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This is to certify that the
thesis entitled
INTERPOIATION OF STATIONARY RANDOM FIELDS
OVER LOCALLY COMPACT ABELIAN GROUPS

presented by

John Karl Scheidt

has been accepted towards fulfillment
of the requirements for

H101). degreein StatiStlcs and
Probability

 

lezM f/L/LLQ

Major professor

Date August 11, 1971

0-7639

LIBRARY

 

—. 1

\\

ABSTRACT

INTERPOLATION OF STATIONARY RANDOM FIEIDS
OVER IDCALLY COMPACT ABELIAN GROUPS

By

John Karl Scheidt

let G be a locally compact abelian group. let (xg) be
a stationary random process indexed by elements g of G.

A.N. Kolmogorov, P. Masani, and H. Salehi derived numerous results
on the minimality and interpolation of random processes indexed by
integers. The main efforts of this thesis are to derive similar
results for processes indexed by the group G. Although the ideas
and concepts used here are similar to the ones used by Salehi in
his work, some of the techniques are different, since the integers
are ordered and singly generated whereas an arbitrary group need
not be.

First, the univariate case is considered. Results comparable
to Kolmogorov's'Minimality Theorem, the Wold Decomposition Theorem,
and the Wold-Cramer Concordance Theorem are obtained. In addition,
results similar to the work of H. Salehi on interpolation of
stationary random processes are established. This subsumes a correct
version of the recent work of L. Bruckner whose main theorem is in
error.

Secondly, the multivariate case is considered. Under extra

assumptions, most of the results of the univariate case are extended.

John Karl Scheidt

Finally, there is a discussion of some open problems of the

multivariate case, as well as infinite dimensional random processes.

INTERPOLATION OF STATIONARY RANDOM FIELDS
OVER IDCALLY COMPACT ABELIAN GROUPS

BY

John Karl Scheidt

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Statistics and Probability

1971

TO PATTY

ii

ACKNOWIEDGMENTS

I wish to express my sincere gratitude to Professor H. Salehi
for his guidance and encouragement in the preparation of this
dissertation. His patience and willingness to discuss any problem
at any time are deeply appreciated.

I also wish to thank Professor V.S. Mandrekar for his critical
reading of the thesis. Special thanks are due to Mrs. Noralee Barnes
for her excellent typing of the manuscript and the cheerful attitude
with which she did it.

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

University.

iii

INTRODUCTION

filq-VALUED c.a.o.s. MEASURES AND STOCHASTIC

INTEGRALS

PRELIMINARY RESULTS ON
OF STATIONARY RANDOM FIELDS

TABLE OF CONTENTS

OOOOOOOOOOOOOOOOOOOOOOOO

.0

PREDICTION AND INTERPOIATION

MINIMALITY AND INTERPOLATION OF UNIVARIATE WSRF'S

MINIMALITY AND INTERPOLATION OF q-VARIATE WSRF's

SOME EXAMPLES AND FURTHER REMARKS ON FINITE AND

INFINITE DIMENSIONAL STATIONARY RANDOM FIELDS

REFEREN CES

iv

Page

12
22

63

92

98

1. INTRODUCTION

The study of stationary stochastic processes was originated by
A. Khintchine in 1934 [ 7]. In subsequent years the theory of sta-
tionary stochastic processes has undergone a remarkable development.
The basic contribution to the theory of prediction of stationary
stochastic processes is due to A.N. Kolmogorov [£3], H. Cramer [2 j,
and N. Wiener [31]. Kolmogorov was the first to formulate the basic
problems of prediction and minimality of stationary stochastic processes
indexed by integers. One of his most famous theorems is on the char-
acterization of minimality for univariate processes in terms of
Spectral properties. P. Masani [13] extended the concept of minimality
to multivariate stationary stochastic processes indexed by integers
and obtained a similar characterization of minimality for such pro-
cesses. H. Salehi [28] extended Masani's work, using generalized
inverses, to deal with interpolation of multivariate stationary
stochastic processes indexed by integers.

The idea of having processes indexed by elements of a group,
instead of the integers, has attracted the attention of several
mathematicians. Wang Shou-JenDO] considered stationary random fields
indexed by the lattice points of the plane. He was able to generalize
Kolmogorov's minimality theorem for univariate stationary random fields
indexed by these lattice points. Later, L. Bruckner [l ] studied

the question of minimality and interpolation of univariate stationary

stochastic processes indexed by elements of a discrete locally compact
abelian group (LCAG). Some of the proofs of Bruckner seem to be in
error. An example to justify this claim will be given later (cf. 6.3 ).
Other aspects of the theory of stationary stochastic processes over
LCAG'S were studied by M. Rosenberg in [22].

In this paper the questions of minimality and interpolation
of univariate, as well as multivariate, Stationary random fields over
LCAG'S are systematically studied. Results motivated by those of
H. Salehi's on minimality and interpolation of stationary stochastic
processes over the integers will be established for stationary random
fields over LCAG'S. Although many of the ideas and concepts are
Similar to the ones used in Salehi's paper, some of the techniques
used here will be different, since the integers are ordered and singly
generated whereas an arbitrary group need not be.

With reference to this background we may now summarize the
contents of this thesis and indicate the new results established.

In §2 we first recall the Hilbertian structure of the Space
kg, and then introduce the notion of a non-negative, hermitian, q x q
matrix-valued measure g_ over an arbitrary measurable Space (0A6).
M. Rosenberg [21] defined the integral. Agdggf for any measure ‘M
in such a way that the space L2 (0,5,1!) of q x q matrix-valued
functions Q for which IQdM§* exists becomes a Hilbert Space under
the inner product ((§ ,Y)) = tr(g§dMY* ). We will quote some of his
results here. We then, following Rosenberg, introduce the concept
of' flg-valued countably additive orthogonally scattered (c.a.o.s.)
measures, and study briefly the theory of integration with reSpect

to such measures.

In §3, we first review the theory of q-variate stationary
random fields over LCAG'S. We introduce several definitions and
notations needed in the following sections. The notion of the rank
of a process with respect to a given family of sets, as given here,
is a direct generalization of the one given previously by Wiener and
Mssani [32] and will turn out to be very fruitful in studying several
aspeCtS of the theory of q-variate stationary random fields over
LCAG'S. We then state the Wold Decomposition Theorem, the Wold-
Cramer Concordance Theorem, and some basic results due to Kolmogorov,
Masani, Salehi, and others in their work on minimality and interpolation
of Stationary stochastic processes, Since we will be primarily inter-
ested in these topics.

In §4 we consider a univariate stationary random field over a
LCAG C. First, we state the Wold Decomposition Theorem for any
family,,J, of non-empty Borel sets of G. Under the assumption
that G is discrete, we establish several important results, Such as
Kolmogorov's minimality theorem and the Wold-Cramer Concordance
Theorem. We also extend the work of H. Salehi on interpolation of
stationary stochastic processes Specialized to the univariate case to
any stationary random field over any discrete LCAG.

In §5 the same problems as considered in §4 are studied for
qdvariate (1 s q < m) stationary random fields over LCAG'S. Most
of the results of the univariate case are extended, though, in some
instances, extra assumptions are needed. The results of §5 extend
those contained in §4 in the same Spirit that Masani and Salehi's
work generalized Kolmogorov's work from the univariate case to the

multivariate case when the proceSs is indexed by integers.

In §6 we will include several examples which were mentioned
in the earlier sections. There will be a brief discussion on the open
problems related to minimality and interpolation of q-variate sta-
tionary random fields over LCAG'S. Also, a few remarks will be made
on the minimality and interpolation of infinite-dimensional stationary

random fields over LCAG'S.

2. Mq-VALUED c.a.o.s. MEASURES AND STOCHASTIC INTEGRALS

In the first part of this section we review the theory of the
Spaces kg, where u’ is a Hilbert Space. In the second part we shall
consider the Special cases ”q = L2(0,B,Ifi), where 1:1 is a non-negative,
hermitian, q x q matrix-valued measure, 1’ being the Space L2(Q,B,I~1)
of q-dimensional (row) vector-valued functions on n. In the third
part we define the notion of countably additive orthogonally scattered
(c.a.o.s.) measures and study briefly the theory of integration with
respect to such measures. These results will be used in later sections
of this work.

2.1 Notation. Small underscored letters 5, y, etc. will
denote q-dimensional column vectors with complex components Xi’ yi,

etc. Large underscored letters A) B) etc. will denote q x q

matrices with complex entries a b , etc. and E) 9, etc. will

ij’ ij
denote q X q matrix-valued functions.

2.2 Definition. Let R’ be a complex Hilbert Space with inner

 

product ( , ) and norm l l. The Cartesian product R9 is defined.

to be the set of all q-dimensional (column) vectors x. with components

in 1V,

efaxiey i=1,2,...,q

i.e., x_= (x

Addition of vectOrs in RB’ and multiplication by q X q complex-

valued matrices are defined as usual (cf. [32]).

5

q )q 6.x”.

i=1 1 i=1
Then (a) the Gramian (x) y) of x_ and y_ is defined by

2.3 Definition. Let x_= (xi) and 2': (y

(is D = [(xiayj)] 1 S I,j S q

where (xi,yj) is the inner product in. Rd (The Gramian may be
thought of as a matrix-valued inner product.)
(b) The inner product of x and y and the norm of x_

are defined by:
((23.9) = tr(1_r_.x) and “EH =Jtr(i.§).
(C) We say that 23 4,1: (23,1) = Q

i.e., for all 1 s i,j s q, (xi,yj) = O

{For the definition given in (c), we refer the reader to [32].}

2.4 Definition. (a) A linear manifold in RA is a non-

 

void subset 721 such that if 5,1 6 7.71: then A i + 11 y_ E 'm for all
q X q matrices A, B.

(b) A subspace of a“ is a linear manifold which is closed
in the topology of the norm H n.

(c) Let T be an operator on. N’ into N2 Then the

inflation T. of T to HA is defined as follows:
= q = q
for all 5 (“Bi-=1 e #1. 1(5) (minfl .

(d) For a given 5 E liq and a subSpace ')_71 C Nq, (gig) will
denote the orthogonal projection of x. onto 21 (cf. [32], p. 132).
It is easy to see that T is a bounded linear operator on

H a T is a bounded linear operator on liq.

We shall now turn to a brief discussion of the Lebesgue

integrals for q X q matrix-valued functions on a Space n.

2.5 Definition. let (0,8,u.) be a measure Space with p.
a non-negative measure. Then for all 5, 0 < 6 s on, we define
L5(Q,B,p,) as follows: L6(fl,/3,p,) consists of all q X q matrix-
valued functions E = [fij] on n with complex-valued entries

fij E L6(Q:B:U«)-

2.6 Definition. The integral of a function E

[fij] e
1.1“],45») is defined by

gamma») = [I fij (w)p.(dw)] .
a

The following is a well-known result (cf. [32]).

2.7 Theorem. (8) F E L6 «LEAD, 0 < 6 < 0° c: E has measur-
able entries and 11:11:11) E L6(O,B,u.). L66],B,p,), l s 6 < co is a
Banach space under the uSYal algebraic operations and the norm
HFHM = Avian; mm?

(b) £2(Q,B,p,) is a Hilbert space under the same operations
and the inner Product (0:36))“. = tr (E’le, where OLE)” =
£§(w)g*(w)p.(dm) is thematricial inner product of E and g.

(c) E. e Lm(n,8,p.) :0 I: has measurable entries and ‘E‘E
is essentially bounded. Lw(fl,6,p.) is a Banach algebra under the

usual algebraic operations and the norm “5““) = ess.l.u.b. ‘F_(<D)‘E.

 

1
)If _A_ is a p X q matrix, then the Euclidean norm of A is

 

P q 2
defined to be ‘A‘E = z 2 \aij‘
i=1 j=l

2.8 Remark. It is easy to see that the matricialinner product
Q39)“. is the Gramian of E and Q (cf. 2.3 (a)), when we lbok
upon E and g as elements of AA, where if is the Space
L2(0,B,p.) of q-dimensional (row) vector-valued functions on 0
whose entries are in L2(0,B,u.). This remark leads us to the following

definition.

2.9 Definition. (a) If P_‘, Q E L2(0,B,p,), then we say that
E 4-9.” (E. a) = .9.

(b) Asequence (Euro in L2(0,B,p,) is called orthonormal
-m —

2.10 Remark. In the Hilbert Space 1.2 (0,8,p.) considered
in 2.7 (b), we took u. to be a non-negative measure over (0,6).
For many purposes it is necessary to consider _I_.2 (0,8,11), where I!
is a non-negative, hermitian, q x q matrix-valued measure over
(0,5). Rosenberg [21] and Mandrekar and Salehi [9 ] have studied
this question. For ease of reference, we will State the main result

here.

2.11 Definition. Let (0,6) be ameasurable Space. Then

fl = [M 1 s i,j s q is called a (bounded) countably additive,

 

A]:

non-negative, hermitian, q x q matrix-valued measure over (0,6) c:

(i) l s i,j s q, M is a countably additive (c.a.)

1.1
(complex-valued) measure on 8.

(ii) M is a non-negative, hermitian, q x q matrix-

valued function on B.

2.12 Remark. Let M be a non—negative, hermitian, q X q

matrix-valued measure over (0,6), and let 3, g be B-measurable

q x q matrix-valued functions on 0. Rosenberg has shown [21] that
the integral gédM‘f may be so defined that the Space L2 (0 ,B,M)

of q x q matrix-valued functions M such that gMdM‘l’f exists is
a Hilbert Space under the usual algebraic operations (cf. [32]), the

inner product (( , ))M and the norm H where

“2,!

(2.13) “MM = trmDM; Mm = «MDM

(2,914 =fi(w>M(dw>i* (w) .

He observed that M << tr M1) and defined

at d! *
01 = ——
<2 4) gym. fidtrMidtrri
n .—

d!
where d_t_r—M is the MatricialRadon-Nikodym derivative of M with
respect to tr M and the R.H.S. of (2.14) is defined by (2.6).

We now turn to the discussion of MCI-valued c.a.o.s. measures
and study briefly the theory of integration with respect to such

measures. Rosenberg has Studied this topic in detail. We will State

several of his results which will be needed later.

2.15 Definition. let (i) N be a complex Hilbert Space.
(ii) M be a c.a., non-negative, hermitian q X q matrix-
valued measure over (0,8) (cf. 2.11). Then a function E on B

into 1v“ such that for all 3,0 6 5

(3(3). S(C)) = MB 0 C)

 

l

)The symbol << stands for absolute continuity. If M is a matrix-
valued measure and p, is a scalar measure, then M << p. means each
entry Mij of M is absolutely continuous with respect to p.

10

is calledéuihflevalued countably additive orthogonally scattered (c.a.o.s.)

measure where M. is a non-negative, hermitian, matrix-valued measure.

When necessary, we Shall write Mg instead of M, M. is called the

associated measure of Q.

= m, and

It easily follows that §(B) i.§(C) if B 0 C

gnu Bk) = Z fifiBk) if the Bk's are disjoint, where the convergence
k k

on the right is in the Ng-norm.

