a) observed that
each ¢r(X) defines a certain r-conjugate field for H(t), which

admits the spectral representation (2.2.15). Now suppose that

H:(t) is an arbitary r-conjugate field to the random field H(t), t e R".
Let {ak(X)} be an orthonormal basis in H:(0). Put bk(X) = Q*(X)ak(X).
Define the operator-valued function ¢(X) as cp(X)yk = ukbk(X)

where {yk} is an orthonormal basis in K and u 's are scalars

k
subject to 2 pi < w and “k = 0 for surplus yk. Similar to

the necessity part of part (a) one can show that ¢r(X) can be
extended to an r-exponential B(K, X*)-va1ued function satisfying the
properties 2.2.l4 (i), (ii), (iii) (c.f. p. 49 ). Evidently Hr(t)
admits the spectral representation (2.2.15) with the help of this
corresponding r-exponential function 6. The proof of the Theorem
is now complete.

We recall from Definition 2.0.3 that the field H(t) is minimal
if it is r-regular,for r + 0. By using Theorem 2.2.l4 we arrive
at the following Theorem which gives necessary and sufficient condtions

for a Banach space valued homogeneous random field H(t), t E Rn

to be minimal.

2.2.19 THEOREM. A B(X, Y)-valued homogeneous random field H(t),
t E Rn with spectral density f(X) satisfying (2.2.1) is minimal
*
if and only if there exists a system of r-exponential B(K, X )-valued

functions {¢r(X), r + 0} such that each ¢r(X) satisfies 2.2.l4 (i),

54

 

.. ... * iXt *“1
(11) and (111). Furthermore Hr(t) = e Q (X)$r(X)K, r + O is
a corresponding conjugate system to the field H(t).
Let us for an instant take t 6 Zn, where Z is the set of all integers
then Definition 2.0.3 for minimality is equivalent to
(2.2.20) v n H(t: t # s)l = v e‘52 H(t: t # 0)i = H(z").
SEZ
As a corollary to Theorem 2.2.19 we obtain the following Theorem which
gives necessary and sufficient conditions for a discrete parameter

random field H(t), t 6 Z", to be minimal (c.f. Makagon [l4]; Miamee
and Salehi [26]).

2.2.21 THEOREM. A B(X, Y)-valued homogeneous random field H(t),
t 6 Z", with spectral density f(X) satisfying (2.2.1) is minimal
if and only if there exists a constant subspace B<: X*(X), a.e. X, such that

T 1

Q (X)B = K(X) a.e. X and fHQ*- (X)y“ZdX < m for any y E B.

*
Indeed there exists a constant B(K, X )-valued function 6, which

satisfies 2.2.l4 (i), (ii), (iii).

Proof. Note that r-exponential B(K, X*)-valued functions ¢r(X) in
Theorem 2.2.19 are identical to a constatn B(K, X*)-valued function,
say m, a.e. X for r < 1.

Take 8 = 6K, then B is a constant subspace of X*(X), a.e. X
and by 2.2.14 (ii), (iii) it satisfies the required properties. The
proof is complete.

We recall from Definiton 2.0.4 that the field H(t), t 6 Rn
is called regular if

n H(t: |t| > r) = {0} .
r>O

55

The following Theorem is an extension to our work in Chapter
1, Theorenll.l.13. It gives necessary and sufficient conditions in
terms of the spectral density of a B(X, Y)-valued homogeneous random
field H(t), t 6 Rn to be regular. The proof is similar to the

proof of Theorem 1.1.13, and is only sketched.

2.2.22 THEOREM. A B(X, Y)-va1ued homogeneous random field H(t),
t 6 R", with spectral density f(X) satisfying (2.2.1) is regular
if and only if there exists a family of r—exponential B(K, X*)-

valued functions mr(X), r + m, such that

(i) ¢r(X)K : X*(X) a.e. X.
-1 -1

 

 

 

 

* *

(11) Q (X)¢r1(X)K : Q (X)¢r2(X)K a.e. wlth r.l < r2 and
*-l
u Q (X)<p,.(X)K = K(X) a e X
r
-1
(iii) For each r, Q* (X)or(X) is a Hilbert-Schmidt operator
-1

a.e. X from K to K and fflQ* (X)¢r(X)H§dX < m.

Proof. Proof is similar to the one give for Theorem l.l.l3. Indeed

since V H(s: |sl > r)l is a doubly invariant subspace, by Lemma

 

 

 

 

r>0
2.2.12, regularity is equivalent to U H(s: [s] > r)1 (X) = K(X) a.e.X.
r>0
The latter with the help of Lemma 2.2.6 is equivalent to
'ZFWF—
U (Q B)(X) = K(X) a.e. X (c.f. Lemmas 1.1.6, 1.1.7).
r>0 (5: (5| > r)

Now with a similar technique as one given in the proof of Theorem
1.1.13 one can show that the latter condition is equivalent to the

set of conditions given above. Proof is complete.

56

2.3 Completely Minimal Fields. Necessary and sufficient conditons

 

for a B(X, Y)-valued homogeneous random field H(t), t e R", to be
minimal was obtained in Section 2, Theorem 2.2.19. In this Section

we will consider a sub-calss of the class of the minimal fields,

namely completely minimal fields, which plays an important role in
charactrizing the L-Markov and Markov properties in terms of the spectral
density. Such a charatrization is the subject of Section 4.

The notion of complete minimality for the Hilbert space-valued
random fields was introduced by Rozanov [39], where sufficient
conditions for a minimal field to be completely minimal were obtained
in that work. The main attempt in this section is to extend his result
to the B(X, Y)-valued random fields. The key to this extension is
Theorem 2.3.8 which says that every B(X, Y)-va1ued r-regular field
with spectral representation (2.1.6) admits a spectral representation
in the form H(t) = eiAt hg(X)K, where h(X): K + K is a.e. X
nuclear with integrable nuclear norm. We will also introduce the
notion of complete minimality for discrete parameter random fields.
Necessary and sufficient conditions for a discrete parameter random
field to be completely minimal is given (c.f. Theorem 2.3.17). In
particular we prove that every minimal field satifies the geometric

property (2.3.4). First we introduce some notations.

NOTATIONS. Let S : Rn be a bounded open region and T the complement
of S, T = Sc. SE denotes an e-neighborhood of S' and 5-6 denotes
the complement of the closure of TE. Also aS denotes the boundary

of S and 368 denotes the e-neighborhood of the 3S.

57

2.3.1 DEFINITION. With the same notations as Section 2, we call a

 

minimal field completely minimal if there exists a conjugate system
H:(t), r + 0, or by Theorem 2.2.19, equivalently. A corresponding
system of r-exponential functions ¢r(X), r + 0, for which
(2.3.2) h(T)i : v u(X)Q (X)¢r(X)K s H(T'5)i, for any, a > o,

r<6

Supp UCS
where T could be a bounded or unbounded region in Rn with S = TC
and a(X) is a Lebesgue integrable scalar-valued function which is

-l
*

also square integrable with respect to “Q (X)¢r(X)y“2, for all

y 6 K, with Fourier transform u(t), t 6 Rn (supp=support).

NOTE. We will show in Lemma 2.3.5 that the second inclusion in
(2.3.2) is always true, and in the case that S is bounded
~ *' * * -6 J.
u(X)Q (X)¢r(X)y E Hr(S). Furthermore we always have Hr(S) ; H(T )
for r < 6. Therefore (2.3.2) reduces to
* -
(2.3.3) H(T)i c v Hr(S) c H(T 5)* , for any a > o.
r<6

(2.3.3) gives a better picture of the notion of complete minimality.
In fact in the discrete case, t 6 Zn, minimality is equivalent to

. * L
(2.2.20) and the conjugate system Hr(t) = H(s: |s-t| > r) , r + O,

*

reduces to the conjugate field H (t) = H(s: s f t)i. In this case

since H(T) : n H(s: s f t), we obtain that always
tGS

*
H(T)l 2 v H(s: s # t)l = H (5). Now if (2.3.3) is satisfied,
tEs ,,
it implies that H(T)i : H (5). Therefore complete minimality in

the sense of (2.3.3) for the discrete parameter random field is

58

equivalent to

(2.3.4) or

H(T) = fl H(S: S f t),

where S is a bounded domain in Zn with complementary domain T.
(2.3.3), for a bounded domain, and in general (2.3.2) are reasonable

substituted for (2.3.4) in continuous parameter case.

2.3.5 LEMMA. Let H(t), t e R", be a B(X, Y)-valued homogeneous

minimal random field. Then the following statements (a), (b) and (c)
are satisfied.

(a) Let a(X) be an integrable scalar-valued function, square
integrable with respect to the weight ”0*-](X)¢r(X)yH2, y 6 K
with supp u c S, then U(X)Q*- (X)or(X)y e H(T‘r)i, where
T = 5°, T7" = (s?)c and u(t), t E R”, is the Fourier
transform of U(X).

(b) For a bounded domain S and U(X) as in (a)

*-1

U<X10 (X)o,<X)v e H:(S)

(c) Always H:(S) C H(T'r)i : H(T'd)i for r < 6 .

~ -1 _
Proof. (a) Note that in order to show u(X)Q* (X)¢r(X)y E H(T r)l
-1
~ * -
it suffices to show u(X)Q (X)¢r(X)y L H(T ") or equivalently

show that
-1

In (B(X)o* (X)¢r(X)y, e

1AtQ(X)y')dX = o for all t E T'r and y, y' E K.

59

But by Lemma 2.1.8 (a) we have

~ *‘1 - _.‘ ~ * *-]
én (u(X)Q <X)w,<X)y. e‘*t0(X)v')dX = (n e lit u(X)(Q (X)Q (X)o,(X)v)(y')dX
=én 6'1\tu~(U(X)(¢r(X)y)(y')dX.

Now let ur(s) be the Fourier transform of the scalar-valued r-
exponential function (mr(X)y)(y'). Then u r.(s) = O for Is! > r.

By using Plancherel identity we obtain (n e :1At (X )(cpr (X )y )(y')dX =
R

(s + t)ds . But u(s) = 0 for s E T and ur(s + t) = O

 

1
f u(s)u
for Is + tl > r. Therefore for t 6 T ur(s + t) = O for s e S.

Thus whenever t e T'r u(s)ur(s + t) = 0 on R", and this finishes

the proof of part (a).
iXt
Mfe e1Atu t)dt = lim 2 e k u(tk)A

(b) Note that K(X) = k
(2n)n Rn

 

The Riemann sums Z e1Atk u(tk)Ak are bounded. In fact supp u E S
k
which implies X Ak 5 volum of S, and u(t) is a bounded function.
k

Therefore [2 en‘tk u(tk)Ak| f C (volum of S). But
k

~ *‘1 n lth *-1 2
HU(X)Q (X)vr(X)y - kg] U(tk)Ak e Q (X)¢r(X)yH
-l

2 (volum of 3)2 “0* (X)¢r(X)y“2.

~ 2 *‘1 2
5 2 |U(X)| HQ (X)vr(X)yh + 2 c

The bound given above is independent of n and is integrable. Therefore

by the bounded convergence theorem 2 u(tk)Ak e Q (X)¢r(X)y
~ *-1 k 2 n
converges to u(X)Q (X)mr(X)y in L (R , X).
-l
*

But since Q (X)wr(X) is the square root of the density of the

60

*
r-conjugate field Hr(t)’

 

*-l iXt 1

* = iXt k *' *
Hr(t) e Q (X)4r(X)K, e O (X)vr(X)y é Hr(S) for tk 6 S.
ith ,-1
For tk é S, u(tk)6k e Q (X)¢r(X)y = 0, therefore
iAtk *‘ll *

Z u(tk)Ak e Q (X)¢r(X)y belongs to Hr(S) and this implies that
k
N *‘1 *
u(X)Q (X)¢r(X)y 6 Hr(S) which gives (b).

* -
(c) Note that Hr(t) ; H(s: ls-tl > r)l. Therefore H:(t) I H(T r)
with t 6 S which implies H:(S) C H(T'r)l. The second inclusion is
equivalent to H(T'G) C H(T-r) with r < 6 and the second inclusion

in (c) is always satisfied.

2.3.6 LEMMA. Suppose H:(t) is an r-conjugate field of H(t), i.e.,
H:(t) is a homogeneous random field with H:(Rn = H(R"), H:(0),G

*
H(s: |s| > r)1 and dim H (O) 5 dim H(O). Then H(t), t E R”, is an

. . * n
r-conjugate field of Hr(t)’ t 6 R .

*
Proof. All we need is to show H(O) 5 Hr (5: ISI > r)‘. For this it is
* *
enough to show that H(O) L Hr(t) for It] > r since Hr(Rn) = H(R").
Let fix t0 with |t0| > r. Then since H:(t) is an r-conjugate
field Of H(t) we have H:(t0) G H(s: |s-t0| > r)l which says
H:(t0) L H(s) for all s with |s-tol > r. But ltol > r, therefore
*
Hr(t0) L H(O).

2.3.7 COROLLARY. H:(t), an r-conjugate field of the field H(t) is

r-regular .

 

 

PM

tin

hC

61

Proof. It is an immediate consequence of the Lemma 2.3.6 and the fact
that every field withaulr-conjugate field is r-regular.

According to Theorem.2.2.l4, a conjugate field of every
homogeneous random field with spectral density satisfying (2.2.1) has
a nuclear density on K to K. But by Lemma 2.3.6 if H*(t) is a
conjugate field of H(t), then H(t) itself is a conjugate field of
H*(t). Therefore we may apply Theorem 2.2.14 to H*(t). Consequently
we Obtain a nuclear density on K for H(t), i.e., H(t) = eiAfEEEXMZ,
where h(X) is a B(K, K)-valued function; h(X) is nuclear a.e. X.
This result enables us to establish sufficient conditions for complete
minimality. It is by itself an interesting result, saying that every
B(X, Y)-valued homogeneous r-regular random field admits the spectral

representation (2.3.9) given below. Here is the detail.

2.3.8 THEOREM, Let H(t) = eiXt Q(XTX, t 6 R", be a spectral representation
Of an r-regular B(X, Y)-valued homogeneous random field with spectral
density f(X) e B+(X, X*) satisfying (2.2.l). Then there exists a

separable Hilbert space K (dim K 5 dim X) and a B+(K, K)-valued

nuclear function h(X) with Legesgue integrable nuclear norm such that

(2.3.9) H(t) = ellt h2(X)k.

*
Indeed there exists a B(K, X )-valued r-exponential function wr(X)

and a B(K, K)-valued r-exponential function wr(X), such that
t _ -1
(2.3.10) h (X) - [Q (X)CR,.(X):I WU).

Analogous to 2.2.14 (i), (ii), (iii) the following properties are easily

read Off.

Pr:

‘IU

55

 

 

(1) v (X)K 9 [Q (X)9r(X)1K a e X
*-1
(ll) 0%(X)K - Q (X)Wr(X)K = K(X) a e X
*‘1 ~k *‘1

(X)wr(X)]-]pr(X)) is a nuclear

(iii) h(X) = (IQ (A)9r(k)l']w(X)) (IQ

operator a.e. X, and its nuclear norm is integrable.

Proof. As we mentioned above the Theorem follows by applying Theorem
2.2.l4 to the H*(t) with the help of Lemma 2.3.6. But let us give some
details for the construction of the r-exponential B(K, K)-valued function
pr(X). The method of constructing ur(X) is similar to the one given

for ¢r(?) in Theorem 2.2.1?. All we need is t? replace H*(O), Q*(X)

and 0* (X) by H(O), Q* (X)¢r(X) and [Q* (X)§(>r,(X)]'1 respectively.
Indeed let {ak(X)} be an orthonormal basis in H(O) = Q(X1X; closure

in L2(K). Define bk(X) = 0*- (X)mr(X)ak(X). Then define

ur(X)yk = qu*-](X)wr(X)ak(X), where {yk} is an orthonormal basis in

K and pk, k = l, 2,... are scalars subject to Z luklz < m, where
k

pk = 0 only for surplus yk's. Properties 2.3.8 (i), (ii), (iii) and
the fact that pr(X) is an r-exponential B(K, K)-valued function with
Or(X) being of Hilbert-Schmidt type, a.e. , can be carried out in a
similar way as 2.2.l4 (i), (ii), (iii).