There is a well established theory of integration with reSpect

to such measures for q 2 1. For ease of reference we shall restate

the definition of integration and the main theorem.

2.16 Definition. (Step 1). For a simple function

n
§.= Z ékxE : where ék are q X q matrices,
l k
n
Aid; = i _A_k soak)

a)

1

(Q ) is a sequence of
—n

A direct computation shows that if

simple functions then

“£21,513 - Fax = \Em - inn“,

Hence the following definition is unambiguous.
2.17 Definition. (Step 2). let g6 L2(0,B,Mg). It is
known (cf. [21], p. 296 ) that there exists a sequence (gm): of
simple functions such that 2n a 2' 1n IQ(0yB,M§). We define

{£ng = lim find; .
new 0

The following is an important theorem on the subject.

11

2.18 Theorem. Let (i) g be aan-valued c.a.o.s. measure
with associated measure Mg over (0,6).

(11) s
S n

Then (a) (gymgdw), gymgdw» = (Lb—mg.

(b) The correSpondence g -. £2<w)§.(dw) is an isomorphism on

. . Cl
L2(0,6,M§) onto SS. In particular, SE. is a closed subSpace of N

be the set of all stochastic integrals jg...) S(du)).

3. PRELIMINARY RESULTS ON PREDICTION AND INTERPOLATION
OF STATIONARY RANDOM FIELDS

Let G be a locally compact abelian group (LCAG) and 6*
the dual group of G. Then G* is also a LCAG under the compact-
open topology. We will denote the elements of G by g and those
of 6* by A. The value of A E 6* at g E G will be denoted by
(g,x). The Borel field ‘6 of G is the o-field generated by the
open subsets of G. Similarly,.6#, the Borel field of 6*, is de-
fined. It is well known that G is discrete if and only if 6* is
compact. Furthermore, it is known that there exists a regular Haar
measure m defined on. 532 Without loss of generality, when G
is discrete, we will assume m(G*) = 1.

We now give the definition of a qdvariate stationary random

field over G.

3.1 Definition. (a) A q-variate mean continuous weakly

 

Stationary random field (WSRF) over a LCAG C (under the operation

+) is a function such that

(as)gec
(i) Mg E N9 for each g E G 0V is a fixed Hilbert Space)
(ii) the q X q Gramian matrix x x ) = F( - ')
(_g:_gt _8 8

depends only on g - g'.
iii x - x x - x a O as - ' a 0 mean con-
( ) (’8 —g" ‘3 _gr) 8 8 (

tinuity).

(b) The q-dimensional temporal domain QM: of a q-variate

is the closed subspace of fig Spanned by the
12

WSRF (Mg)gEG

13

Kg 6 H”, g E G with q X q matrix coefficients.

(e) Let ) and (yg)gEG be Hg-valued WSRF'S

(’38 SEC
over the same G. We 88 that x and are

y (1566 (xg) g€G
mutually homogeneously correlated if (Eg’xg') depends only on g - g'.

The following is contained in.[22] and will be Stated here

for completeness.

3.2 Lemma. Let (Mg)gEG be an.Ng-va1ued WSRF. Then there
exists a strongly continuous group of unitary operators (Ug)gEG on
N’ Such that for each g E G, we have

x U X
-g ‘3-0’
where 9g is the inflation of Ug to RA (cf. 2.4c).

We shall now recall the generalization of Stone's theorem

([22], Theorem 2.3) and of Bochner's theoren1([22], Theorem 2.4).

3.3 Stone's Theorem. Let (Ug) be a weakly continuous

 

g€G
family of unitary operators on a Hilbert Space k’ over a LCAG G.

Then there exists a unique Spectral measure E(-) defined on the

*
Borel subsets of the dual group G such that

Ug = [*(g.>.)E(d>.) .
G

3.4 Bochner's Theorem. (a) f is a continuous positive

 

definite complex-valued function on the LCAG G if, and only if,
there exists a bounded non-negative regular measure m on the Borel

* *
Subsets ,6 of the dual group G Such that for all g E G

f(s) = j*(s.i)m(di) .
G

14
(b) If for all g 6 G,

J”*(g.>.)m(dx) = j*(g.1)u(di)
G G

*
where m and u are bounded complex-valued regular measures on .6 ,

then m = u.

3.5 Remark. It is known (cf. [22]) that if E(-) is a
Spectral measure defined on the Borel field 66' for a Hilbert Space
A! and y.) is the inflation of E(-) to if“, then g.) _——.§(.)s_50
is an Ng-valued c.a.o.s. measure. The non-negative, hermitian, q x q

matrix-valued measure §_ defined by:

12(3) (5(B) , §(B))

*
where B 6.6 , is called the Spectral distribution of the WSRF (Eg)gEG°

With this in mind, we state the following lemma (cf. [22],

p. 339)-

3.6 Lemma. Let (Ug) be the shift group of the Hg-valued

gEG
WSRF (Mg)gEG and let E(-) be the associated Spectral measure. Let

g. be defined by:

_ q . _ 1r *
s — {y_ 6 N . y_ -j*i(1)§(di)go. g 6 £2“; ,8 .9}.
G

Then

Now, applying (2.18) together with this lemma, we obtain the

following important theorem.

15

3.7 Isomorphism Theorem. With the above notation, we have:
* 'k

(a) For 1. 1 6 12m J3 ,1).

(Lymydux . J”*1(i>1(dx>x_o> = (1,1) =

11
G G
*
J" 1(A)I:(di)1 (A)-
*
G
(b) The correSpondence g a]. 2(A)§(d)\)xo is an isomorphism
*

‘k * G
on _I_._2(G ,6 ,E) onto 711x.

We will now introduce some new notations.

3.8 Notations. Let J be any family of Borel subsets of
6 closed under translations (i.e., if I E .9, then I + g E J for
all g EC). Let I be an arbitrary element of J and ()ig)gEG
be an Nq-valued WSRF over G.

(i) We will let ml,x denote 601g, g E I); i.e., the
closed subspace of ”q Spanned by Mg, g E I.

so o J. . o o
(11) We W111 let 22 c denote 731,): fl 7%, i.e., 71 C 18
I ,X I ,X
the closed subSpace of Ex orthogonal to m1,x°
iii We will let = n , .
( ) mJ,x IEJ MLX

The following definition is a generalization of the concept

of rank given by Wiener and Masani (cf. [32], p. 136).

3.9 Definition. Let I E B and g E G. Then the rank of

the yq-valued WSRF (Mg) with reSpect to I and g, denoted by

gEG

pl 3’ is defined to be the rank of the Gramian matrix of
’

x-(x

_g filml’x) with itself; i.e.,

pl 8 = rankoig - (58ml.x)’ is ' (égmlixn'

16

3.10 Remark. If G is a discrete LCAG, the families .J

of Borel subsets of 6 with which we will be concerned are:

(1) “”0

(ii) J = {80,g1,...,gn] C G. 4%0’81""’gn is the family

is the family of complements of singletons of G.

of the complements of the translates of J; i.e.,.J =
gO,Ooo,gn
{JC + 8: 8 E G}, where JC is the complement of J in G. For
simplicity, when there is no danger of confusion,.J will
g0,ooo’gn
be denoted by 4%.
(iii) 4;, is the family of complements of finite subsets
of G.
(iv) For C = Z, the integers,._ap is the family of In's
where I = {k : k S n}.
n

We introduce here the following definitions which arose in

this Study.

3.11 Definition. Let .J be a family of Borel subsets of
G. Then
(i) An yq-vaiued WSRF (1g, g E G) is called J-Singular

if for all I 6.0, 211,): =m(; i.e. MAX =7_7{x.
(ii) An irq-valued WSRF

721.0,}: = 1.0.} '

For G = Z, the integers, Masani, Salehi, and others have

(Mg)86G is called.J-regular if

introduced some of these notions under somewhat different terminology.
To make the relation between their work and ours clear, we will State

some of their results and make the appropriate comparisons.

3.12 Definition (Kolmogorov, Masani). Anwflq-valued WSRF

(5n)fm is said to be minimal if, and only if, £0 4 Z& x’ where

l7

I={000 -2, -1, 1, 2,000}.

3.13 Remark. It is eaSy to see that if G = 2, an K’q'valued
WSRF (Mn)f; is not minimal if, and only if, (Mn):an is Jb-Singular.
Obviously, a.WSRF is either minimal or not minimal.

However, in general, it is not true that a WSRF is either
regular or singular, as the example in §6 (cf. (6.3 ) shows. The
statement of the main theorem (4.1) of L.Brucknen which he considers
his extension of Kolmogorov's minimality theorem, as well as its proof,
is in error. The error stems from the fact that he claims that a
WSRF is either Jb-regular or Jb-singular. The exact relationship
between the two concepts of regularity and minimality for a WSRF
will be given in Theorem 4.8.

In §4, we will define the concept of minimality for any dis-
crete LCAG, and extend Kolmogorov's minimality theorem.

To give the flavor of the types of theorems proved in §5,
we give some existing results for integers. The following, Kolmogorov's
minimality theorem (cf. [13], Theorem 2.8) is one of the most funda-

mental results of this theory.

3.14 Theorem (Masani). Let (Mn)co be anHHg-valued WSRF

. on
with Spectral distribution 2, Let 91,0 be the rank of (Mn)_m
m 0
with respect to I = {..., -2, -1, 1, 2,...} and 0. Then (Mn)_m is

minimal and pI 0 8 q if, and only if, E} is invertible a.e. on C,
9

and {"1 e _1_.1.

Also, in.[28], H. Salehi introduces the notion of interpola-

tion for REL-valued WSRF'S over the integers. He proves several

theorems on the interpolability of a given WSRF in terms of the spectral

18

distribution of the random field. From one of his theorems (cf.
[28], Theorem 2) he deduces the following, from which Masani's

multivariate extension of the minimality theorem follows.

3.15 Theorem (Salehi). Let (Mk)fm be an.HA-va1ued WSRF.
Let I = {..., -2, -l, l, 2,...} and let go be the orthogonal

k

' J. =

projection of £0 onto the subSpace Z&,x° Let 3k U E0 where
k m . . T
(U )_00 1S the assoc1ated shift group of (Mk)_m. Let
1k = (50,2 )#§%L where (30,30)# is the generalized inverse of
(20.2) (cf-[20], p. 355 ). Then
2
# n =1m) _J_ 1)
(3) (50’50) “10th =§;l T
-n ._

where g. is the projection matrix on the space Cq of q-tuples of

complex numbers onto the range of (20,50) in the privileged basis

of Cq. 2
m n .1011) i
(b) (518%) is m1n1mal 1ff £1 T 9‘ _(_)_ .

In §5, we will extend Salehi's theorems of interpolation of
WSRF'S with reSpect to the integers to Ng-valued WSRF'S over any LCAG.
Our extension will yield a generalization of Masani's (Kolmogorov's)
minimality theorem from integers to any LCAG.

We may add that our notion of JM-singularity coincides with
the concept of interpolation of a finite set of integers introduced
by Salehi [28]. Similarly, our notion of JM-Singularity and Salehi's

concept of interpolability of the entire random field are the same.

 

2
1) 11 1011) i
For the definition of n.-_EET—- , see ( [28], p. 308 ).

19

We will now state some known results (most of which concern
WSRF'S over integers) which we will use in the later sections in
connection with our results about the concordance of the Wold and

Cramer decompositions. We Start with Cramer's decomposition.

3.16 Remark. Let M_ be a non-negative, hermitian, q X q
matrix-valued measure defined on the family of Borel subsets of
(-m,m). Let u be a g-finite, non-negative, scalar-valued measure

on the same family. Then there exist unique matricial measures

a

IZ

a
and M? such that M_= M? +-M§, M_ << u , MI l-p 1) and

a
M_ and NS

are non-negative, hermitian measures. This was proved
by Cramer [2] and goes by his name.

The following, a finite dimensional Cramer decomposition
theorem, can be derived from Mandrekar and Salehi's result ([12],

Theorem 3.15).

3.17 Theorem (Cramer's decomposition). Let E} be the

sepctral distribution associated with the Hn-valued WSRF (Mg)gEG
*

where G is a LCAG. Let m be the Haar measure on 6’, the family

*
of Borel sets of G . Then

a
i=1 +18
a s a s .
where E_ << m, E. A,m, and both E_ and E_ are non-negative,
hermitian. For simplicity, the RadonrNikodym derivative of {I
with reSpect to the Haar measure m is called the Spectral density

of the WSRF )

(is 366'

 

l
)M? i.uI means each component Nij of M? is singular with respect
to the mEasure u.

20

On the other hand, for multivariate processes indexed by
integers, the Wold decomposition theorem was proved by Wiener and

Masani ([ 32], Theorem 6.11).

3.18 Theorem. Let (x )°° be an Rfl-valued WSRF. If
———-— —n -oo

( )on is non deterministiC' i e fo so e n E
Mn _°° : 0 0s r m 91‘“ m1 ,X,
n-l
I = [k : k S n}, then

n

x u+v
-n -n -n

where (1) En 1.x“ for all n,

(11) (2n)fm is purely non—deterministic; i.e., Ehz,u
P

= {9}.
(iii) (3“): is deterministic; i.e., m], “ft/(V.
In §5 we will prove a Wold decomposition theorem with respect
to a given family .J, closed under translations, of Borel subsets
of G, a LCAG.
Under certain assumptions, there is concordance between the
Wold decomposition in the time domain and the Cramer decomposition
in the spectral domain. The following theorem is due to Wiener and

Masani ([32], Theorem 7.11).

3.19 Theorem. Let (i) (3511):, be an Vq-valued WSRF and

IO = {... -2, -l, 0}; (ii) p 1 = q; (iii) in = En +-yn be 1ts

I
0,
Wold decomposition; (iv) En Eu, Ev be the Spectral distributions

. m m m . . a
of the random f1elds (Mn)_m, (2n)-m’ (2n)-m respectively, (v) E_,

E? be the absolutely continuous and singular parts of E_ reSpectively.

Then

21

Later, Robertson ([19], Theorem 5.2) obtained the concordance

result under a weaker assumption.

3.20 Theorem. For any kg-valued WSRF (Mn)0° with

pI 1 = p (0 s p s q), there is concordance between the Wold de-

3

cogposition and the Cramer decomposition if, and only if, rank
Ef(e) = p a.e. (Leb.), where M_ is the spectral distribution of
the random field (Lin): and g' is the derivative of g with
reSpect to the Haar measure on the circle.

In §4 and §5, we will prove a.Wold-Cramer concordance theorem
for the cases J = .00, .0 = Jim, and J = .000.

3.21 Remark. As we already mentioned in the first paragraph
of this section, the group G is discrete if, and only if, its dual
G* is compact. In this case, the Haar measure of G* will be finite.
These facts allow us to stay in the framework of the theory of Fourier
series rather than Fourier transforms.

For a non-discrete group G the notion of minimality of a
WSRF over G has not been treated in the literature. This may be
due to the fact that in a discrete group every point is a neighbor-
hood of itself while in a general group this may not be the case.
However, one may consider the problem of interpolation for WSRF'S
over a group which is not necessarily discrete (See [25], [29] and
[35]). The results for the non-discrete case do not follow from those
of the discrete case and need a separate discussion involving the theory
of Fourier transforms.

For these reasons and in order to keep this thesis within its
previously stated confines we are not including the theory of minimality

and interpolation of'WSRF's over a non-discrete group.

4. MINIMALITY AND INTERPOIATION 0F UNIVARIATE WSRF'S

In this section we will extend Kolmogorov's minimality
theorem (Theorem 4.7) for univariate WSRF'S over the integers to
LCAG'S. L. Bruckner has considered this case. As we mentioned in
Remark 3.13, his main theorem is in error. (The earlier work of
Rozanov on WSRF'S indexed by integers [24] also indicates Bruckner's
mistake.) Among other things we will give a corrected version of
his theorem (Corollary 4.9) and give the exact relationship between
the concepts of Singularity and regularity defined by Bruckner and
the notion of minimality given by Kolmogorov and Masani (Theorem 4.8).

We will state the Wold decomposition theorem (Theorem 4.2)
for univariate WSRF'S with respect to a family .J, closed under
translations, of nonempty Borel subsets of a LCAG G. This provides
an extension of the usual Wold decomposition theorem given for WSRF'S
over the integers. Using our result on minimality we will then
establish the concordance relation between the Cramer decomposition
and the Wold decomposition theorems (Theorem 4.13). This will con-
stitute a natural extension for the univariate case of the same result
given by Wiener and Masani ([32], p. 146) and Doob ([13], p. 576)
for WSRF'S over the integers.

We will Specialize our result on minimality to the case where
the random field is over the integers. In this case, the notions of
past, future, and past & future are well defined. Using our results,

we will examine the relationship between the past and the past &
22

23

future of such a random field.

We finally give an extension of Salehi's results on inter-
polation of random processes over the integers to univariate WSRF'S
over LCAG'S. This will provide a natural extension of earlier
results of this section in the same way that Salehi's work on inter-
polation provided a generalization of Kolmogorov and Masani's work
on minimality. Although the ideas and concepts used here are Similar
to the ones used by Salehi in his work, some of our techniques are
different, since the integers are ordered and singly generated whereas
an arbitrary group need not be.

The main reason we have considered the univariate case
separately is that Kolmogorov's minimality theorem and most of the
results of Salehi on interpolation theory can be extended without
any further assumptions. To get the correSponding results in the
multivariate case we have to make certain assumptions, under which
we are able to carry out our work.

Throughout this section only univariate WSRF'S will be con-
sidered. We first state the Wold decomposition theorem whose proof
is given in L. Bruckner's paper for the univariate case. First,
though, we will need the following definition.

4.1 Definition. Let (xg) (yg) be univariate

s60 366
WSRF'S over a LCAG G. Let .J be any family of non-empty Borel

sets of G. Then is said to be -subordinate to x
(yg)g€G J ( g)gec

if
(1) W3, 9 77g, ;
(ii) 'mI’y cml,x for all I E J ;

(iii) (xg)gEG and (yg)gEG

are mutually homogeneously correlated.

24

We will now state the Wold decomposition theorem for a uni-
variate WSRF over a LCAG G.

4.2 Theorem (Wold decomposition). Let .9 be any family
of non-empty Borel sets of G closed under translations. Let
(xg)gEG be a univariate WSRF with values in N. Then there exists

a unique decomposition of (xg)g with respect to .0 in the form

EC

x= +w
3 ya 3

where

(i) (y ) EC and (w ) are V—valued WSRF'S on G;

g s s EEG
" and w a - b d' at t ;
(11) (yg)86G ( g)gEG re .9 su or 1n e o (xg)gEG
(iii) (yg)gEG and (wg)gEG are orthogonal; i.e.,

(yg.wg.) = 0 for any 8.8' E G;

(iv) (y ) 6G is J-regular; (w ) is J-singular.

s s s EEG

We will now state the definition of minimality for a uni-
variate WSRF over a discrete LCAG.
4.3 Definition. Let G be a discrete LCAG. Then the

univariate fil-valued WSRF (xg)gEG is minimal if, and only if,

x0 € 7711 x where I = {01C 1)-

The proof of the minimality theorem for WSRF'S over a dis-
crete LCAG will depend on the following leumas.

4.4 Leanna. Let (xg)gEG be an V-valued WSRF over a dis-

rete LCAG G. let ft de te - whee I = Oc.
c 3 no xg (xg‘mI-l-g,x)’ r { ]

Then is an ll-valued WSRF over G. Also, (xg) and

(“shes gEG

l){0)c will stand for the complement of the zero element of G.

25

l

5‘: have the same shift.
( s)sEG

Proof. Let (U8) be the unitary group of shift operators

SEC

associated with x . Then U R = U x - U x =
( 8)8€G s s' g s' s( s"W&+s'.x)

- a t. 1a A = - 0
Ug(x In p r 1cu r, ng0 x8 Ug(x0‘7RI,x)

8"m1+g'sx).
=mI , it follows that Ug(x0‘7711,x) = ( ‘771 )°

x
482x 8 I'l'gsx
Hence x ) is a WSRF which 1m lies that it
<< 317711 )86G . p < g>g

x
8'42
8‘ e
inc Ung,x
+g,x EC
also a WSRF. Q.E.D.

4.5 Leanna. Let (xg)g€G be any-valued WSRF. Then

A 's ' ' al if a d l ‘f f all C,
(xg)gEG i m1n1m , n on y 1 , or g E xg E 771
I = {0}°.

I+g.x’

Proof. The proof of sufficiency follows trivially. If

(x8)g€G is minimal, then by (4.3) x0 4 ”[1“. But Ung,x = 7721

im lies that x E . Therefore x E
p g ”5+g.x g ”E

+8.X

+3 x for all g E G. Q.E.D.
’

The next lemma plays an important role in the theory of
minimality of WSRF'S.

4.6 Lemma (Main Lemma 1). Let (xg)gEG be an fl-valued

WSRF over a discrete LCAG G with the shift group of unitary operators

(U8) and E be the Spectral measure of (U8)8

EEG ec'

.. c x 2
x8 = x8 - (xg‘mI-I-g,x)’ I = {0] and a = [x0] . Let F be the

Spectral distribution of and f be its spectral density;

x
( s>g€G
i.e., f is the Radon Nikodym derivative of Fa {absolutely continuous

component of F as in 3.17} with respect to the Haar measure. Then

x8 = {*(g.i)mb(i)E(di)xo
G
where ‘90 is defined by

26

a
o/f, on the carrier of F

“’0: . s
O , on the carrier of F .

Proof. Without any loss of generality, it suffices to prove

that RO is given by

so = L ch(A)E(d>.)XO -
G

Since %0 E ”9’ by the Isomorphism Theorem (3.7)

so = Leone (dnxo
G

for some m0 E L2(G*,6f,F). Also, (cf. 3.2),

"g = f*(g.i>E(di>x0 .
G

Therefore,

(x .520) = j*<g.i>$0(i>F(di>

8

C

(I)
=j <s.i>cp0<i>f(i>m<di> +f (gmmomFSwM-
* *
C G
Recalling that x0 = x0 - (xO|Wz,x), we see that (xg,x0) =
6g 0° where 6g 0 is the usual Kronecker delta and o is as above.
From [26], p. 10 , we know that I (g,A)m(dA) = 6g 0' Therefore,
* 3
C

(II) (xgsxo) = U [*(SAMWM-

G

Combining (I) and (II), we get the following equation:

o j*(g.i)m(di) = j *<g.i)$o<i>f(i)m(di) + j‘*<t.x)$0(i)1~*s (d1)
G G G

27

which in turn is equivalent to:

(III) J‘*(g.i)[o - Booms!» = j*(g.i)$0(i)FS(di)
G C

By Bochner's Theorem (cf. Theorem 3.4 ), we get that

(IV) J‘*(o - $O(i)£(i))m(dx) = j c-Po().)Fs(dA). 3* e 5*.
‘k
B B

. S . . . .
Since F is Singular with respect to m, it follows that each of

the integrals in (IV) is equal to zero. Therefore,

(V) 0 =1“ (o - E (i)£(i))m(di) = I E (A)Fs(d>.) for all 3* e 5*.
* 0 * O
B B

From (V) and [5 ], p. 105, it follows that

Therefore, if 0 ¢ 0, we have

o/f, on the carrier of F8
To = . s
O , on the carrier of F
If C = 0 then ‘R \2 = 0 and hence I ‘ \ZdF = 0 There-
’ o ’ ’ * “Po '
fore, m = o a.e. F. G Q.E.D.
We are now ready to state and prove the minimality theorem

for a univariate WSRF (x8)86G

4.7 Theorem (Kolmogorov minimality theorem). Let G be a

over a LCAG G.

discrete LCAG and (x8)gEG a univariate flkvalued WSRF over C with

Spectral distribution F. Then (xg)gEG is minimal if, and only if,

28

l * * , .
-f- e L1(G ,6 ,m) where f is the Spectral denSity of (xg)gEG

Proof. Sufficiency. Set m(A) = '% (A), on the carrier of F8

O , on the carrier of FS.

Then

2 _ 1 2 2 s
£*l(P(>.)l F(di) — 15—15(1)] f(A)m(dA) + [*0 F (d),)
c c

= [*E%Xy.m(dA) < m (by assumption).
G

* *
Hence, m E L2(G AB ,F). Now, by the Isomorphism Theorem (3.7), there

exists a y E 772x Such that

y = j*<p(i)E(di)x0 -
C
._ 2 = 1 .
Note that (y,y) —.I*‘¢(X)| F(dA) I*f(A) m(dA) # 0. (Otherw1se,

G G
= O on every set of positive Haar measure which is impossible be-

 

HIIH

cause F is a finite measure.)

We will next Show that y l-xg, g f 0. Let xg be an

arbitrary element of the WSRF (x ) . Then
SgEG

(Xgfl) = f*(s.>.)cp(>.)F(dA) = j‘*(s.i)m(di) = 6g“) .
G G

Since {xg, g ¢ 0] is dense in. ”E x’ 1 = {o}c’ we have that
3
y = CXO where c is a non-zero constant. Hence, xO E W&,x which

implies that (xg) is minimal.

gEG

Necessity. Suppose is minimal. Then R i O

(xg)gEG 0
(cf. Lemma 4.5). Now, by Main Lemma 4.6,

520 = [*cpo(i)E(d).)xo
C

29
where
(£0,RO)/f, on the carrier of Fa
0 , on the carrier of FS

Hence,

A x _ x 2 1
(XO’XO) ’ (kosxo) I* f >\) m(dx)°
G

Therefore,

(xo,§<O)-1 mm).

- _1__
- f* f(1)
G

But, since £0 ¥ 0, we have (RO,§O)-1 is finite and hence

I
-——- d m
G
1 * a:
or f E L1(G ,6 ,m). Q.E.D.

Next, we will establish a theorem on the relationship be-

tween the concept of minimality and that of Jb-regularity introduced

in 3.11.

4.8 Theorem. Let (xg) be a (non-trivial)1) univariate

gEG
R%valued WSRF over a discrete LCAG G. Then (xg)gEG is Jb-regular

if, and only if, (x ) is minimal and F, the Spectral distribu-

8 SEC
tion of (xg)gEC is absolutely continuous with reSpect to the Haar

measure m.

 

Proof. Necessity. Since (xg)g€G is Jb-regular,
c
771 = fl = {0], where I = {0] . Hence, x f 772
1)

i.e. Wk * [0}.

30

for g E G, and so is minimal.

(Kg)86G
We now wish to Show that F is absolutely continuous.

*
Define the function W on G as follows:

, a
O, on the carrier F

1(1) = S
1, on the carrier F .
Then
I \‘l!().)l2F(di) =‘[ (O)f(A)m(dA) «+]‘ 1 FS(dA)
G* G* G*

3*
F(G)<oo.

* *
Hence 1 E L2(C A6 ,F). By the Isomorphism Theorem (3.7) there exists
2 EEWQ such that z = I ¢(A)E(dA)x0. From Lemma 4.6 (Main Lemma I),
*

.. a 0
x8 [*(g.i)cp0(i)E(di)xo where

G
(R0,fio)/f, on carrier of F8
¢b = s
0 , on carrier of F .
Hence,
(I) (xg.z) = j*<g.i)cp0(i)1(i)F(di)
G
(§0’§0) S
= -—-—- '0. d + . .
Um) m) f(>.)m( 1) Us.» 0 1 F (di)
G G
= O for all g E G.
= ' ‘L = ° 'L =
Note that Wub,x [0} 1ff ”(o’x Wk. Since Wub,x closure
1337?]g,x = closure : 0(Rg) = C(fig’ g E C), we have

(11) 6638. 8 E G) =77(x-

31

From (I) and (II) it follows that z J.W&. But 2 E Wh’ and hence

z = 0. Therefore, 0 = ‘2‘2 = I ‘W(1)‘2F(d1) = f Fs(dk) = FS(G*))
6* *

G
which shows FS = 0. Hence, F is absolutely continuous.

Sufficiency. Let 2 E_W&. If we can Show that z 1.§

8
for all g E G implies that z = 0, then we will have shown that

{xg, g E G] is dense in 77(x, and hence that 6(xg, g E G) =71“
But, as above, 66:8, g 6 G) =771x iff 771420,]: = [O]; i.e.,

(X )

g 366 is Jb-regular.

From Main Lemma 1, we get that

fig = j*(g.i)m0(i)E(di)x0 for all g E G .
C
where Yb is as in the lemma. Since 2 EEWQ, z = f ¢(A)E(dA)xO
*

* *
where w E L2(C ,6 ,F). Hence, if we assume that G(z,$‘<g) = 0 for

all g E G, we get that

o = (z,fig) [*v(i)(-g.i)mb(i)F(di)

G
(so.so)j*v(i)(-g.i)m(di) for all g e c.
G

From minimality, we have R0 ¥ 0. Therefore, we obtain the following:

f ¢(i)(—g.x)m(di) = o for all g e c .
*
G

This implies that all the Fourier coefficients of A are zero, and
hence, by Bochner's Theorem 3.4, W = O a.e. m. Hence, Since F

is absolutely continuous,

2 2
(2.2) = j |¢(i)l F(di) =.f 0 f(i)m(di) = 0 .
* *
G G
which shows that z = 0. Q.E.D.

32

From the above theorem and Theorem 4.7, we obtain the follow-
ing corollary.
4.9 Corollary. Let (xg) be as in 4.8. Then (x )

SEC g 360
is Jb-regular if, and only if, F, the Spectral distribution of

 

m(dA) < m, where f is

(x )

is absolutely continuous and
s SEG’ I*

G

f(A)

th t al d s't f .
e Spec r en 1 y o (xg)gEG

AS in the case where C = Z, there is a definite relation
between the concept of Jb-singularity and non-minimality, as the
following remark shows.