We are now in a position tO investigate conditons under which

a minimal field is completely minimal.

2.3.11 THEOREM. A B(X, Y)-valued homogeneous minimal random field
H(t), t E R", with density f(X) is completely minimal if

1 -1 1

(i) 0*- (X)w,(X)h%(X) = [0* (X)w,(X)iIo*' (X)9r(X)]-]wr(A) = v,(X) + I.

as r + 0 in the sense of strong convergence in K(X).

63

3%-] 1
(ii) -10 (x)d,(X)nz~uhi(X)n 5 c

Proof. By Lemma 2.3.5 all we need is to show the first inclusion in
(2.3.2), i.e.,
... *
H(T)*.g v u(X)Q (X)Tr(X)Ko

r<6
SUPP ufiB

The inclusion follows by proving the following steps.

Step 1: Every function a(X) of the space H(R") can be approximated

by functions Or(X)a(X) as r + O in L2(Rn, K).

Step 2: For each r and a(X) e H(T)l, ur(X)a(X) can be approximated

*

by X Uk(X)Q (X)wr(X)yk, where {yk} is an orthonormal basis in K
k

and Uk(X) = (hi(X)a(X). yk).

Proof of step 1. Any function a(X) é H(R") takes values in K(X)

a.e. X and 2.3.11 (i) implies pointwise convergence, i.e., wr(X)a(X) + a(X)

as r + 0 a.e., X in K norm.

But luv,(X)a<X) - a(X)u§ 11(v,(X) - man)“2

4 {Xv,(X)I2 + X112} 11am)“2

IA

5 4 {c2 + 1} ((a(XM)2 by 2.3.11 (ii).

Now since a(X) é H(R"), 4{C2 + l} “a(X)h2 is integrable function.

Therefore by bounded convergence theorem

én‘“¢r(4)a(l) - a(X)“2dX + 0 as r + 0,

64

Proof of step 2. Let a(X) 6 H(T)l, then by using spectral representation
(2.3.9) we get
iXt %
In (a(X), e h (X)y)dX = O for all t 6 T.
R
This implies that the function u(t), the Fourier transform Of the function

B(X) = (h%(X)a(X), y), has support inside S = Tc.
But h‘i(X)a(X) E K(X) : K. Thus

h*(X)a(X) = Z (h%(X)a(X), yk)yk, which implies

k
*‘1 I *-1
o (X)f,(X)h*(X)a(X) = E (h2(X)a(X). yk)Q (X)¢r(X)yk
~ *‘1
= E Uk(4)Q (A)?F(A)‘yk
~ *-1 2
Each uk(X)Q (X)¢Pr(X)yk e L (K), because

*-1

*-1
Ecnfik(X)o (X)w,(X)ykw2 5 E lfiknw2 “Q (X)w,<X)w2hvkh2

= 10* (X)w (AN)2 E IUk(X)IZ

1

5:no*' (X)<p,.(X)uzllh*i(X)a(X)ll2

*‘1 I
10 (X)vr(X)h§'Hh‘(X)h2IHa(X)H2

IA

5 CZ ha(x)h2 .

|~ *- o c '
which implies Z-huk(X)Q (X)cpr(X)yk112 is an integrable functlon and

consequently f “Uk(X)Q* (X)¢r(X)yk“2dX < m.

65

Now in order to use the bounded convergence theorem what is left is to

n ~ *-1
show N Z uk(X)Q (X)¢r(X)ykM? is uniformly bounded by an integrable
k=1

function. But

n ~ ‘ *-l ‘2 n ~ . -1 . 2
V kg] uk(X)Q (X)y,(X)ykl 5 (k;1 luk(X)lihQ (X)fr(X)ykh)
n ~ 2 n -l 2
5 kg] luk(X)1 - kgl=uo (K)Wr(k)yk“
m ~ -1
f k=1 luk(X)|2 - kg] “Q (X)i»,.(X)yk112

I, *‘1
vllh’2(X)llzllQ morn)“; ((a(XMlZ

IA

c2 llaIXMlZ .

IA

.8

and evidently fnlfla(X)“2dX <
R

-l -1
~ *
Therefore E uk(X)Q (X)<pr(X)yk converges to Q* (X)¢r(X)h%(X)a(X) =
Or(X)a(X) in L2(K) and the proof Of the theorem is complete.

NOTE 1. In case Of Hilbert space, h(X) = f(X) and the functions 4, u

are related by u(X) = f-%(X)¢(X)f%(X), ¢(X) = f%(X)w(X)f-%(X), a.e. X.

NOTE 2. A conjugate system with the properties 2.3.11 (i) and (ii)
for finite dimensional random fields with certain properties on the
spectral density (including some growth conditions on f'] ) always

exists [10], [28], [33].

66

EXAMPLE. Let us give a simple example to the Theorem 2.3.11. Let

*

¢r(X) be r-exponential scalar-valued function corresponding to the density
f](X) of a univariate field satisfying 2.3.11 (i). Note 2.3.11 (ii)

for the univariate case is trival. Now let X = Lp(EO,1], dX),l-+ l—= l

p q

2 5 p < m then x* = L9(IO,ll, dX). Define the density function f(X)
to be f(X) = f1(X)I where I is the identity on X into X*.

2(

K, K(X), x*(X) are L [0.11, dX), f?(X)Lp, f1(X)Lp respectively.
—1

The operator-valued functions Q(X), 0*(X), 0* (X), ¢r(X), ¢r(X) are
f?(X)I1, f%(X)Iz, f'%(X)I3, ¢r(X)Iz, mr(X)I4 respectively where
I : Lp-+ L2, I : L2.+ L”, I : L2 C;Lq +-L2 and I : L2 +»L2 are
l 2 3 4
identity operators.
Let us now consider the discrete case, i.e. t 6 Z". We start

with the following lemma.

2.3.12 LEMMA. Let T be any domain in Zn with complementary domain
5. Then for a minimal field H(t), t E 2",

(2.3.13) v u(X)Q (X)°PK.§H(T)'Ls
supp uCS

where X? is a constant B(K, X*)-valued function given in the proof of
the Theorem 2.2.21 and K(X) is a Lebesgue integrable scalar-valued
function which is also square integrable w.r.t. the weight MQ*-1(X)4yH2,
for y 6 K -u(t) is the Fourier transform of K(X).

-1
~ *
Proof. It is sufficient to show u(X)Q (X)Qy L H(t), t E K. This

-1 ,
is equivalent to {n (U(X)Q* (X)Qy, e1AtQ(X)x)dX.= 0 for all

t 6 T, x 6 X, y 6 K or In e'IAt U(X)(?y)x dX = O for all t E T
T

67

x E x, y E K. But In e'W‘ U(X)(<Py)de = (pr)x (n e‘”t a(X)dX =
T T

(my)x u(t) = 0 for t 6 T. Proof of the Lemma is complete.
In view of this lemma it is appropriate to make the following

definition.

2.3.14 DEFINITION. A B(X, Y)-valued homogeneous discrete parameter

 

minimal random f1€1d H(t). t 6 Z", is called completely minimal if
equality occurs in (2.3.13), i.e.,
*‘1

(2.3.15) V U(X)Q (X)¢K = H(T)‘.
supp u G S

2.3.16 REMARK. When S is bounded U(X) = Z u(t)eiAt is a poly-

-l tES
*
nomial, and since 0 (X)? is the density of the conjugate field
* _ L . * _ iXt *- .
H (t) - H(s: s f t) , i.e., H (t) - e Q (X)¢K, we obtain
~ *' *
that V u(X)Q (X)¢K = H (S). Therefore in the case that S
SUPP PCS

is bounded (2.3.15) is equivalent to H*(S) = H(T)‘, which is equivalent
to the geometrical property (2.3.4).