4.10 Remark. Let (x8)g€G be a univariate RZValued WSRF

over G, a discrete LCAG. Then is Jb-Singular if, and

x

( s)s€G
* *

only if, %E L1(G ,8 ,m) where f is the Spectral density of

or e ivalentl , is not minimal .

G
Proof. Let I = {O}C. Then (xg)gEG is Jb-singular iff

, g E G iff ( ) is not

as. W 8 E G x. E A... x. gt.
minimal iff ‘% E L1(G*,Er,m), by Theorem 4.7. Q.E.D.
As we have mentioned earlier (cf. Remark 3.13), the main
theorem 4.1 of L. Bruckner []_] and the proof of this theorem are
in error. Using results of 4.7 - 4.9, we first give a characteriza-
tion of a WSRF over a discrete LCAG which is neither Jb-regular nor
Jb-singular in terms of its Spectral distribution. An example of
such a random field will be given in §6. We will then give a con-
dition under which a WSRF (xg)gEG over a discrete LCAG must be

either Jb-Singular or Jb-regular, as Bruckner claims.

4.11 Theorem. Let be a univariate flzvalued WSRF

x
( s>g€G
over G, a discrete LCAG. Let F be the spectral distribution of

*
(xg)gEC and f be its Spectral density. Then ‘% E L1(C ,6r,m)

33

S . . C2
and F 5‘ 0 if, and only if, [0} $771.00,x gm.

Pr f. Ne essit . S se = . Th is
00 c y uppo 771]2 ,x 77g, en (xg)gEG
* *
JO-singular and, hence, by Remark 4.10, Isl-E L1(G ,6 ,m), which is a
contradiction. Now S se = 0 . Then x is .0 -
a UPPO mJ0,x { } (8)86G 0

regular and, hence, by Theorem 4.8, F is absolutely continuous,

. . . . C2 C2
which is a contradiction. Hence, 0 s 7]) v m.
JOSX

Sufficiency. If 77142 ,x ’9 77k, then (x8)g€G is not '00-
‘ * *
singular, and, hence, by Remark 4.10, %E L1(C ,6 ,m). If
771—00,), 1‘ {0}: then (x8)gEC is not JO-regular and, hence, by
*
Corollary 4.9, either %4 L1(C ,6*,m) or F is not absolutely

* *
continuous. But, above, we showed that %E L1(C ,6 ,m). There-
fore, F is not absolutely continuous; i.e., FS # 0. Q.E.D.

4.12 The e . Let be as ’ 4.11. let F the
orm (xg)gEG in ,

Spectral distribution of (xg) be absolutely continuous with

SEG’

respect to the Haar meaSure m. Then (xg)gEG is either JO-singular
or JO-regu lar.

Proof. If ( ) is not JO-singular, then 772.0

xg gEC o’x #m.

' h tha = O C;
Then there exists an xg, g E G, suc t xg E mI+g,x’ I { }

*
i.e., (xg) is minimal. By Theorem 4.7, % E L1(C*,6 ,m). This

86G

fact, together with the aSSumption that F is absolutely continuous,

imply, by Theorem 4.8, that (xg) is Jo-regular. Q.E.D.

86G

Our next objective will be to establish the Wold-Cramer con-

cordance relation for a univariate V-valued WSRF (xg) over C,

366
a discrete LCAG, with reSpect to .00. The proof of this theorem

will depend on results on minimality and regularity of the WSRF

(

xg)gEG over G, a discrete LCAG, as established in this section.

34

Our proof will resemble the proof given by Wiener and Masani ([32],
p. 146) and Doob ([3 ], p. 576) for the Wold-Cramer Concordance
Theorem with reSpect to the past of a process. The following is

our Wold-Cramer Concordance Theorem.

4.13 Theorem (Wold-Cramer concordance for Jb). Let
(i) (xg)gEG be a univariate fl9va1ued WSRF over G, a discrete
LCAG;
ii w and ( be the components of (x ) as
( ) (g)gEG yg)gEG g gEC

occurred in the Wold Decomposition Theorem with respect to 4b;

(iii) F, Fy, and Fw be the spectral distributions of (xg)gEG’

) and w ) res ectivel and f, f and f their
(3'8 366’ (g gEC p y y. w

correSponding Spectral densities;

(iv) Fa, F8 be the absolutely continuous and singular components

of F with respect to the Haar measure m, as in the Cramer Decomposi-

tion Theorem ;

(v) % e L1(G*,B*,m).

Then
a 5
FY — F , Fw — F .

Proof. From Lemma 4.6 (Main Lemma I), R = I m (AE(dA)x
0 * 0 0

where G

(§0,RO)/f, on the carrier of F8
To =
0 , on the carrier of Fs .

. 1 * * . .
Since f E L1(G ,6 ,m), by Theorem 4.7, (xg)86G is minimal and,

hence, (£0,20) > 0. We will need this fact later in the proof of

the theorem.

35

NOW, (xg,xo) = (Wg + yg. W0 + yo) = (wgswo) + (ygsyo)° A180,

(xg.x0) = f*(g.x)F(di); (wgmo) = f*(g.A)Fw(dx);
G G

(ygayO) g j*(g:A)Fy(dh) -
G

Hence,

j (g.i)F(di) =j (1.1)(1‘w +Fy)(d>.) for all g e c.
6* 0*

Therefore, by Bochner's Theorem 3.4, we get that
(I) dF = dF + dF
w y

Since (yg) is non-trivial by ( v) and is Jb-regular, by Theorem

gEG

4.8, we see that Fy is absolutely continuous, and hence, from (I),

that
(II) dF = de + fydm .
From the Cramer decomposition theorem 3.15, we have
(III) dF = f dm + dFS .
Combining (II) and (III), we obtain
(IV) fwdm + dF: + fydm = f dm + dFS
which is equivalent to
s

S
(V) (f - f, - fw)dm - de - or .

Since the left-hand side of (V) is absolutely continuous and the

right-hand side is singular with respect to the Haar measure m,

36

it follows that

f=f+f a.e. m,
y w

dFS = dFS
W

We now wish to Show that f = fy a.e. m, which will complete our

. A C
proof. Since x8 = x8 - (xg‘m1+g,x)’ I = [O] , we see that
xg LmI-l-ggt for all g E G and, hence, x8 1 32677)1+8,X = ”1.00:"

for all g E C. However, from the Wold Decomposition Theorem

' that J- = . He e A f all C. 8'
1t follow: * 'Ilz‘lpo’x 77S, nc , xg E 778, or * g E ince
<90 6 L2(G ,6 ,F), by (I) it follows that (p0 E L2(G ,6 ’Fy) n

* *
L2(C ,cp ’Fw)' Hence, the integrals in the following equations are

well defined .

(VI) A, = f~*<p0(x)l:(di)xo = f*cpo(i)E(di)(yo + wo)
G G

= E‘EicchOJE(dim), + £*cpo(l)E(dk)W0 .

But by Lemma 3.6 f*mO(A)E(dA)yO 6 my and [*mO(A)E(dA)w0 e 77),.
c G
.. = 1
Also, x0 E 77) . Therefore, by (VI), J‘*tpo(),)E(dA)wo E 77) 77) ,

G
Hence, J‘*tpo(A)E(dA)wo = 0 and

G

(v11) £0 = f*¢b(i)E(di)yo -
c

Combining (VII) with the result in Main Lemma I, we get

(VIII) x0 = j*¢b(x)E(di)xo =.f*¢o(i)E(di)yo-
C C
Using the first equality in (VIII), we get

37

(IX) (£0.90) = f*|m0(i)\2F(dx) = (&O.so)zj f(T) m(d i) .
C

Using the second equality in (VIII), we obtain
f (A)

(x) (10,90) =£*\e0(i)\% Fy(dk) = (x03O ) 2j*% f, (x) m(dx) .

Combining (IX) and (X) and recalling that (R0,RO) > O, we get

f(k)

1
(XI) m(di) =.f -1———-m(di) .
£** f(*) 6* I2 (1)
Hence,
1 .—
(XII) f* f(,) [1 - £,(i)/f(i)]m(di) —
G
But, since f = f + f a.e. m, f 2 f a.e. m and, hence,
y W Y

1 - fy/f 2 0 a.e. m. Therefore, by (XII)
l/f[l - fy/f] = o a.e. m .

But 1/f > 0 a.e. m. Therefore, 1 - fy/f = 0 a.e. m, and hence,
f = f a.e. m. Q.E.D.
* *
4.14 Remark. If l/f E L1(G ,6 ,m), we note that Remark

4.10 implies that (xg)gEG

x = w for all g, and, hence, that F = F . In this case F
g g w w

could be absolutely continuous with respect to the Haar measure.

is Jb-Singular, which Shows that

Using Theorem 4.8, we can easily Show that if (yg)gEG is non-trivial

and the Wold-Cramer Concordance Theorem holds (i.e., F8 = F ;

y

*
F8 = Fw)’ then l/f must be in L1(G ,Br,m).

In general,any analytic condition on the spectral distribution

of a WSRF gives rise to certain geometric pr0perties of the random

38

field itself. AS we have seen, the analytic condition
l/f E L1(C*JB*,m) is equivalent to minimality for the process.
For a WSRF over the integers, the analytic condition
log f E L1(G*,6$,m) implies, among other things, that the process
is non-deterministic (for the definition of a non-deterministic
processes, cf. Theorem 3.18).

From our work one may suSpect that the analytic condition
1/f E 11(GIA6r,m) will have a definite relation to the past &
future of the process in the same way that the weaker analytic con-
dition log f E L1(G*,6r,m) had a close tie with the past of the
process. For this reason we will temporarily digress to a Short
discussion of WSRF'S over the integers where these notions of
past and past & future are meaningful. Using our results on
minimality we will then make appropriate comparisons between the past
and the past & future of a WSRF over the integers.

We will first set up some notations.

4.15 Notations. Let Z be the integers. Let ‘(xn)f

be a univariate flfivalued WSRF over Z. Let Jk = {k}c and

Ik = {11: n S k}. Then

(i) ka,x will denote 6(xn, n 94 k)

(ii) 971118}t will denote 6(xn, n s k) (of. Remark 3.10, (iv)).

With these notations, we see that 77) C 771 for all
Ik-l’x Jk’x
m as
k. Hence 7) = f) 77) g n 77) = .
Jp’x k=... Ik’x k=-.. Jk’x Jo,“

In the following theorems we will examine conditions under

C
or map ,X ’mJ0,x°

which 77) = 771
JP ,X .00 ,X P

39

4.16 Theorem. Let (xn):O be a univariate N-valued WSRF
* *
over the integers. let %E L1(C ,6 ,m), where f is the Spectral

I m g
dens 1ty of (xn) _m. Then ”2.0),," 771.00 ,x'
Proof. From Theorem 4.13, we see that Fw = FS; Fy = Fa,

where Fw and Fy are the Spectral distributions of the components

(D

co co . .
(wn)_m and (yn)_‘JD of (xn)_m given by the Wold decomp051tion

theorem with reSpect to the family {Jk}: (cf. Theorem 4.2). In

* *
addition, Since l/f and f are in L1(C ,6 ,m), then log f is

* * a
also in L1(G ,6 ,m). Hence, by Theorem 3.17, Fv = FS; Fu = F ,

where Fv and Fu are the spectral distributions of the components
Q CD . . .
(vn)_m and (un)_m given by the usual Wold decomp051tion theorem

with reSpect to the family {1k}: (cf. Theorem 3.16). Combining

these two results, we obtain

(I) Fw = Fv; Fy = Fu .

From (I) and the fact that 7790’" 2779‘)“, we W111 Show that wn = v
for all n. We observe that

(II) (xn‘m'ap,x) = ((xn‘mJ0,x)‘m.ap,x)

' = a d = ‘
Hence, Since vn (xnlmJ ,x) n Wn (xn‘mJ ,x)’ it follows that
v = (w ‘77) ). Using this last relation, we can easily Show that
n n Jp,x

VII .1. (wn - vn) for each n, from which it follows that

(III) ‘wn\2 = lunl2 + ‘wn - Vn‘z for all n.
Now. (w,.w,) = j (g.l)(g,i) Ewan) = Fw(c*) and
G*
(v,.vn) = he.» (3.1) F,(di) = F,(c*)

G

4O

‘2 = 0 for all n

Combining (I) and (III),;..we get that ‘wn - vn
and, hence, wn = vn for all n, which implies that 77)w = 77)v. Since

a and = we see that = . .E.D.
m 771.0 :X 771d ”(J ,X , may ,x 771.9 ,X Q
P 0 p O
In the above theorem we aSSumed that the analytic condition

*
l/f E L (G ,6*,m) held and saw that 77) = 77) . The natural
1 Jp’x Jo’x

question to ask is what happens when we assume that the weaker

analytic condition of log f E L1(C*,6*,m) holds, but not the con-

dition 1/f E L1(C*,6*,m). The answer is given in the next theorem.
4.17 Theorem. let (xn)m be as in Theorem 4.16. let

* * * * c:
1/f 4 L (C ,5 ,m). Let log f e L 1(C ,/3 ,m). Then 77) 77) =77)<.
Proof. Since 1/f E L 1*(C ,6 ,m), by Remark 4.10,x(xn)_m
is JO-singular, and, hence 77).“,0,x = W[(. It is well known ([3 ],
6* a)
p. 577) that if log f E L 1*(G ,6 ,m), then (xn)_m is non-
deterministic and, hence, 77).,p “$77k. Q.E.D.

In §6, we will give an example of a process for which

log f E L1(G*,6*,m), but LEI-3 E L1(G*,6*,n), where P is any given

polynomial. In particular, l/f will not be in L1(G*,6*,m). (cf. 6.4).
4.18 gm. If we assume log f E L1(C*,6*,m), then

l/f will not be in L1(G*,6*,m). By Remark 4.10 and ([3 ], p. 577),

we have me =77)“00’x = 77))(. We also note that when

1/f 6 L1 (Gk ,6 ,m) then 7:)Jp x =77)“,0 x = [0} iff FS = 0. Further-

*
6 ,m), then

9

*
more, in case l/f E L 1*(G ,6"r ,m), bat log f E L1(C
’x a {0} iff F8 = o.
4.19 Remark. The rest of this section will be devoted to
an extension of Salehi's work on the interpolation of WSRF'S indexed

by integers to univariate WSRF'S over discrete LCAG'S. (This concept

41

for WSRFIS indexed by integers or real numbers was first studied by
A.M. Yaglom [35] and later by Y.A. Rozanov [24].) This will provide

a natural extension of earlier results of this section in the same
way that Salehi's work on interpolation provided a generalization of
Kolmogorov and Masani's work on minimality. Although the ideas and
concepts used here are Similar to the ones used by Salehi in his work,
some of the techniques are different, since the integers are ordered
and singly generated whereas an arbitrary group need not be.

Our main reason, besides the historical one, for treating
the minimality problem separately is that in this case the reSults
are obtained in a more closed, compact, clear, and Simplified form.
In addition some of the results on interpolation are obtained under
the added assumption that the group G is endowed with an order
relation compatable with the structure of G.

We will now recall some of our notations and introduce some
new ones needed in the rest of this section.

4.20 Notation. Let (xg)gEG be a univariate flzvalued WSRF
over the discrete LCAG G with Spectral density f. Let
J = {g0,g1,...,gn] be a fixed set of n+1 elements of G. Then,

we will denote by:

. = i .
(1) 7),,x 77) c 077), (cf. 3.8).
J ,X n
(ii) 65 = [P : P(A) = Z ck(gk,x), c0,c1,...,cn arbitrary

2 k=0 * *
complex numbers, and ‘P(A)‘ /f(A) E L1(G J6 ,m);

(in) J, = {1° + g. g e c).

4.21 Remark. The set of polynomials 63 and the subSpace

were introduced in Salehi's work and will play an important

72J x

42

role in the theory of interpolation of WSRF'S. It is obvious that

z E NJ x if, and only if, z l-xg for all g E JC and that
S
= 6 R ... R where it is defined b
”Jsx ( go, , 8n) gi y
R = x - (x Am ).
81 8i 31 Jc,x

We will now make the following definition which is an exten-
sion of non-minimality for a WSRF over a discrete LCAG.
4.22 Definition. Let J be as above and (x )

g gEG
univariate flkvalued WSRF over G, a discrete LCAG. We say that

 

(x )

is interpolable with respect to J if

8 SEC
772%,, 97R C
J ,x
or, equivalently,
= 0 .
72%, M

It is clear that fig x is a subSpace of ”1' It is also

obvious that the set 53 is a linear Subset of all polynomials.

We introduce an inner product in 63 in the following manner.
P1(A)P2(k)
C

 

The proof of the following lemma is straightforward and thus
will be omitted.

4.23 Leggg, With the above notation, 63 is an inner pro-
duct Space over the complex numbeniwith the inner product

P1(1)P,(i)

* f(i)
c

 

(13,112)“f = m(di). P1.P2 e 63.

43

The fact that this inner product Space is finite-dimensional
and, hence, complete will follow from the following important lemma.

This lemma will be used repeatedly in the interpolation of WSRF'S

over C.

4.24 Lemma (Main Lemma II). With the above setting the
finite-dimensional subSpace ”J x (cf. Remark 4.21) and the inner
3

product Space 63 are isometric; i.e., there exists a linear Operator

T on fid,x onto 63 such that
(21,22) = (T217T22)1/f, 21:22 6 71],), '

Proof. Let 2 E ”J x' We define the polynomial Pz by
n
(I) P (I) = Z (2.x )(g .1)-
7‘ k=0 8k k

We claim that P2 is an element of 63. In view of the fact that

the subspace ”J x is spanned by [R ,Rg ,...,R ], it suffices
’

8O l 8n
to prove that Pk E 65, 0 s i S n. For simplicity, Pi(A) will
gin
denote P. (A) = Z (R ,x )(g ,A). Since R E , by the
"g1 k=0 8i 8k 1‘ gi 779‘

Isomorphism Theorem (cf. Theorem 3.7), there exists

* *
mi 6 L2(G as ,F) Such that Rgi = I*¢i(1)E(dX)XO- From the fact

G
that Rg 1.xg, g E JC, we get the following:
i
A = .. = C
(II) (xgiacg) [*cpi(>.)( s.>.)F(d>.) 0 . g 6 J

G a
(X ,X),86J
Ai 8

Let ck = (R8 ,xg ), O S k s n. Then we have the following equations.
1 k

44

n
(III) j*P,(i)(-g.i)m(di) = kEock_j*(gk.i)(-g.i)m(di)
G C

n
z ck ) (gk-g.i)m(dl)
k=0 9*

0 . g e J°

ck’gEJ.

From (II) and (III) we see

(IV) j*Pi(i)(-g.i)m(di) f*qq(i)(-g.i)F(di)
G G

which is equivalent to

(V) ) (~g.i)[P,(i) - c,(x)f(i))m(dx) = F (-g.i)c,(i)FS(di) .
* ”*
G

S . . . . .
But F is Singular w1th reSpect to m. U31ng measure theoretical

arguments, we get

I (~s.i)[Pi(i) - ei(i)f(i)]m(dA) = 0
*

(VI) G

)*(-g.i)o,(l)Fs(dx) = 0
G

which imply by Bochner's theorem 3.4

Pi(A) = mi(A)f(A), on the carrier of F8
(VII) S
mi(A) = O , on the carrier of F .
2
\Pi(i)\

But l .(A) 2F(d),) < m and, hence, m(dA) < m. There-
* $1

G C
fore Pi E 63.

* f(A)

We now define the operator T on fl] into 93 by

9

(VIII) T2 = P2 , z E flj,x .

45

Clearly, T is linear and it is not hard to Show that it preserves
the inner product.

It remains to prove that T is onto. To do that, we Show
n
that for any given P E«9 , P = 2 c (g ,A), there exists a z E n
k=0 k k J,x
such that P = Pz° We remark that the function

P(A)/f(A), on the carrier of F8
¢(K) =

0 , on the carrier of FS
0 O * * s
is in L2(G ,6 ,F). Define z me by
z = cp(x)E(d>.)X -
l, o

G

We now examine Tz:

(Iz)(>.) T()*<p(i)l:(d))x0)())

G

n
13:30 ()*cp(i)E(di)x0. j*(gk.i)E(di)x0>. (ska)
C G

n
:30 )*m(i)(-gk.).)F(di) - (gkm

n
P
1.30 L f—Efi (-gk.i)£(i)m(dx) . (gk.>.)
G

n n

2 2 c. (g.-g .x)m(d).) ° (3 .1)
k=0 j=0 3 £11 J k k

n

z c (g .i) = 13(1)
k=0 k k

Therefore, P2 = P. Q.E.D.
The following is the analogue of Kolmogorov's minimality

theorem (cf. Theorem 4.7) for the case when J has n+1 elements.

46

4.25 Theorem. Let J = [go,...,gn] be a fixed set of n+1

elements of G, a discrete LCAG. Let (x8)8€G be a univariate R“

valued WSRF over G with f its Spectral density. Then (xg)gEC

is not interpolable with reSpect to J if, and only if, there

*
exists a non-zero trigonometric polynomial P(A) on G of the
n

2 'k *
form P0,) '= Z ck(gk,),) Such that ‘P‘ /f E L1(G ,6 ,m); i.e.,
k=0
9, f {0}.

Proof. Necessity. Since (x ) is non-interpolable,

s EEG

771 c #mx- Hence, there exists a g E J SUCh that x g 771 c
J .X 8 J ’x

‘Without loss of generality, let g = g0. Since x E W)C , it

80 J ,x
follows that Rg E 0. By Main Lemma II, the function
n 0
a . 2 * *
P(A) = z (x ,x )(gk,x) is such that )P) /f e L1(G ,5 ,m).
k=0 go 8k 2
Clearly, P is a non-zero polynomial Since (R ,x ) = ‘R \ > O.
80 80 go
Sufficiency. Now Suppose there exists a non-zero polynomial
n
* *
P of the form P(A) = z ck(gk,A) such that ‘PIZ/f E L1(G A6 ,m).
k=0

Then P E 95 and, hence, by Lemma 4.24 , there exists a z E flJ x
’

such that z = f m(A)E(dA)x where
* O
G
P(A)/f(A) , on the carrier of F8

$0.) =

s
0 , on the carrier of F .

2
Hence, ‘2) = f )P(A)‘2/f(x)m(dx). Since P is a non-zero polynomial
*
C
and f is finite-valued a.e. m, it easily follows that
I ‘P(A)\2/f(x)m(dx) > 0. Hence, ‘2‘2 > O, and so 2 E 0. Therefore,
*

G
we have exhibited a non-zero element in Nb x; namely, 2, and, hence,
9

(xg)gEG is not interpolable with respect to J. Q.E.D.

47

The proof of the following corollary is immediate.
4.26 Corollary. With the same setting as in the above

theorem, we have: (xg) is interpolable with respect to J if,

366
. 2 * * .
and only if, ‘P‘ If 1 L1(G ,6 ,m) for any non-zero trigonometric
n
*
polynomial P on G of the form P(x) = Z ck(gk,x); i.e.,
k=O

= O .
a, H
The following lemma will be used in the proof of the next
theorem. Its proof is very similar to the proofs used in Lemmas 4.4
and 4.5 and, hence, will be omitted.
4.27 Lemma. Let J = {g0:gl,...,gn}. Let x8 6 fl be

such that fig ¥ 0 for some fixed gi E J. Then
i

A

file ) ¥ 0 for all g E G.
J +g,x

Next, we will establish a theorem on the relationship between

(xg+gi‘

i

the concept of non-interpolability and that of Jk-regularity intro-
duced in 3.11.

4.28 Theorem. Let (xg)gEG be a univariate fl9va1ued WSRF

over G, a discrete LCAG, F be its Spectral distribution, and f
. . __ C
be 1tS spectral denSity. Let J = {g0,g1,...,gn}, 4% — {J +3, g E a}.

(a) If (Kg)

is non-trivial and is Jk-regular, then

BEG

is not interpolable with reSpect to J, {hence with reSpect

( )

x
8 SEC
to J + g, fdr all g E G}, and F is absolutely continuous.

(b) Let G be ordered. If F is absolutely continuous

and (xg)g€G is not interpolable with reSpect to J, then
is - 1a .
(xg)g€G Jk regu r
Proof (a). Since (xg)gEG 18 J%-regular, then. mU%:X = {0}.
In particular, for some g E G, 771 c 1‘ 779‘. Then, by Lemma 4.27,
J +8:x

72J

x # {0}; i.e., (xg)g€G is not interpolable with respect to J.

48

We now wish to Show that F is absolutely continuous.

= J. =
Since mJn’x {0}, it follows that 771.0 ,1: Wk. But

n
”2"” = closure U 7711' = closijre U =
42“,): 866 Jc+8.x gEG J+g’x
6<§gi+g, 0 S i 5 n, g E G). Hence,
(I) mx=6(xgi+g,OSiSn,g€G)

In the proof of Main Lemma II (Lemma 4.24), we saw that

igi = f*wi(x)E(dx)x0 where

G
Pi/f’ on the carrier of F6
$1 = , S
O , on the carrier of F ,
0 S i S n. Hence, from a = U fi , it follows that
8i+8 g 81
(11) fig +g = f*(g.x>¢i(x)E(dx)xo. o s i s n, g e c .

G
Now, as in the proof of the necessity part of Theorem 4.8, define

*
the function W on G as follows:

a
O , on the carrier of F

(III) I = S
l , on the carrier of F .

* *
Then W E L2(G “B ,F) and, hence, by the Isomorphism Theorem, there

exists a z 6:”; such that z = I ¢(x)E(dx)x0. Hence,
*
C

(IV) (fig +g.z) = f*(g.x)¢i(x)w(x)F(dx)

1.

G
( ——P1 (k) 0 f d S
= , . O + ’ . 0 . ]-F d
£* 8 x) f(x) (x)m( x) £*(g x) < x)

= 0 .

49

From (I) and (IV), we see that z 4.7%. But 2 E‘Wh and hence

z a 0. Therefore, 0 = \2‘2 = j \¢(x>|2F(dx) = j Fs(dx) = Fs(c*).
* *

G G
which shows FS = 0. Hence, F is absolutely continuous.

(b). The proof of this part of the theorem will closely
resemble the proof of the sufficiency part of Theorem 4.8. Let

2 EEWQ. If we can Show that z l'fig +3 for all i 6 {O,l,...,n}

1
and all g E G implies that z = 0, then we will have shown that

{fig +8, 0 S i S n, g 6 G} is dense in. ”Q and, hence, that
i .
65‘: OSiSn €G= .But 65? ,

if, and only if, ”ngx = {0}; i.e., (xg)86G is Jk-regular.

O S i S n, g E G) ==Wk

In the proof of Main Lemma II (Lemma 4.24), we showed that

Sig =J‘ tpi(7()E(d)()xO where
i *
G
Pi/f , on the carrier of F8
$1 = , S
0 , on the carrier of F ,
0 S i S n. Since x = U fi we et
si+8 g 81’ g
A = k . o .
(I) xgi+g (*(g, )¢l(k)E(dl)xO . o s I s n, g e c

G
Since 2 6 m: z =J~ ¢(),)E(d)\)x0, Where w E L2(G*:B*9F)° Hence:
*

if we assume that G(2,5! ) = O, O S i S n, g E G, we get
gi+g

(II) 0 = (fig +g.z) = f*ei(x)(-g,x)¢(x)F(dx)

1 G

= j Pi(x)(-g,x)¢(x)m(dx). 0 s i s n, g e c .
*
G
This implies that all the Fourier coefficients of Pi o E' are zero,

0 S i S n, and, hence, by Bochner's Theorem (3.4),

50
(III) Pi - $'= 0 a.e. m , o s i s n .

Since (xg)gEG is not interpolable with respect to J, by Theorem
4.25, there exists some gi E J such that fi . ¥ 0. This implies
that the correSponding polynomial Pi is non-:ero. Since G is
ordered and Pi is a non-zero polynomial, it will be Shown in the
following lemma (lemma 4.29) that Pi cannot vanish on a set of

positive Haar measure. Hence, by (III), W = 0 a.e. m. Therefore,

2
‘2‘ = I ‘¢(x)‘2F(dx). But, F is absolutely continuous and, hence,
*

c
\z|2 = f \¢(x)\2f(x)m(dx) = 0, which shows 2 = o. Q.E.D.
*
G

4.29 Lemma. Let G be endowed with an order relation
compatable with its structure. Let P be a non-zero trigonometric
polynomial on G*. Then P cannot vanish on a set of positive
Haar measure.

Proof. Because G iS ordered one can Show that there exists

some gi E {g0,...,gn} such that

(I) P(x) = (girk)P1(X)r

n

where P1(x) = 'Eodj(g ,x), With gO = 0, dO # 0, and gj 2 0,

j
1 S j S n. It follows (cf. [26], Theorem 8.4.1) that

* * * *
log ‘PI‘ 6 L1(G ,B ,m) and, from (I), log ‘P‘ 6 L1(G ,B ,m). But
this implies P # 0 on every set of positive Haar measure. Q.E.D.

An immediate consequence of Theorem 4.25 and Theorem 4.28

is the following corollary.

4.30 Corollary. Let (x8)g€G be as in 4.28. Then

(a) If (xg)gEG IS non-triV1a1 and is Jh-regular, then F,

the Spectral distribution of (xg)gEG’ is absolutely continuous and

51

for all J +-g E 4%, J = {30""’gn}’ 63+g ¢ {0}.

(b) Let G be ordered. If F is absolutely continuous

and for J 6 JE, J = {g0,g1,...,gn}, 63 ¢ {0}, then (xg)86G is
karegular.

4.31 Remark. If G is not ordered, one can easily con-
struct a non-zero polynomial P on G* such that P = O on some
set of positive Haar measure. An example of Such a polynomial will
be provided in §6, Example 6.5.