The following Theorem is a Corollary to Theorem 2.3.11, and it
gives sufficient conditions for a B(X, Y)-valued discrete parameter

minimal field to be completely minimal.

2.3.17 THEOREM. A B(X, Y)-valued homogeneous discrete parameter minimal
field is completely minimal if the constant B(K, K)-valued function
u, given in Theorem 2.3.8, satisfies the conditons (i) w is identity

on K(X) and (ii) Ng%(X)N2 Ng'*(X)pu < c, where g11 = 0* (X)e, and g is
the density Of the conjugate field H*(t) = H(s: s f t)*.

68

NQI§5_ When the dimension of a field H(t), t 6 Zn is finite the
condition 2.3.17 (i) as we show in the proof Of Theorem 2.3.18 given
below is not needed and a sufficient condition for a minimal field to
be completely minimal is that l“f(X)“1Nf-](X)M < C, a.e., X, where f']
is the generalized inverse Of f. In the following we show that every
finite dimensional minimal random field H(t), t 6 Zn satisfies the
geometric property (2.3.4) or by Remark 2.3.16 equivalently satisfies

(2.3.15) for bounded domains S with complementary domains T. This

interesting result is the content Of the next Theorem.

2.3.18 THEOREM. Let H(t), t e 2", be a finite dimensional homogeneous
discrete parameter minimal random field. Then for any bounded domain

5 in Zn with complementary domain T,

H(T) = ,n H(t: t f s)
563

NOTE. Theorem 2.3.18 says that every discrete parameter minimal field is

completely minimal in the weak sense (restricted to bounded sets).

Proof. By Remark 2.3.16 Theorem follows by showing that (2.3.15) holds
in this case. We recall from Theorem 2.2.21 that the field H(t), t E z",
is minimal if and only if there exists a constant B(X, X)-valued function

.9 such that

1

(i) CPX.Cf;i(X)X, a.e. X, (ii) f-;5(X)<PX = (5(X)X a.e. X,
(iii) fnvflf'%(X)mҤdX < m. These conditons are equivalent to (i)
T

rang f = mX is constant, a.e. , (ii) [nuflf‘IHdX < m, where
T

69

' -= (f‘%)*f'% and X is a finite dimensional Hilbert space (c.f.

[15]). Now assume the field H(t), t 6 Z", is minimal. Define the Operator
P on X onto ex to be the projection onto ¢X. Clearly PX = WX.
Also since any a(X) 6 H(Z") takes values in f%(X)X, a.e. X, we have
f'%(X)P f8(X)a(X) = a(X), a.e. X. Take a(X).€ H(T)‘, then ux(t) =
f e'ikt (f%(X)a(X), x) = O for t 6 T, x E X. Therefore for any

x 6 X UX(X)= (f%(X )a(X) , x) is Lebesgue integrable with supp ux(t) C S.
But 5 is bounded, therefore UX(X) is bounded. This implies that

U x(X) is square integrable w.r.t. the weight '“f-%(X)Qy“2, y.€ X.

Now let {xk}k=1 be an orthonormal basis in X. Then
I n 1‘5
f3(X)a(X) = kg] (f (X)a(X). xk)xk
which implies that
n
a(X) = f‘*(X)p f%(X)a(X) = z (f3(X)a(X), xk)f‘%(X)pxk, a.e. X.
k=1

Put Uk(X) = (f%(X)a(X), xk) and note that ka 6 ex. Then the expression

given above implies that a(X) E V U(X)f-%(X)¢X. Therefore
SUPP W95

H(T)1,G V U(X)f'3(X)¢X. Proof is complete by applying Lemma 2.3.12.
supp USS

2.4 Markov Minimal Fields. In this section we will discuss Markov and

L-Markov properties of a homogeneous random field. Necessary and sufficient

conditions for a completely minimal B(X, Y)-valued field to be Markov

or L-Markov will be given. The result extends the work of Rozanov [39]

to the B(X, Y)-valued fields. Similar new results for discrete case

are also obtained. With dwesame notations as in Section 3 (c.f. Notations

preceding Definition 2.3.1), in this Section we shall consider bounded

70

open regions S.g Rn for which

 

(2.4.1) (s€)'e = (s'e)e = s

for sufficiently small 6 > 0 (For instance, this property is possessed

by regions with finite piecewise smooth boundries).

2.4.2 DEFINITION. we call a field H(t), t e R", Markov if for any

 

bounded domain 5 (satisfying (2.4.l)) with complementary domain T,

6

(2.4.3) P(S)H(T),g H(a S)

for sufficiently small .6 > 0, where P(S) stands for projection onto
H(S).

The following results are due to Rozanov [39], which can be
observed also from the work of Kallianpur and Mandrekar [8 ] and
Mandrekar [16].

(a) Markov property (2.4.3) implies

(2.4.4) P(55)H(T5),s H(BES) . e > 5
(b) (2.4.4) implies that
(2.4.5) P+(S)H+(T) = H+(aS),

where “+(5) = .n ”(56) and P+(S) is projection onto H+(S);
€>0

(c) (2.4.5) implies that
(2.4.6) H+(5)i L H+(T)l ;

(d) (2-4-5) holds if and only if

(2.4.7) H(Sé) L H(SS) for any 6 > O ;
(e) if the following condition

(2.4.8) H(S])le(T ), : H((S n T2)€)

2 1

holds for sufficiently small 6 > 0 and bounded domains S], S2 with
$24: $1, i.e., T1 and S2 are disjoint, then (2.4.3) through (2.4.7)
are equivalent.

Let us now also consider L-Markov fields. L-Markov property can
be described as follows: Let L be a sufficiently good neighborhood
of zero, namely such that the regions SL = §'+ L = {s + t: s e S’,
t 6 L} satisfies (2.4.1). We denote a S = as + L the thickened boundry

L

of S and by ‘355 its é-neighborhood.

2.4.9 DEFINITION. A homogeneous field H(t), t e R" is called L-Markov

 

if for every bounded domain S satisfying (2.4.l),
E
(2.4.l0) P(SL)H(TL),g H(aLS) ,

for sufficiently small 6 > 0.
In analogy with the case of Markov property, assuming (2.4.8)
we have (c.f. Rozanov [39]) the following equivalent properties for the

determination of L-Markov property.

H(t) is L-Markov e P(SE)H(TE).C H(af s) e > 5

(2.4.11)

72

.L .. J.

L H(TO) , 5 > 0

5) L

a H(SL

2.4.l2 LEMMA. The property (2.4.8) is satisfied for completely minimal
fields.

Proof. Let S1 and S2 be bounded domains with T130 52 = Q. Then

(T1 U 52)"6 = T{5 u 536. From (2.3.2) we have

6 L ~ *-1
H((s1.n l2) ) .c v U(X)Q (4)4,(X)K .

a<6 _€
supp u9(T1U 32)

where u(t) is the Fourier transform of a(X) and can be represented

as a sum u(t) = u](t) + u2(t), where

-G

‘6 u(t), t e 52

u(t) , t 6 T
u](t) = and u2(t) =
o .tilf 0 .12452

But u2(t) is a bounded function with bounded support 556 , therefore
its Fourier transform fié(X) is a well defined bounded function. Thus
by Lemma 2.3.5(b) U2(X)Q* (X)¢a(X)y belongs to H(TS'G) for

a < 5 while the function U](X) = B(X) - 32(X) is Lebesgue integrable
and is square integrable with respect to the weight NQ*- (X)¢h(k)yflz

with supp u],5 Tie . Therefore by Lemma 2.3.5(a)

~ *-1 E-a l 6-6 i
u1(X)Q (X)¢&(X)y e H(s1 ) : H(s1 ) for o < o .
Now by taking 5 =.€ we obtain
1 *-l ~ *-l

Gnm”(ngon=fifioo uw<oy+oom (oguneHnJNHopt

73

Therefore

J. .L .L

H((S ..f112)€) ,c H(TZ) v H(S]) ,

l

which is equivalent to (3.1.6) and the proof of lemma is complete.