We see that the assumption that G is ordered was used in
the proof of Theorem 4.28(b). As we saw earlier (cf. Theorem 4.8),
this aSSumption is not needed when J consists of a Single point.
It may be that the conclusion of part (b) of Theorem 4.28 is true

even without the assumption that G is ordered. However, our
proof does not demonstrate this.

Just as there was a definite relation between the concepts
of Jb-Singularity and non-minimality, there is also a relation be-
tween the concepts of Jk-Singularity and interpolability, as the

following remark shows.

4.32 Remark. Let ( ) be a univariate WSRF over G,

x
8 EEG
a discrete LCAG. Let J = {g0,g1,...,gn} be a fixed set of n+1

elements in G. Then ( ) is Jh-Singular if, and only if,

x
8 SEC
for all g E G, n = {0}, or, equivalently, for all g E G,

J+srx
9H3 = {0}.

Proof. (x8)gEG is J%-singular iff Wgc+g x =‘Wk: 8 E G,
nh+g,x = {0}, g E G, or, equivalently 463+g = {0}, g E G. Q.E.D.

Now, using Theorem 4.28, Corollary 4.30 and Remark 4.32, we

iff

will first give a characterization of a.WSRF over a discrete LCAG

52

which is neither Jk-Bingular nor Jh-regular in terms of its Spectral

distribution. We will then give conditions under which a WSRF

(x8)g€G over a discrete LCAG must be either Jk-singular or 4%-
regular.
4.33 Theorem. Let (xg)gEG be a univariate NQValued.WSRF

over G, a discrete LCAG. Let F be the spectral distribution of

8 SEC
be a fixed set of elements in G and 4% = {JC + g, g E G}.

(x ) and f be its Spectral density. Let J = {g0,g1,...,gn}

(a) If OJ # {0} and Fs 5‘ 0, then {0] Gimme $7786
(b) let G be ordered. If {0}$m.an.x$7’9<’ then

S
éh+g # [0} for all g 6 G and F ¥ 0.

PfO f a o S Se = 0 Th OS "'

0 ( ) UPPO Wln}x W; en (Kg)gEG 1 J;
singular and, hence, by Remark 4.32, 65 = {0], which is a contradic-
tion. Now, suppose WUESX = [0}. Then (xg)gEG is 4%-regular
and, hence, by Corollary 4.30, F is absolutely continuous, which is
a t ad' t' He 0} g 9

con r 1c ion. nce, { ”Ukfix W9.
(b) If 77) ,x $779!, then (xg)gEG is not Jn-singular and,

n

hence, by Remark 4.32, 9‘ [0} for all g E G. If 77).! x 5‘ {0},
n,

63+g
then (xg)gEG is not Jk-regular . By Corollary 4.30(b), this implies
that either F is not absolutely continuous or for all g 6 G, or
65+g = {0}. But the latter cannot happen. Hence, F is not absolutely
continuous; i.e., FS * 0. Q.E.D.
4.34 Theorem. Let G be a discrete LCAG which is ordered.

Let and be as i 4.33. Let be a univariate H“

J Jh n (xg)g€G
valuedNWSRF over G and F, its spectral distribution, be absolutely

continuous with reSpect to m. Then either (xg)gEG is Jk-singular

or Jk-regular.

53

Proof. If (x8)gEG is not Jgfsingular. then. ”5%)3 #‘Wk.

Hence, W1 ¥ , which implies that (x )
Jc,x Wk g BEG

with reSpect to J. Based on this and the fact that F is absolutely

is not interpolable

continuous and G is ordered, Theorem 4.28(b) implies that

(xg)geG is JE-regular. Q.E.D.

Our next objective will be to establish the Wold-Cramer

concordance relation for a univariate fl9valued WSRF (xg) over

gEG
G, a discrete LCAG with reSpect to 4%. The proof of this theorem

will depend on results on interpolation and Jk-regularity of

(X)

that e t Stabl‘shed.
g gEG wer jus e 1

4.35 Theorem.(Wold-Cramer concordance for 4%). Let

(i) ( )

x be a univariate fl-valued WSRF over G, a
8 BEG
discrete LCAG, which is ordered; J = {g0,...,gn} and
c
Jn={J +g, gEG}.;
o o d
(11) (wg)86G an (yg)g€G be the components of (xg)gEG

as occurred in the Wold decomposition theorem with reSpect to 4%;
(iii) F, Fy’ and Fw be the spectral distributions of

, and w res ectivel and f, f , and f
( )gec (yg)gEG (g)g€G p y y w

x

8
their corresponding Spectral densities;

a
(iv) F , FS the absolutely continuous and singular com-

ponents of F with reSpect to the Haar measure m, as in the Cramer
decomposition theorem;
(v) 63 ¥ {0}.

Then

54

Proof. By Main Lemma II, 4.24, W5 i {0}. Without loss of

generality we may assume that fi * 0 . In the proof of Lemma 4.24
0
(Main Lemma II): We saw that xgo = I*¢b(l)E(dk)xo where

G

PO/f, on the carrier of Fa

(P0 = . S
0 , on the carrier of F ,
n
P (I) = 2 (3: .x )(g .I). Also. since a. I77: and m =
0 k=0 80 3k 1‘ go J°,g Jn’x
0 7n , we see that 5‘: 4,77) ; i.e., x E 77pL . However,
BEG Jc,g go Jn’x go Jn’x
from the Wold decomposition theorem (Theorem 4.2), it follows that
, sml- . Hence, 5‘: e .
978, “fix 80 778’
Since (yg)gEG is non-trivial and is J%-regular, by Theorem

4.28(b), it follows that Fy is absolutely continuous with reSpect
to m. Then, using the same technique as in the proof of the Wold-
Cramer concordance theorem with reSpect to Jb (Theorem 4.13), we

obtain
(I) f = fy + fw a.e. m,
dFs = dFs .
W

Hence, if we can Show that f = fy a.e. m, our proof will be

finished. In the same manner as in Theorem 4.13, we get

(II) ago = £*oO(I)E(dI)xO = ggPoWE‘dWo -

Using the first equality in (II), we obtain

(III) (I: .2 ) = f 1o0(>.)l2F(dx) = j' \P0(I)|2/£(x)m(d>.) .

55

Using the second equality in (II), we get

52 fl \2 (d I‘Pom‘z f () (d)

(IV) (9 a ) = o (x) F I) = ----- I m I
g0 g0 6* O y 0* £20) y

Combining (III) and (IV), we get

(V) j |P0(x)\ /f(x)m(dx) =j —§——- f (mum).
* * f 0.) y
C G
which is equivalent to
2
IP0().)\ I (x)
.________ -.JL___ =
(VI) * fo) [1 f().)]m(d>‘)
G

But (I) implies that f 2 fy a.e. m and, hence, l - fy/f 2 O a.e. m.

Then, by (VI), we get

f

1
(VII) 2 [1 - E1] = o a.e. m.

 

Since G is ordered, Lemma 4.28 implies that \P0\2 > 0 a.e. m.
Since 1/f > 0 a.e. m, it follows from (VII) that 1 - fy/f = 0 a.e. m,
and, thus, f = fy a.e. m. Q.E.D.
4.36 Remark. If any non-zero trigonometric polynomial P
on G* of the form P(x) = kgock(gk,)‘) satisfies the
condition \P‘2 /f i L 1*(G ,B* ,m), then, by Remark 4. 32, (xg)gEG is
Jh-singular, which Shows that xg = wg for all g and, hence, that
F = Fw. In this case, Fw could be absolutely continuous with reSpect
to m.
We will now Specialize our results on interpolation of WSRF'S
to processes indexed by the integers. As in the case of minimality,

under suitable analytic conditions we will make appropriate comparisons

between the subspaces in the time domain (i.e., 771“) X and 771.9 x).
9 3
P

56

First, we will recall some notation and introduce some new ones.

oo

4.37 Notation. let 2 be the integers. Let (xn) be

-m

a univariate fl-valued WSRF over 2. Let J = {k0,k1,...,kn;

k0~< k1 <...< kn} be a fixed set of n+1 integers and

Ik={j :jSk}. let

(i) mc =6(x.,j¥k,+k,OSiSn).
J +k,x J 1
(ii) MIR“ = 6(xjr J S ‘0-
Obviously, 7n = U C n 772 = 77( °
'0 ,X k=-co mlk’x k=-oo Jc'l'kgx Jn’x

In the following theorems, we will examine conditions under
WhiCh ”(J :x = 771“, ,X or me? :x g 771-0 ,X .
P n p n
4.38 Theorem. Let (xn): be a univariate fl-valued WSRF

over Z, the integers. Suppose there exists a non-zero trigonometric
n

*
polynomial P on G of the form P0,) = Z cj(kj,),) Such that
i=0
* a:
‘P‘2/f E L1(G ,B ,m), where f iS the Spectral density of (xn):o.

,X

Then 771 = 771
up ,X J

p n
Proof. By Theorem 4.35, we see that Fw = FS; Fy = Fa, where

F and Fy are the Spectral distributions of the components

(wn)°° and (y )m of (xn):D given by the Wold decomposition

theorem with reSpect to the family Jn = {J + k, k E 2} (cf. Theorem

*
4.2). Also, since there exists a non-zero P on G of the form

n
P(A) = Z cj
j=0

tells us that (x8)

“‘1’“ such that ‘P‘z/f E L1(G*,B*,m), Theorem 4.25

iS not interpolable with reSpect to

BEG
J '= {k ,...,kn}, which implies that there exists a k E J Such

that xk é 6(xj, j e Jc). But this implies xk é 6(xj, j < 1(0)

and, hence, by stationarity, x f 5(XJ: J < k0)- Thus, (xk):°

ko

57

* *
is non-deterministic and so log f E L1(G J? ,m). Thus, by Theorem

3.17, F = Fs; F = F8 where F and F are the Spectral dis-
v u v u

tributions of the components (vn)fg and (um)?co given by the usual

Wold decomposition theorem*with respect to the family {1k}co . Hence,
-oo
(1) F = F ; F = F .

By (I) and the fact that m‘a x 2 7R."2 x we can use an argument similar
a a
n P

to the one given in the proof of Theorem 4.16 to show that

”I = ° Q.E.D.
Jp’x Jnax

4.39 Theorem. Let (xn)co be as in Theorem 4.38. Let

'G
* *
log f E L1(G ,6 ,m) where f is the Spectral density of (xn)fm.
*

Suppose that for any non-zero trigonometric polynomial P on G

n
of the form P(x) = 2 c (k ,x), we have

jgo j J

2 * * c: _
\P‘ /f E L1(G ,6 ,m). Then m'ap’x $77911," Wk.

Proof. By Remark 4.32, we see that (xn)f is Jk-singular
and hence, W2 ==W&. It is well-known ([3], p. 577) that if

4%:3
* *
log f E L1(G “6 ,m), then (xn)fg is non-deterministic and, hence,
m ,x Em.
P * *

4.40 Remark. If we assume log f é L1(G “B ,m), then
771.01,»)! = 77): and, hence, mJn’x = 772x.

This concludes our discussion on the problem of interpolation
with respect to 4% = {Jc +-g}, where J is a fixed set of n+1
elements of G, a discrete LCAG. We will devote the rest of §4 to
interpolation theory with respect to 4;, the family of complements
of finite sets of elements in G. First, though, we will recall some

notation, introduced earlier, which is relevant to what we will be

doing.

58

4.41 Notation. Let (xg)g€G be a univariate NQvalued WSRF

over G, a discrete LCAG. Let f be the spectral density of

(x )gEG and J be any finite set of elements of G. AS in 4.20
8

we will set
(i) 7? = 77!" fl ;
J,x Jc,x Wk
(ii) .05 = {P: P(X) = 2 c (g,x), cg's are arbitrary complex

EJ g
2 * *8
numbers, and ‘P‘ /f E L1(G ,B ,m)};

(iii) .J; = family of complements of finite sets of G.
We are now able to give the following definition of inter-
polability.

4.42 Definition. Let ( ) be a univariate RQValued

x
8 SEC
WSRF over G, a discrete LCAG. We say that (x8)

 

‘S i t labl
gEG 1 n erpo e
if (xg)g€G is interpolable with reSpect to every finite set of
elements of G.

The following remark follows immediately from Theorem 4.25.

4.43 Remark. Let be as in 4.42. Then

(xg)g€G (Ks)sEG

is interpolable if, and only if, 63 = {0} for any finite set J C G.
As we mentioned earlier (cf. Remark 3.13), there was an

error in the main Theorem 4.1 of L. Bruckner [1]. A similar type

of error regarding the relation between Jtiregularity and its char-

acterization in terms of the Spectral density of the WSRF is contained

in Theorem 5.2 of Bruckner. The following establishes a relation-

ship between the concept of non-interpolability and that of 4L:

regularity and includes a corrected version of Bruckner's result.

Here, again, part (b) may be true without the assumption that G

is ordered, but at this point we are not able to diSpense with

59

this assumption.

4.44 Theorem. Let (xg)gEG be a univariate ”fivalued WSRF
over G, a discrete LCAG. let F denote the spectral distribution
of (x8)86G and f its spectral density.

(a) If (xg)gEG is non-trivial and is JL-regular, then
there exists a finite subset J of G Such that 63 ¥ {0} and
F is absolutely continuous with reSpect to m;

(b) Let G be ordered. If F is absolutely continuous

and there exists a finite set J of G such that 63 f {0}, then

(x )

8 gEG is Jon-regular.

Proof (a). Trivially, there exists a finite set J of G
such that OJ 4% (0)..
a

Let W = 0 , on the carrier of F

s
l , on the carrier of F .

It is obvious that w E L2(G*¢6*,F). Let 2 E ”h correspond to

y. Using techniques similar to those in the proof of Theorem 4.28(a),
we can show that z l'nJ,x for all finite subsets J of G. But
this implies, because (x )

8 SEC
2 = 0, which implies (2,2) = FS(G*) = O.

is Jab-regular, that Z .L'mx. Hence,

(b) Trivially,

an

_ C
(I) ”400,1: 771.0an . Jn - {J + g, g E G} .
By (I) and Theorem 4.28(b), we have ”ULRX = {0}. Thus, (xg)gEG
is JL-regular. Q.E.D.

In the following remark we will state a characterization of

.JL-singularity for a.WSRF over a discrete LCAG.

60

4.45 Remark. Let (x ) be as in 4.44. Then (x )
"““' 8 g g€G

gEG
is.4”-singular if, and only if, any non-zero trigonometric polynomial
P on G* satisfies the condition that \P‘zlf é L1(G*,5r,m); i.e.,
for all finite sets, J, of elements of G, 65 = {0}.

Next, we will give conditions under which a WSRF over a dis-
crete LCAG is neither JLfsingular nor Jg-regular. We will then give
conditions under which a prOceSS must be either Jtisingular or 4;:
regular. The proofs of these results follow from 4.43 and 4.44 in
the same manner that the proofs of 4.33 and 4.34 were derived from
4.28 and 4.32.