2.4.13 REMARK. Before we state the main result of this Section, by using
I

1
Lemma 2.2.6, we observe that the orthogonality of H(SE) and H(TE) ,
o > 0 which gives the L-Markov property for completely minimal fields
is equivalent to

,-l ,-l
(2.4.14) 4" (Q (X)b1(X), Q (X)b2(X))dX = 0, b1 6 B 5’ b2 6 B

5
SL T

L
for an arbitary bounded domain S with complementry domain T. If by
convention (c.f. Rozanov [38] and the note given below) we speak of the
generalized Fourier transform f‘](t) as a linear operator from X*
to X defined by

-1 -l

é" (0* (X)b](X). 0* (noznnaA e f I

~ ~-1 ~
Rn\SE Rn\Ti b2(t)(f (s-t)b1(s))dsdt,

1

then (2.4.14) is equivalent to say that the supp f' lies in the domain

L - L = {t-s, t,s 6 L},

supp f-],G L - L .

NOTE. When complete minimality is assumed the following Theorems:
Theorem 2.4.15 and Theorem 2.4.18, give necessary and sufficient conditions
for L-Markov and Markov properties respectively. The conditions are in

terms of the conjugate fields. However it would be interesting to obtain

74

1

effective L-Markov and Markov conditions in terms of the f' , the inverse

of the spectral density f(X). For the univariate fields over R such

conditions for Markov property is obtained by N. Levinson and H.P. McKean

[12]. Indeed they proved that under some condition on the growth of f'],

the univariate homogeneous field H(t), t E R, is Markov if and only if

-1

f is an entire function of minimal exponential type. This result was

extended to the univarite homogeneous reandom fields with parameter t

in R", by Kotani [10]. We remark that under the condition that f"1

has a polynomial growth, the Markov and L-Markov properties for the
univariate case, t 6 R", also were studied by G. M. Molchan C28] and
L. D. Pitt [33]. Kotani introduced the concept of generalized Fourier

transform for a sufficiently large class of functions. Kotani showed

1 1

that f' the generalized Fourier transform of f' exists under some

growth condition on f'], and is defined as an "ultradistribution". We

note that when complete minimality is assumed and f.1 exists, Markov

property is equivalent to supp f'1 = {0}. The characterization of

Markov property in terms of f'] for the multivariate random fields with
parmeter t in Rn is discussed by Rozanov [39], p. 16. In view of
these observations, the last paragraph of Remark 2.4.13 is in need of
further scrutiny. To our knowledge,conditions for the existence of

f-l

have not been studied for the infinite dimensional case, and deserves
serious investioation.
The following Theorems are the main results of this section which

extends the work of Rozanov [39] to the Banach space case.

2.4.15 THEOREM. A B(X, Y)-valued homogeneous completely minimal field

H(t), t 6 R", is L-Markov if and only if for each r > 0, the Fourier

75

transform §r(t) of the density of conjugate field H:(t)

*-l -l

g,<4) = [Q (A)Pr(4)l*[0* (4)4 (4)].

r
satisfies the condition

Y‘

(2.4.16) supp Er.: Lr - L , r > 0

Proof. Because of the Lemma 2.4.12 it is sufficient to establish that
(2.4.16) is equivalent to the last condition of (2.4.11), namely

1 .L

(2.4.17) H(sf) I H(Tf) , o > 0.

Let us assume that (2.4.17) is satisfied. By Lemma 2.3.5(c)
I I
H:((sf+'”)c),c H(SE) and H:((Tf+")c).s:H(Tf) . Therefore (2.3.16)

*
implies that Hr((TE+r)C) I H:((Si+r)c). This is equivalent to

,-l -l

I e'i*(5't)(o (x)4,(x)y. 0* (4)2,(4)x)dx = o for s e (5fi*r)c. t e (lf+r)°.

x, y 6 K.
Now let v be outside the closed region Lr - Lr.

6+r
L

Then there exist a

5 > 0 such that v = s-t with s 6 (S )c, t 6 (T6+r)c’ which implies

L

fe'ikv(gr(X)x, y)dX = 0.

This implies (2.4.16).
Now suppose (2.4.16) is satisfied. Since the field H(t) is
completely minimal according to (2.3.2) we have
I -l I *-l
6 ~ * 5 ~
H(TL) .9 V u](X)Q (A)¢r(X)K. H(SL) .9 V u2(X)Q (X)¢r(X)K .
r,<€ r.<€

where u1(t) = 0 for t.€ TE and u2(t) = 0 for t 6 SE .

76

u](t) has compact support and by (2.4.16) §r(t) also has compact support,

u1(X) and (gr(X)x, y) are bounded functions, therefore

4" (5](X)Q (A)Wr(4)xa 52(X)Q (4)4 (X)y)dX

= 4" 31(A)(9r(X)X, y) E;(X)dX
= In Him In H (S)(g( 5- -t)x. y)ds dt = o
R

because for t 6 SL , u2(t )= O and for t ¢ SL , In u s)(gr (s- t)x,y)dsdt=
The latter follows from the fact that by (2.4.16) (gr(s- -t)x, y) is

only different from zero on {5: s-t e L6 - Le} with r < 6, but for

t 4 SE we have {5: s-t 6 Le - L6 }c T356 9 TE and by using the
fact that u](t) = O for t 6 TL we obtain u1(t)(§a(s-t)x, y) =

for t 4 SE , s e R". Proof is complete by noting that the elements of
H(SE)‘ and H(TE)l can be approximated by the elements of

.V 31(X)Q* (X)Wr(X)K and V 32(X)Q* (X)¢r(X)K as r + 0 respectively.
ul u2

Markov property is equivalent to the L-Markov property with
respect to all arbitary small neighborhoods L. The following theorem

is a stright consequence of the Theorem above.

2.4.18 THEOREM. A B(X, Y)-va1ued homogeneous completely minimal field
is Markov if and only if for each r > O, the spectral density
*‘1 * *‘1
gr(X) = [Q (4)4r(X)1 [Q (X)¢r(X)J

*
of the conjugate field Hr(t) is a 2r-exponential function, i.e.,

-iXt _
é" e gr(X)dX - O , It] > 2r

0.

77

Remark 2.4.13 and the fact that for completely minimal fields
(2.4.16) and (2.4.17) are equivalent imply that if the generalized
Fourier transform exists, a completely minimal field is L-Markov or

1 S L - L or supp f']

Markov if supp f' = {0} respectively. As we
mentioned in the note preceding the Theorem 2.4.15, the existence of the
generalized Fourier transform has not recieved satisfactory attention
for the Hilbert space case as well as the Banach space case, and is need
of further study.

Let us now consider the L-Markov property for discrete parameter
random fields, i.e., t 6 Z". We start by introducing the following

notations.

NOTATIONS. Let L be a fixed finite neighborhood of zero in Z".
We are assuming that O E L. For any bounded domain S 9 Zn, define

L

S = S + L = {s + 2, s 6 S, 2 E L}. Also by L-boundry of S we mean

aLS = SL\S. Note that aLS<: T, where T is the complementary domain

of s in 2".

2.4.19 DEFINITION. A discrete parameter random field H(t), t e Z",

is called L-Markov if

(2.4.20) P(T)H($)_c H(aLS),

where S is a bounded domain in Zn with complementary domain T and

P(T) stands for projection onto T.
2.4.21 LEMMA. (a) Under the assumption

(2.4.22) H(SL)Q H(T) = H(aLS),

78

the following condition is equivalent to (2.4.20)

(2.4.23) P(T)H(S) = H(SL) n H(T)

(b) (2.4.23) is always equivalent to

(2.4.24) H(S

Proof. (a) (2.4.20) implies (2.4.23) from the following expression
L L _ L _ L
H(S )_n H(T) S P(T)H(S ) - P(T)H(S) C H(a S) - H(S ) u H(T).

It is clear that (2.4.23) implies (2.4.20).
(b) Note that for two closed subspaces A and 8, since P(A)B is a
splitting subspace [40], we always have A V B e A = B e P(A)B.