4.46 Theorem. Let (xg)g€G be a univariate N9valued WSRF
over G, a discrete LCAG. Let F be the Spectral distribution of
(xg)gEG and f be its Spectral density.

(a) If there exists a finite set J of G such that
OJ 9‘ {0} and FS 5‘ 0, then {0} $771420!”x $77126

(b) let G be ordered. If {0} $77142“),X gm, then there
exists a finite set J of G such that 63 ¥ {0} and FS # 0.

4.47 Theorem. Let be as in 4.46. Let G be

(Xg)g€G

ordered and F, the Spectral distribution of (xg) be absolutely

gEG’

continuous with respect to m. Then either (xg) is thsingular

gEG
or JL-regular.

4.48 Remark. Let G be ordered. We have proved that if
F8 = 0, then we have either.J-regu1arity or Jksingularity for the cases
J = .90, Jn’ and Jon (cf. Theorems 4.12, 4.34, and 4.47). We remark
that if the WSRF is Jb-regular, then it is also 4% and Jtrregular;

or, equivalently, if the WSRF is JLfsingular, then it is also J%

and Jb-singular. Other cases of interest may happen; e.g., a WSRF

61

may be 4b-singular and yet 4% and hence 4Lfregular.

We will now establish the Wold-Cramer concordance theorem
for JQ' Since the proof of this theorem.is very Similar to the
proof of Wold-Cramer concordance theorem for 4% (Theorem 4.35),
we will only Sketch it.

4.49 Theorem.(Wold-Cramer concordance for 4;). Let

(i) ( )

x be a univariate N—valued WSRF over G, a
8 SEC

discrete LCAG, which is ordered; 4; = family of complements of finite

sets of G;

i’ w be the com onents of
( 1) ( g) (yg) p (xg)gEG

866 866

as occurred in the Wold decomposition theorem with respect to 4&3
(iii) F, Fy, and Fw be the Spectral distributions of

)

( respectively and f, fy, and fw

xs sEG’ (ys)gEG’ and (wg)s6G
their correSponding spectral densities;

(iv) Fa, F8 the absolutely continuous and singular com-
ponents of F with respect to the Haar measure m, as in the Cramer
decomposition theorem;

(v) there exist a finite set J in G such that 95 # {0}.

Then

Proof. Let 2 be a non-zero element in W5 x' Trivially,
3

z .Lm‘a ,x and, hence, from the Wold decomposition theorem with
a
reSpect to 4m, 2 E Wg’. A similar proof to that of Theorem 4.35

shows that fy = f a.e. m. Hence, Fy = Fa; Fw = F8. Q.E.D.

4.50 Remark. If any non-zero trigonometric polynomial P

* *
on 6* satisfies the condition that ‘Plzlf é L1(G ,6 ,m), then,

62

b Remark 4.45 x is -sin ular. Hence x = w for all
y a ( g)gec .000 s r g g
g, and, thus, F = Fw' In this case, Fw could be absolutely con-

tinuous with reSpect to m.

In specializing our results for 4; to the case when G = Z,

the integers, we will simply state the results comparing ”U7 x and
p,
WU, x’ since their proofs follow closely the correSponding proofs
3
Q
for 4%.

4.51 Theorem. Let (xn)0° be a univariate.fl9va1ued WSRF
over Z, with spectral distribution F and Spectral density f. If,

for all n E Z, there exists a finite set J containing n Such

that eJ 9‘ {0}, then 779 ,x =771J
P

oo’x.
4.52 Theorem. Let (xn):°co be as in 4.51. Let
* *
log f E L1(G ,6 ,m). Suppose for all finite sets, J, of elements

- C =
of 2, OJ - {0}. Then ”up“ as m'lm’x 779‘ (see Example 6.4).

4.53 Remark. If log f e L1(c*,e*,m), then

meex = "lama '7 W‘x'

5. MINIMALITY AND INTERPOIATION OF q-VARIATE WSRF'S

In this section we will consider the problems of minimality
and interpolation for q-variate oquvalued) WSRF'S over a discrete
LCAG. In the univariate case the fact that the Spectral density is
a non-zero Scalar a.e. m and, hence, has a well-defined inverse
Simplifies the work considerably. Since in the multivariate case
the spectral density is matrix-valued and, hence, does not have an
inverse in general, the results on minimality and interpolation
become harder to handle.

By employing the notion of the generalized inverse of a
matrix, we can extend several of our results on the univariate case
to the multivariate case. The notion of a generalized inverse in
connection with the minimality and interpolation of a WSRF indexed
by integers was first introduced and exploited by H. Salehi [28].

We shall use his ideas. However, in some cases, it will be necessary
to use actual inverSes. In these cases, we will make the assumption
that certain matrices have full rank.

To avoid any duplication between our work on the univariate
case (as presented in §4) and on the multivariate case (as presented
in this section) we will omit the proofs of those results in the
multivariate case which are analogous to the proofs of the correspond-
ing works in the univariate case. Hence, we will provide proofs

only for those Statements which involve new techniques or new ideas.

63

64

Our first objective will be to establish the Wold decomposi-
tion theorem for the multivariate case. Since the proof is similar
to the proof for the univariate case, we will only outline the main
steps of the proof for the benefit of the reader. First, we intro-

duce the following definition.

 

5.1 Definition. Let (fig) (XB)gEG be Hg-valued

gEG
WSRF's over G, a LCAG. Let .4 be any family of non-empty Borel

sets of G. Then (Zg)g€G is said to be J-Subordinate to

'f
(Emacs 1

(i) 29 CEZQ ;
.. c .
(11) 731’), 7111,)! for all I E 42,
(iii) (x ) and (yg) are mutually homogeneously
-s sec 366

correlated.

5.2 Theorem (Wold decomposition). Let .4 be any family
of non-empty Borel sets of G closed under translations. Let
(£8)gEG be an.flg-valued WSRF over G, a LCAG. Then there exists

a unique decomposition of (x ) with respect to .J in the form

gEG

x= +W
‘8 XS ‘8
where
(i) (1g)gec and (Eg)gEG are wq-valued wsap's on 0;
(ii) (yg)g€G and (Eg)g¢G are.J-subordinate to
)gEG’
(iii) (2g)86G and (wg)86G are orthogonal; i.e.,

(a.

,w = 0 for an ' G;
(28 _g.) _, y 8.8 6

is 42-3 ingu lar .

(iv) (yg)86G is.4-regular; (wg)gEG

65

Proof. Let U ) be the group of unitary Operators on
(‘8 EEG

q .
Y . ass ciated with . It can be shown that U =

g€G

.L = 1 , =
and hence gng,x) 71(4),): for all gEG Let ‘13 (358mm)

and lg = £8 - Hg for all g 6 G. Then it is easy to see that
7.71" = mJ,x and my Its-71);“. The argument that (Xg)gEG and
(I13)gEG are WSRF'S and the proofs that (Zg)gEG and (1g)gEG are
J-subordinate to (358)86G are straightforward.
To prove that (1g)gEG 'is J-regular, we observe that EU”
is both perpendicular to 7114,x and contained in it, so that
721.0,), = {9}. Therefore, (yg)gEG is J-regular. An argument similar
to the classical one (cf. [32], p. 137) Shows that 711.0,x =29” @Wz'paw

and hence 719 w = 740. Therefore, (I13)g is J-Singular.

6G
We remark that for any decomposition of ()_(_g)gEG into
and w Satisf in conditions ' - iv we have that
(18)86G (_g)86G y 8 (1) ( ).

m = 711 . This important relation makes the decomposition unique.
Jaw Jsx

We will now State the definition of minimality for a q-variate
WSRF over a discrete LCAG.

5.3 Definition. let G be adiscrete LCAG. Then the ”q-

valued WSRF (gig)gEG is minimal if, and only if, £0 6 Z&,x’ where
I = {0}°.

The proof of the minimality theorem for q-variate WSRF'S
over a discrete LCAG will depend on the following lemmas. The proofs
of these lemmas are analogous to the proofs of Lemmas 4.4 and 4.5
and hence will be omitted.

5.4 lemma. let (5g) be an liq-valued WSRF over a dis-

gEG
crete LCAG G. let 5? denote x - x where I = 0 C.
-s 1 (“sml+s.x)’ { }
Then 5? is a yq-valued WSRF over G. In addition x
(-g)g€G ’ (-g)gEG

66

and ) have the same shift.

36G

5.5 Lemma. Let (x ) be an Hq-valued WSRF over G, a
""“ ‘8 86C

is minimal if, and only if, for all

s
('8

di t . The
scre e LCAG n (35g)gEG

c
= 0 .
g E G’ fig é ZlI-fg,x’ I { }
The next lemma plays an important role in the theory of

minimality of.Ng-valued WSRF'S.
5.6 Lemma (Main Lemma 1). Let (345g)gEG be an Ng-valued
WSRF over G, a discrete LCAG, with the Shift group of unitary
o erators U ) and E be the spectral measure of (U ) .
p (ggec ggec

Let R = 5g - (xg‘ ), I = {0}c. Let E_ be the Spectral

‘8
distribution of

21133,):

(§8)86G’ if its Spectral density, and Ef# the

generalized inverse of El (cf. [17] and [20]). Then

3g = f*(s.).)go().)§(d).)§o
c

where Q

‘0 is defined by

(30,$O)Ef# , on the carrier of F?

20

IO

, S
, on the carrler of §_.

Proof. Without loss of generality, it suffices to prove

A

that x is given by

8

120 = f*§O(x)§(dl)2gO -
G
Since 50 622;, by the Isomorphism Theorem (3.7),

:30 = f* goorydmo
G

67

-k *
for some £0 E L2(G ,6 ,F). Also (cf. 3.2),

£8 = f*(8sk)§(d7\)’_€0 for all g 6 G.
C

Using the same techniques as in the proof of the univariate case

(leuma 4.6), we arrive at the following

(1) (30,30) = 202' , on the carrier of Fa

20 = Q , on the carrier of [S .

Letting Y. = 9'02? ', where £1? ' denotes the projection operator
F F

onto RF" the range of 11', we can easily Show that g = 20 in

* *
112(G ,6 ,1_“). Therefore, (I) can be written as

A " . a
(II) 3; 1;" = (150,150) , on the carrier of F
‘1 = Q , on the carrier of F_‘3 .
Hence, by (II), it is clear that
(530,530)§'# , on the carrier of F_‘a
X = S
Q , on the carrier of F_ . Q.E.D.

For the proof of Kolmogorov's minimality theorem, we need
the following lemma, whose proof is found in [20].

5.7 m. let 6 be a a-algebra of Subsets of a Space
0 and p. be a non-negative a-finite measure on 6. Let g be a
non-negative, hermitian, q X q matrix-valued function on n such

that 2. E Elm,6,u) 0 Then

rank (IE gap.) 2 rank g a.e. u .

68

The following theorem on the minimality of a WSRF over a
discrete LCAG is an analogue of a theorem by Masani (cf. 3.14) for
processes indexed by integers.

5.8 Theorem (Kolmogorov minimality theorem). Let (xg)gEG
be a q-variate WSRF over G, a discrete LCAG, F. its Spectral dis-
tribution, and if its spectral density. Then (5g) is minimal

sec
and pI 0 = q, I = {0}C if, and only if, Ff-l exists a.e. m and

- * *
g' 1Egla: ,6 ,m).
Proof. Sufficiency. Set
0'1 - a
F_ , on the carrier of F
g: S
Q_ , on the carrier of F .
Then
* * *
(I) jsdii =jit'2dm+fidts_
* x *
G G G
= I Ef-ldm .
*
G

* *
Hence, g_€ L2(G “B ,F). Now, by the Isomorphism Theorem (3.7) there

exists y_€ 2% such that

an x=jgnmmm%.
*
G

Following the proof of Lemma 4.6, we can Show that

(III) (53,1) = 6g,01'°

Hence, y_= A_ Note that by (I), y_# 9. and thus “ #‘Q.

50- ’io

Therefore, £0 4 ELX’ which implies (ig)gEG . is minimal.

69

By (II) and the fact that y_= é_x , we get

(Iv) (bl) = j‘*§_"1(i)m(dx) = escape"

C

By assumption, rank(§f-1) = q a.e. m and hence, by Lemma 5.6,

rank (y,y) = q. Thus, rank (30,30) = q. Hence, 91,0 = q.
Necessity. Since 91,0 = q, then rank (EO’EO) = q. But,

in the proof of Main Lemma I (5.6), we had

||
A
X
X
v
a)
m
B

F!
20..
Hence, rank (Ff) = q a.e. m, which implies Ff-l exists a.e. m.

. A _ * = a ['1 A
since (50.30) — 1,9612 3 $550.30): (soarOmm. and

.-1 * *
and hence F; E L1(G ,6 ,m). Q.E.D.
The following is a partial analogue of Theorem 4.8. We note

that in part (b), we need a Stronger assumption than minimality;

namely, pI 0 = q, I = {O}c.
’

5.9 Theorem. Let (x ) be a non-trivial.Ng-va1ued
‘—'—-—' ‘8 86G
WSRF over a discrete LCAG G.
(a) If (35g)86G is 4b-regular, then (x3)gEG is minimal

and F the S ectral distribution of x is absolutel con-
tinuous with respect to m;
(b) If F_ is absolutely continuous and pI 0 = q, I = {0}c
9

then is 4b-regular.

(359366

70

’ a .L I: '
Proof (a). Since 23.00% {0}, 23-00,), Zg‘. But, as in the

proof of Theorem 4.8, we have 7114- = 66? , 3 EC). Let
Jo’x 8

a
Q , on the carrier of F_

, S
I , on the carrier of F .

AS in the proof of Theorem 4.8 we can Show ‘_i'_ 6 L2 (G*,6*,I_7_). By
the Isomorphism Theorem (3.7), there exists a y 6 Ex Such that
(y,y) = I}! d_ 1*. It is not hard to Show that (1,13%) = Q for all
g 6 G, End hence I = 9_. Thus, since (y,y) = _F_S(G*), ES = Q.

Trivially, since 711.00,x = {Q}, Jig for all g,

4 mI-l-g ,x

and hence is minimal.

x

(‘8)BEG
(b) let a me' If we can Show that z 13g for all

g 6 G implies that g = Q, then we will have shown that

{33, g 66} is dense in 171x, and hence that 6(58, g EG) =7ZLK.

But 6(539 g E G) =2; lff mJO’x = {9.}; i°e°a (5g)gEG is ‘00-

regular.

From Main Lemma I (5.6), we get that

3g = j*(g.).)go(l)§(d).)2go for all g e G,
G

where go is as in 5.6. Since 3 6 Ex, 5 = J‘*g_(x)§(d)\)xo where
* G

*
1 E 112(G ,6 ,D. AS in the univariate case,

9 = (5,53) = ()josfio)‘f*l('8sk)m(dk) for all 8 E G-
G
Since pI 0 = q; i.e., rank ($50,530) = q, hence,
,

Q = j” I().)(-s.).)m(d).) for all g e o.
*
G

71

As in the proof of Theorem 4.8, we get g'= g a.e. m. Therefore,

Since F. is absolutely continuous,

(zz)= “I! E'Yd =0
i,— =,.v.)" - _
G G*
which shows that E.=.Q° Q.E.D.