Assuming (2.4.23) with the help of expression given above we obtain that

H(T)VH(S) e H(T)

H(T)i = H(z”) e H(T)

H(S) e P(T)H(S)

H(S) e H(SL) n H(T)

CM§19HBHIHWUCHBH.

J.
Therefore H(T)i I H(SL) .
H(SL) = H(T)‘ e H(T) n H(SL), which implies that P(T)H(SL) = H(T) IlH(SL)

or P(T)H(S) = H(T) IIH(SL), which is (2 4.23). Proof of the Lemma is

Now suppose (2.4.24) holds. Then

complete.

The following Lemma is similar to Lemma 2.4.12 with a similar

proof.

79

2.4.25 LEMMA. Let H(t), t 6 Zn be a B(X, Y)-valued homogeneous

completely minimal random field. Then

(2.4.26) H(S ),n H(T = H(S u T

1.: )

l 2) 2 ’

where 51 and S2 are bounded domains in Zn with complementary domains
T1 and T2 respectively satisfying T]_n $2 = B.
Let s1 = 5L and s2 = s in (2.4.26). Then we obtain that
(2.4.22) is satisfied for discrete parameter completely minimal fields.
The following Theorem gives necessary and sufficient condition
for a Banach space-valued completely minimal field to be L-Markov. This

Theorem in the univariate case implies Rozanov's Theorem (c.f.

[37] Theorem 3).

2.4.25 THEOREM. A B(X, Y)-valued homogeneous discrete parameter completely

minimal random field H(t), t e z”, is L-Markov if and only if
supp 5.9 L.

~ * *
where g is the Fourier transform of the density g(X) = [Q $3 [Q o];
'k
of the conjugate field H (t) = H(s: s f t)‘.

Proof. Proof is similar to the proof of Theorem 2.4.18.

2.4.26 COROLLARY. Let H(t), t e 2", be a finiate dimensional

homogeneous minimal random field. Suppose the density of the field,

1

f(X), satisfies Hf“ Hf'1h < c, where f' is the generalized inverse.

The H(t), t 6 Z", is L-Markov if and only if

(2.4.27) f'1(X) = A(I - Z A(t)ei*t)
teL\O

80
where A(t), t e L are matrices with A= A(O),“Au > 0.

Proof. This follows from Theorem given above and the note to the
Theorem 2.3.17. Also note that the point zero always has to be in the

supp f']. Proof is complete.

2.4.28 COROLLARY. Let H(t), t 6 Zn, be a finite dimensional homogeneous
minimal random field. If the spectral density f(X) is bounded i.e.,
“f(X)“ <C a.e. X, then H(t), t 6 Zn is L-Markov if and only if

(2.4.27) holds.

Proof. With a similar proof to the one given for Lemma 2.4.21 (b),
L-Markov property (2.4.20) implies (2.4.24), and the latter implies that
supp f",9 L, which implies that (2.4.27) holds. For the other part
suppose (2.4.27) holds. This implies aflf'IH < c a.e. X. But

lflfu < c a.e. X. Therefore (hfhlhf'lfl < c a.e. X. Now apply Corollary
2.4.26. Proof is complete.

CHAPTER III
A RECIPE FORMULA FOR THE LINEAR INTERPOLATOR

Introduction. A set of reals or complex-valued random variables gx(t)
over a probability space (a, B, P) depending on a parameter t in
Zn = the Cartesian product of 2 (set of integers) with itself n-times, where
the index x runs through a set X, is called a (discrete parameter)
random field. Let E denote the expected value. We assume that
E gx(t) = O and E|£x(t)|2 < w as elements of a Hilbert space
H = L2(n, B, P) of random variable a, Elgl2 < a, with the scalar product
E 533 E, n 6 H.
In this chapter we assume that X is a finite dimensional
Hilbert space and for each t‘e Z", gx(t) is linear in the variable x.

As such, one can .express gx(t) in the form
€x(t) = a(t)x; x e x. t e z".

where for each t, g(t) is a bounded linear operator on X into the
Hilbert space H. In view of the relation above, the operator-valued
g(t), t 6 Z", is also called a random field. These two versions of a
random field will enterchangbly be used throughout this chapter. A
random field g(t) is called homogeneous if E gx(t)E;T§)' depends

only on t-s for all x, y e X. Let xk, k = l,...,q be an orthonormal

basis in X, then we will refer to gk(t) = 5k (t), k = l,...,q, as the
x

81

82

kth component of the homogeneous random field g(t). We will also
call g(t), a q-variate homogeneous random field denoted by
(c.f. Notations 3.1).

€(t) = {gk(t)}1fqu

The main purpose of this Chapter is to derive a recipe formula
for the linear interpolator in the time domain which seems to have
potential applications. To be more precise let a(t) = {£k(t)}1<k<q,
t E Z", be a q-variate homogeneous random field and let ‘ '
Tk’ k = l,...,q, be finite subsets of Z". Let us assume that all

the values gk(t), k = l,...,q, are known except for the values

£k(t), t 6 Tk, k = l,...,q. The problem which is the subject of this
Chapter is to express the linear projection of the unknown component
5k(t0), t0 6 Tk, in terms of the known components g£(t), t 4 T2’

2 = l,...,q, as an infinite series expansion. In a recent paper

[42] H. Salehi derived an explicit representation for the

linear interpolator of a univariate random field under the boundedness
assumption on the spectral density of the field. Earlier works in

this area were carried out by A.M. Yaglom [48] and Yu. A. Rozanov [35].
Our intention in this Chapter is to carry out an algorithm for the linear
interpolator of a q-variate random field under a more relaxed assumption
on the spectral density. This algorithm reduces to the one givenby Salehi
in [42] for the univariate case. We replace the boundedness

condition on the spectral density by its square norm interability.

3J1.Notations and Priliminaries. A matrix a. consisting of elements

 

a k = 1,... n, j = 1,...,m) have n rows and m columns, will

kj(

be denoted by (ak.}1535m

or simply {ak.}m . A matrix a consisting
3 lfkfn 3

n

83

of a single row of elements a1,...,am, will be called a row vector

i}n

l<j<m m

d . - - . ' .‘ .

enoted by {33} or {aJ} , and by {b11151fn or {b
we mean a matrix b consisting of a single column of elements
b1""’bn’ such a matrix is called a column vector. As usual the

= n . .
and b {bj£}n 15 a matrix

. n
product of two matrlces {akjlm
"I

c = {Ck2}m where Ckz =j§1 akjbj£° Further, the adjoint of a

matrix a = {a .}m is defined to be the matrix. a*= {5- }n.
k3 n 3k m

where ask is the complex conjugate of ajk' Also the Euclidean norm,

“a”, of a matrix a = {akjlg is defined to be

n m 2 P
(3-1-1) 113” = Z Z lakjl
k=1 j=l

It is known that g(t) has the spectral representation
- - iXt _ .
a(t) - fe o(dX), where 4 - {4k}1fqu, is the random spectral
measure associated with the multi-dimensional homogeneous random
field [19], [34], [36] (whenever the domain of integration is missing,
it is understood that the integration is over Tn’ the n-dimensional
torus, which is the Cartesian product of the unit circle T with
itself n times). Define the measures Fk£(o) = E(Ok(A)O£(A)),
k, 2 = l,...,q, then the square matrix {Fk2}g is called the spectral
measure of the process €(t). In the case that all the elements Fkt
are absolutely continuous with respect (w.r.t.) to the Lebesgue

dF
= q ' = ———k£ °

measure X, f(X) {fk£(X)}q wlth fk£(X) dX (X) 15 called the
spectral density of the q-variate homogeneous random field g(t).

In this work we assume that the process has a density f(X).

84

The Hilbert space L2(F), which plays an important role in
the analysis of a homogeneous random field, consists of all vector-

valued functions b(X) = {pk(X)}q with [(b(X)f(X)p*(X) dX =

I I ¢k(X)fk£(X)$k(X)dX <w. The inner product between m and
k,£=l

O in L2(F) is given by < o,u > = f o(X)f(X)u*(X)dX. Let H

E
be the span clousure of the random variable gk(t), t 6 Zn

and l< = l,...,q, in H, i.e.,

HE = v {gk(t), t e z", k = l,...,q} .

Then there is an isometry between HE and L2(F) which to any

h 6 HS corresponds a unique 4 = {mqu in L2(F) with
h = fw(X)¢(dX). For the definition of the integral f¢(X)¢(dX)
and the isometry in a general setting see [17], [34]. In our case
the integral foo(dX) reduces to 2 fok(X)oK(X). For
addional information on q-variate homogeneous processes see [19].
Let N(T) be the closed subspace of Hg spanned by the
differences gk(t) - Ek(t), t E Tk, k = l,...,q, where Ek(t) is
the projection of gk(t), t e Tk’ onto the V{€k(t), t ¢ Tk,
k = l,...,q}, and let A(T) be the closed subspace of L2(F)
corresponding to N(T) under the isometry map between L2(F) and
H,. Let B(T) be the space of vector-valued functions b(X) =

{bk(X)}q whose components bk(X) are trigonometric polynomials

of the form

85

*

satisfying (i) b (X) 6 Range f(X) a.e.X; (ii) the integral
[b(X)f-1(X)b*(X)dX < x, where f'1(X) is the inverse of the restriction
of f(X) to the Range f(X).

The following lemma is a special case of Lemma 1.1.3 or 2.2.6.

It is also given in [38] and [44].

B(T)f'1, meaning that for any 6 e A(T), there
-1(

3.1.2 LEMMA. A(T)

exists a unique b(X) in B(T) such that g(X) = b(X)f X).

The following theorem gives a necessary and sufficient
condition for gk(t0) - Ek(t0) to be different form zero for each
k, I 5 k 5 q, i.e., the interpolation is imperfect (see McKean [51]).
This theorem was originally proved by Kolmogorov for the univariate
case [9 ]. Extensions to multivariate case were carried out by

Rozanov [36] and Masani [18].

31.3 'MEOREM. In order to have imperfect interpolation for the
q-variate homogeneous random field, it is necessary and sufficient

that f is non-singular a.e.X, and that
(3.1.4) [ nf'ludX < m.

It should be noted that the notion of ”imperfect interpolation"
introduced here is equivalent to the concept of "minimality” given
by Rozanov [36] and "minimality of full rank" given by Masani [18].

The following theorem gives a recipe for Ek(t0), t 6 T

0 k’
the projection of the gk(t0), t0 6 Tk, onto the l/{gj(t): t 4 Tj,

j = l,...,q}, as a spectral integral.

86

r - r 1 n _ °
3.1.5 THEOREM. Let s(t) - {,k(t))lfqu, t E Z , be a q varlate
homogeneous Gaussian random field which has imperfect interpolation.
Also let ak(t0), t0 6 Tk, be the projection of gk(t0), t0 6 Tk,

(a fixed but arbitrary component of g(t )) onto V{€k(t),

0
t 4 Tk, k = 1,. . q}, where Tk, k = l,...,q are finite subsets
of 2". Then
(3.l.6) Ek(t0) = j6k(X)o(dX).

A = A q
¢k(4) L¢kj(A)} has the form

(3.1.7) ¢k(X) = e 0 - b (X)f'1(X),

_ q - - =' .
where 5k - {okj} wlth 5kj O for 3 f k and Gkk 1,

U'
7'?
A
>’
v
II

C] .
(3.1.8)

a 'Xt . .
b .(X) = 2 a .(t)e‘ , J = l,...,q .
k3 teT. K3

and the coefficients {akj(t): t E T. j = 1,.. ,q} can be obtained

J,
from the following systenlof'equations:

q x
. - . = O f t T , 2 k,
jél SET. pJ£(s t)akJ(S) or e 2 2
J
(3,1,9) jgl SET. pjk(s-t)akj(s) = I for t = t0 and
J
2 pjk(s-t)akj(s) = O for t E Tk\t0.

.= T.
j 1 56 J

87

where f-1(X) = {p.,(X)}q and p.8(t), t 6 Z", is the Fourier

JL 4 J _ t
. . ~ = 1X
coefflc1ent of pjg at t, namely, pj£(t) [e p

n
., A CIA, At = . ..

Note. We point out that this Theorem can be found in [36] p. 101 where

the third equation in (3.1.9) is missing. For completness we will

give the proof below.

Proof. By the isomorphism between the time domain Hg and spectral

. 2 r : _
domaln L (F), we have 5k(to) - ak(t0) - fhko(dX) where hk 6 A(T).

Therefore by LEMMA:&1.2 hk(x) = bk())f-1(A), where bk(A) = {bkj(4)}q

with b .(X) = Z a .(X)e‘t*, j = l,...,q. This implies that the
k3 tET. k3 2 iXt0
corresponding imagé of §k(t0) in L (F) is e 6k
iXt
which is orthogonal to A(T), i.e., <e 06k - bk(X)f-1(X), g(X)> = O
..1(

- bleif‘lm

iXt
52
in view of (3.1.5) is in A(T). The orthogonality can be expressed

for any g(X) 6 A(T). Now take g(X) = e f X), t 6 T3, which

as

f(e 06 - b (X)f“1(X))f(X)(elita f‘1(X))*oX = o for all

C

OY‘

iXt . .
(3.l.lO)}e Ookf(X)l:e”ta£f'1(X)J*dX - jbk(X)(e‘4to

*

zf-1(X)) dX = O,

t 6 T

2’ 2 = I ...,q.

But f(X) = {fij(X)}g and f'1(X) = {p (X)}g, therefore 6kf(X) =

q *1 =
{fki} and 6 f (X)

2

88

Substituting the above quantities in (3.1.10) we obtain

iX(t
(3.1.11) fe

o‘t)

ll MD

-— -iXt. ‘ _. _
ka-(XJPZJ-(Udk - ie jg] bkj(i)p£j(}x)d}x - 0,

j 1

t 6 T

2’ Z = l,...,q.

But since ff'1 = I we have

1 for k = Z

(3.1.12) jgl fkj p3}Z

0 for k f C.

Now let k f 2, then with the help of (3.1.8), (3.1.12) and the fact

that f'1 is self-adjoint, (3.1.9) becomes
3 X a .(s) fe-ikt enS p = 0 t 6 T 2 f k
i=1 $6Tj k3 35 3

or

(3.1.13) 3 Z a .(s) 6. (s-t) = 0 . t e T , 8 f k.
i=1 SETj “3 JK 3

which is the first equation in (3.1.9)
For k = Z and t 6 Tk\t0 similar to (3.1.13) we have

2 akj(s)p.k(s-t) = O, and for k = Z, t = t0 we have

'=1 T. J
J 56 J

89
1 - I Z a .(s)p. (s-t) = O, which gives the last two equations
i=1 SET. k3 3k
J

in (3.1.9). As in [42] we can easily show that the system of

equation (3.1.9) has a unique solution. This completes the proof.

 

:12 ‘A_Recipe Formula. In order to derive a recipe for ék(t0),
t0 6 Tk, as a series expansion involving the known values g£(t),
t 6 T , 1 f K 5 q, we are forced to impose certain additional

condition on the spectral density. We have already mentioned that

under the boundedness of f and the square integrability of f"1

Rozanov [35] and Salehi [42] have obtained such expansions.

In this section we remove the restrictive boundedness assumption

on f and replace it by a more relaxed condition, namely the square
intergrability of f.

We now state the main result of this Chapter.

3.2.1 THEOREM. Suppose that the spectral density f(X) of a q-

variate homogeneous random field g(t), t 6 2". satisfies
. , .2 .. -1 2
(l) [hf(X)“ dX < w and (ll) [Hf (X)H dX < m .

Then the following statements hold:

(a) bf'1 as well as (bf-1) converge to bf'1 in L2(F) as

m m
m + w for any polynomial b(X) and in particular for the poly-

2(d ), g is defined by

nomial bk given in (3.1.8), where for any 9 6 L m

9m = {91'( 94)}15qu With gi.(m,X) = Z

90

s _ iXt , _ n
gij(t) - fe gij(x)dX and Fm - {t 6 Z : [Id] 5 m}, m = 1,2,...

(b) The random variable £k(t0), t0 6 Tk, giving the best linear

interpolator of ak(t0) based on gi(£), 2 6 Ti’ 1 5 i 5 q, can

be obtained from the following formula:
q
(3.2.2) Ek(t0) = z [T s,(s)a,(s).
i

where

and the coefficients akj(t)’ t 6 Tj, I f j 5 q are obtained from

the system of equations (3.1 .9).

The convergence in (3.2.2) is understood to be in the space

Hg‘

Proof. (a) We give the proof for bkfn'l wit the polynomial

bk(X) given in (3.1.8). The proof can be carried out similarly for

2
(

the remaining cases. Note that bk(X)f'1(X) is in L F). We

-1 = 1<J<q
recall that bk(X)fm (X) {.E bki(X)pij(m,X) - - and

l=1
q .
bk(X)f‘1(X) = (1)1 bki(X)Pij(X)}lfqu . Thus
bk(X)f'1(X) - bk(X)f;1(X) = { 2 bki(X)(p,j(X) - Ia,.j(m.X))}15ij .

Therefore

91

= £21 £1 £1 £1bkr(X)BT<1-7'X')fzj(k)(pr£(4) - p (mourn) - 67mm).

Let

I bkr(X)bki(X)f£-(X)pr£(X)ETTlX).

A(X)=
2:1 r=1 i=1 i=1 3

q ....
8mm = £21 (41,241 121 mill—Wm X)f£j(x)pr£(X)PiJ-(m,4).

q q
Cm(X) = £21 r21 jgl 121 bkr(X)5ki(X)f£j(X)pr£(m XTp;3(X) and

q q .___
Dm(X) = £21 r21 jgl iél bkr(X)5kiIX)f£j(X)pr£(m,X)pij(m,X) be the

four terms of the expansion of the expression above. Therefore

*

I<bk<X)f'1(Xl-bkulfr'nlO))fm(bk(X)f“1(X)-bk(le;11(X)) m =

(3.2.3)
fA (X)dX - me(X)dX - [Cm(X)dX + [Dm(X)dX

A(X) = i I I bkrmrkiong run»

K(X)}E'.".(X). But
r=1 i=1 j=1 {=1

r lj

—h
ll

4
Z

q
I giving 2 f .p = 1 for r = j and
2:1 33 r3 2 1

ffijprfl = 0

92

for r f j

Therefore bkj(X)bki(X)pij

(X)

kj(X>bk,<X)p,j(X)dX

, Therefore we get

q i(t-s)
Z Z akj(t)aki15) fe
j=1 i=1 tGTi tETj

q q
.2 . Z
j=1 l=1 t6T1. tETJ.

Similarly

f

q
5 £4"

Bm(X)

5;;(X){

i I I bkrm

r=1 j=1 i=1

i i

j=1 i=1

bkj(X)5'TlX)6}j(m.X)

kl

a t)a

Z Z O( I
'= .= — kJ k1
j 1 l 1 t6T5 seTi £6Fm

Therefore

A

(slpiJ

‘2‘

. . Z
J=11=1t6Tj

f8 (X)dX

a .(tTE‘T
sETi 26Fm k3 k1

n n

a .(t)a‘f(s)fi..(t—s)
i=1 i=1 tETj $6Ti k3 k‘ ‘3

A(3’..(X)dX

akj(t)E;;15)S}j(s't).

X)pr£(X)}5,j(m,X)

(5)3- (£)eiX(t-S+£)

ii

-(£) fe1A(t-S+£)dA

for m large enough.

93

F)
A
V
v
N

E II I b (\)" (. ( g ——
m {=1 r=1 i=1 kr’ bki ")er m,>\)j=1f£j(>\)p1j()\)

—1

q
E El bkr(4)6ki(X)p .(m,X) , because f isself adjoint.

r=1 i= r3

Therefore
ICm(X)dX = I ,3 z z .2 ak,(t)6;}<s)b,,(i) Ie‘AIt'S‘j)dX
r=1 l=1 t6Tr $6Ti JeFm

= r21 igl thr ngi akr(t)5;1(s)6ri(t-S)

= r=1 i=1 teTr sETi akr(t)3;}(s)6}r(t~s) for sufficiently
large m.
Finally
0mm =2,r,:l%,i=1Tr’Ti x6Fm yeFm akr(t)a—l§i(s)5r2(x)gijmeiW‘s-““96(A)
Let Amn = I Z I z akr(t)éfi(s)fir€(x)6fi(yXei (t‘s'x’f,j(u3(y).

£,r,j,l=1 Tr’Tj x6Fn YEFm

We show below that Amn as, n + on, converges uniformly in m and the

same is true for Am", as m + w. This will allow us to write

1im fDm(X)dX = lim lim Amn ,
m m-won-m '

94

because
(Down I Y Z I a (t)a—.(shs (657.6)
. m . . ~ 5 ~ kr kl r2 lj
'KsraJ91TraTiXEIm .YEFm
tel-i(t's'xh (x)](y)=A
fij mm

Now for any (i,£,j), 1 5 i,£,j f q, since fzj and pij are in

, elX(t-s-x)f

t—s-x) ‘ <
f£j(A)J(Y) + plj 23

>

L2(X) we have 2 pi.(y)[e1x(
y6Fm 3

uniformly in x, where <-> stands for inner product in L2(dX),

the convergence is clear and the uniformity follows from the Cauchy-

. . . g iX(r-s-x)
Schawrtz lnequallty and the fact that hfzjflz He f2j“2
for all x.
iX(t~s-x) = -iX(t-s-x) -——— . . .
But (pij , e f£j> [e pij(X)f J.(X)dX. This implies

A

9
that X X p.

-(y)[e1x(t-S-X)f .(X)](y) converges uniformly in
i=1 yeFm ‘3 33

x to

(X)f£.(X)dA = fe'14(t-S-x)

fe-1A(t‘S‘X)p J

1 ij

"W

ll l‘~/).O

J 1 p,j(X)fj£(X)dX.

J
which is equal to zero when i f C and is equal to

fe'1x(t's'x)dX for i = 2; and the latter is zero for x # t-s

Therefore

95

ID (X)dX 4 E E Y ) akr(t)3;i(s)p (t-s)

m = .= e ri
r 1 l 1 t6Tr SETi

9 .2
= Z I Z Z akr(t)aki(s)p,r(t-s)
r=I i=1 tETr S611
thus fAm(X)dX, [Bm(X)dX, [Cm(X)dX and [Dm(X)dX converge to the
same limit and the proof is complete by (3.2.3).

(b) Recall from Theoremill.51hat Ek(t0) = fmk(X)¢(dX), where

ok(X) = elitak - bk(X)f'1(X).
-1 q . -i-Xs 15ij
But by part (a) (bkf )m = { Z Z Z aki(t)pi.(t+s)e }
56F tET i=1 3
m l
-1 . 2
converges to bkf in L (F) Now let

Then from (3.1.9) we have

iXt

0 -1 iAt0 3 . iXs 1<j<q
e 6 - (b (X)f (X)) = {e o . - Z A Z a .(t)p..(t-s)e } - -
k k m k3 56F i=1 t6T. k‘ ‘3
m l
. 1<j<q
= { Bj(s)eTAS} ' '
$6Fm\Tj
iXt _1 2
Since e (35k_ (bkf )m converges to ok in L (F), by using the

isomorphism betwen the time domain and spectral domain we obtain

that

which completes the proof.

96

REMARK. In this<:hapter our main Theorem 3.2.1 is derived under the

assumption that f(X) is nonsingular a.e.X, and that f"1 is in

L2(X). The problem of obtaining a recipe for the case when f
may be less than full rank remains open (the case of singular f
has been considered in mathematical literature in connection with

regularity of homogenous random field (c.f. [15])).

 

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