From the above theorem and Theorem 5.8, we obtain the follow-
ing corollary.
5.10 Corollar . let x be as in 5.9.
__y_ (1)2366

(a) If (fig)

is 4b-regular, then (x3) is minimal

gEG gEG

and g is absolutely continuous;

(b) If E_ is absolutely continuous, Ff-l exists a.e. m,
ad F"16L G** h 'SJ la
n _ _1( ,6 ,m),ten (§3)g€G 1 0regu r.

5.11 Remark. We can easily give a characterization of

4b-singularity in terms of pI 0. In fact, (x8) is 4h-singu1ar
9

gEG

if, and only if, pI 0 = 0. However, in terms of the spectral domain,
5

we are, at this time, only partially able to extend the characteriza-

tion of Jb-singularity for the univariate case to the multivariate

case, as the following remark indicates. The proof is straight-

forward and, hence, will be omitted.

5.12 Remark. Let (x ) be an Vq-valued WSRF over G,
__-‘—' ‘8 866
a discrete LCAG- If (X ) 15(J ~Singular, then either Ff-l

gEG 0

doesn't exist a.e. m or else {"1 é L1(G*,6*,m).

As in the univariate case, there exist conditions under
which a q-variate WSRF over a discrete LCAG is neither 4b-singu1ar
nor 4b-regular. Once again the proof is not difficult and thus

will be omitted.

72

5.13 Theorem. Let be a q-variate WSRF over a

(Eg)gec
discrete LCAG G. If 91,0 # 0, I = {O}c, and E? # Q, then
(2

{9.} * 271,0, $7.72,.

In the following we prove a theorem on the concordance of
the Wold decomposition with reSpect to 4b and the oximer decomposi-
tion for a q-variate WSRF over a discrete LCAG under the assumption
that the process has full rank. The problem remains open when this
condition is not satisfied. As one can see from Theorem 3.20, for
q-variate processes indexed by the integers Robertson has given a
necessary and Sufficient condition involving the rank of the
spectral density for concordance of the Wold decomposition with
respect to the past of the process and the Cramer decomposition.
For q-variate WSRF'S over a discrete LCAG, it would be interesting
to give a necessary and sufficient condition involving the rank of
the Spectral density for concordance between the Wold decomposition
with respect to 4b and the Cramer decomposition.

5.14 Theorem (Wold-Cramer concordance for 4b). Let

(1) (Eggs; be an yq-valued WSRF over (2, a discrete
LCAG;

(ii) (Eg)g€G and (yg)gEG be the components of (x3)gEG
as occurred in the Wold decomposition theorem with reSpect to 4%;

(iii) F, Ey’ and Ew be the Spectral distributions of

and (33) respectively and E}, E;, and E;

x
(1)36? (l8)8€G sec
their corresponding Spectral densities;

a
(iv) F_, F? be the absolutely continuous and singular

components of F with reSpect to the Haar measure, as in the Cramer

decomposition theorem;

73

- *
(v) F' exist a.e. m. and F' 1 E L (G ,Er,m).
._ —' -—1

Then
a s
Ey — .1: 3 E“ " E 0
Proof. By assumption (v) and Main Lemma I, 5.6, we get
(I) so = j*20(x)§(di)so
G
where

(30,30)§f-1, on the carrier of F?

s
O , on the carrier of §_.

By Theorem 5.9(b), 2y is absolutely continuous with reSpect to m.

Now, using the same type of proof used in Theorem 4,13, we obtain

the following relations:

(II) F' = F' +'F' a.e. m
dFs = dF8 .
- w

A

Since we can easily Show that 50

6.29, again imitating the proof

of Theorem 4.13, we obtain

20(I)F_3(d).)x0 .

(III) so = j*
C

By (I), it follows that

3%

(Iv) (530.510) = j*iod£

-1 A

= ' “

1*(§D’*C)E- (x:,x:)dm .
G

By (III). we set

*

A A = = 3' d .
0”) (£0,350) I*Qod§'yio J‘*(goago)£ (£09 9_O) m

G G

Combining (IV) and (V), we have
(VI) j (a )_ '1(i i )dm = I (R s )F"1F'F"1(s s )dm .

* "0,-0 _09_0 * _Q3_,O __ _y_ ...0:__0

G G
But, by assumption (v) and Theorem 5.8, rank (30,130) = q, and, hence,
(VI) is equivalent to

c'1 _ 1'1 I t’1 _

(VII) j*(E_ E, EYE. )dm - 9 -
By (II), F' 2 F' a.e. m and, hence, F'”1 2 F' 1F'F . m.

-' '7 — '- ‘YF
This fact and (VII) imply, by Ieunna 5.7, that F'1=;1-1;:'
a.e. m and, hence, E' = E; a.e. m. Q.E.D.

5.15 Remark. If 91,0 = o, 1 = {0}c, then (5g)gec is
Jo-singular and hence, 5g = 2g for all g E G. Thus F_‘ = Ew' In
this case, Ew could be absolutely continuous with respect to m.

We will now Specialize our reSults to the discrete group Z,
the integers. We will first recall some notations (cf. 4.15).

5.16 Notation. let Z be the integers. Let (>_(_k)°:m be

a q-variate WSRF over Z. let Jk = {k}c; Ik = {n: n S k}. Then
k
(ii) 71111.: =6(§fl, n S k) .

oo

Clearly, 2341),): =13 oak“; kg-” 7411‘“ = mJo’x.

In the following theorem we will give a condition under
which Zita ,x =EJO’X.

P

5.17 Theorem. let gm)?” be an W-valued WSRF over Z.

,X

1‘1 1'1 * *
Let F exist a.e. m and F E L1(G ,6 ,m). Then 271.0 s: 7.7!
— - - J
0

75

a S
Proof. By Theorem 5.14, Fy = 5,; Ew = F_. Since

: 741 ,x’ it follows that p But, by Theorem
n

21
In-l’x
5.8, pJ

2 .
10.1 pJoro
= q and, hence, p = q. Then by Theorem 3.19,
’0 10,1
are the spectral distributions of the

0

S a
I. E»Eu‘fi- Inez.

components (2n)fm, (En)fm in the usual.Wold decomposition of

a
(gn)_m with respect to the past. Hence, we get

F = F ; F = F .
-u -y —v -w

Using the same techniques as in the proof of Theorem 4.16, by (I)

, we get = Q.E.D.

m o
thx .00 ,X

5.18 Remark. In Theorem 4.17 we gave analytic conditions

and the fdct that m", ’x 3m“, ,x
0 P
. c:
in terms of the spectral densit under which ¢ = .
y me,X mJO,X m
For the q-variate case, in general, a reasonable analytic condition
is not available. However, in terms of the rank of the process we

make the observation that if pJ 0 = O and pI 1 > 0, then

0’ o’
C
me’x ’1‘ mJopc - Ex. On the other hand, if pI O = 0, then

0’
m4)p,x = 17140,): a 7-7'x'

The rest of this section will be devoted to an extension of
Salehi's work on the interpolation of q-variate WSRF'S indexed by
integers to q-variate WSRF'S over discrete LCAG'S. In this connec-
tion we may add that comments Similar to the ones made in Remark
4.19 regarding the minimality and interpolation of a univariate
WSRF can also be made for the multivariate case. To avoid duplica-
tion we will not repeat these comments and will refer the interested

reader to Remark 4.19.

As we saw in the theory on minimality, some of the results

in the univariate case could only be partially extended to the

76

multivariate case. We will find that the same thing happens in our
theory on interpolation. Whenever necessary, we will make the
assumption that certain matrices have full rank.

We will now recall some old notation and introduce some new

ones needed in the rest of this section.
5.19 Notation. Let (x ) be an Hg-valued WSRF over a
‘_—‘—“—' ‘8 SEC

LCAG G. Let F be the Spectral distribution of (x ) , F' its
- 1866 - # 1)
'

spectral density, {'1‘ the generalized inverse of If, and g = 11' E

Let J = {g0,g1,...,gn} be a fixed set of n+1 elements of G. Then

<1) 19.5%} ”719.3
n

,x
(ii) QJ = {3: Bil) = 2 (gk’k)§(, ék's are arbitrary
k=0

q X q complex-valued matrices; P g = g a.e. m and

#

. * * *
.13: 2 611m 6.111)};

(III) .0“ = {J° + g, g e c}.
5.20 Remark. It is easy to see that E.6 fig x if, and
3

onl if z .1. x for all 6 JC and that = e 52 52 s
y a _ ‘8 g flJ ,X (10:11, ’1n)’
where a is defined by s = x - (x ‘21 ).
-gi _gi _gi _gi J ,x

We now make the following definition which is an extension
of non-minimality for a q-variate WSRF over a discrete LCAG.

5.21 Definition. Let J be as above and (x ) be a
‘8 86C

q-variate WSRF over G, a discrete LCAG. We say that (x8)gEG is

interpolable with respect to J if

Zh,x c;Zlc

J ,x

 

1)

Q. is the orthogonal projection onto the range of F'.

77

or, equivalently,

It is clear that is a subs ce of . t is also
21.1 ,x Pa 71%. 1
obvious that the set £5 is a linear subset of all the matrix-

*
valued polynomials on G . We introduce a matricial inner product

in QB in the following manner:

€«9

_ .# *
(31,22) '4, - 1,31“): (i)_1:2().)m(d).). 21.32 _J

E- G

The proof of the following lemma is straightforward and
thus will be omitted.
5.22 Lemma. With the above notation, Q5 is an inner pro-

duct Space over the ring of q X q matrices with the inner product

*
G

((21.22))F'# = tr f 21(x)1:'#(x)f:(i)m(dx), £1.22 6 QJ
The fact that the inner product Space is finite—dimensional
and, hence, complete will follow from the following important lemma.
This lemma will be used repeatedly in the interpolation of'Rg-valued
WSRF'S over G, a discrete LCAG.
5.23 Lemm§_(Main Lemma II). With the above setting the
finite-dimensional subSpace flU,x and the inner product Space Q6

are isometric; i.e., there exists a linear operator T_ on 35 x
3

onto Q3 such that

(31332) = (1 £1: 1 £27F'#a 51,32 6 21.],X.

78
Proof. let 5 €22J,x° We define the polynomial 22 by
n
(I) P (X) " 2 (8 9X)(E_ax ) °
‘2 k=0 1‘ 'gk
We claim that 22 is an element of QJ. In view of the fact that

the subSpace flJ,x is spanned by {38 ,2? ,...,_R_g }, it suffices
0 1 n

to prove that 25‘: E QJ, O S i S n. For simplicity, 2110.) will
8
i
n
denote P. (I) 8 2 (g ,1) (5‘: ,x ). Since 5‘; E , by the
-x81‘ k-O k ‘31 ‘3k 81. Z;

* *
Isomorphism Theorem (3.7), there exists £1 E L2(G ,6 ,g) such

6

that £81 =I*210‘)§(d>‘)§0° Using a similar proof to that of
G

lmuna 4.24, we get

a
(II) P. = i) _F_‘_' , on the carrier of E

Q, = 0 , on the carrier of Fs

_1 - — .

I I I
Thus L2,. L2,: I i 2, dm
G G
a: I
L511]? e dm
G
*
= f sidp o, .
*" - “‘1
c
,1; 'k * * .
Hence, £1: £1 6 21(6 ,6 ,m). Also, by (II) it follows that

2.9.3.21 a.e. m and thus PiGQJ.

We now define the operator 1 on flJ x into QJ by ,
9

(III) gig—PZ’ EEflJ,x °

Clearly, T is linear and it is not hard to Show that it preserves

the matricial inner product.

79

It remains to show that T is onto QJ. To do that, we show
n
that for any given P E Q , P0,) 8 2 (g ,x)A , there exists a
- J - k=0 k -k

z E 21 such that P .- P . We remark that the function
J,x - -z

2 F_' , on the carrier of Fa

pen
ll

9 , on the carrier of F_‘S
. * '1:
is in _I_._2(G ,6 ,F). Define 5 me by

g=jgomwnx
*

G 0

As in the proof of 4.24, by examining 1 3, we get:

IH
IN
A
7
v

u

n
z (3km) (-sk.i)g(i)z'# my 0.)!!!(‘120
k=0 6"

n
2: (8 00 (-g .x)£().)m(d).)
k=0 k £* k

n
2 (g ’7‘) a P()\) I
k=0 k 5k

where the second equality follows because .29. = P a.e. m. Q.E.D.

5.24 Remark. If F" exists a.e. m, then g=la.e. m
and, hence, the condition 29. = _l: a.e. m is automatically
satisfied. In particular, this is true in the univariate case Since
f has an inverse a.e. m.

The following is the analogue of Kolmogorov's minimality
theorem (cf. Theorem 5.8) for the case when J has n+1 elements.
The proof follows directly from Main lewna II (5.23), and thus
will be omitted.

5.25 Theorem. Let J = {go,gl,...,gn} be a fixed set of

n+1 elements of G, a discrete LCAG. let be an yq-valued

(53’366

80

WSRF over G. Then (58)86G is not interpolable with reSpect to
J if, and only if, QB * {Q}.
The following corollary is immediate.

5.26 Corollary. With the same setting as in the above

theorem, (£3) is interpolable with respect to J if, and only

gEG
if, QJ . {Q}‘
We will need the next two lemmas in the proof of Theorem
5.29. The proof of the first is easy and is omitted.
5.27 Lemma, Let J = {g0,g1,...,gn} and g1 E J. Then

it o if, d if, - - o.
e" a“ ..r, 331% age “swarm”

5.28 LEEE§° Let G be an ordered, discrete LCAG and K
be any finite subset of G. let P[ be a non-zero trigonometric
polynomial of the form .§(1) = EK(g,x)Ag, where each A8 is a
q X q complex-valued matrix. Then rank P = constant a.e. m.
Proof. By examining the minors of ‘P of various orders,
one can show there exists a minor of order r, say Ar’ 1 S r S q,
such that Ar is a non-zero polynomial and all minors of higher
order are identically zero. As in the proof of lemma 4.29, we can
prove that Ar i 0 a.e. m. Therefore, rank 2 = r a.e. m. Q.E.D.
The following theorem is an analogue of Theorem 4.28. At this
stage we have only been able to prove it under a full rank condition.

5.29 Theorem. let (:58) be an Ifl-valued WSRF over G,

BEG
a discrete LCAG. Let J - {g0,gl,...,gn} be a fixed set of n+1
c
elements in G; 4% = {J +Ig, g 6 G}.
(a) If (xg)86G is non-trivial and is 4h-regular, then

(leg)BEG is not interpolable with reSpect to J +ig for all g E G,

81

and I: is absolutely continuous with reSpect to the Haar measure m.
(b) Let G be ordered. If E is absolutely continuous

and there exists a polynomial l: in QJ such that rank _l_’ . q

on a set of positive Haar measure, then (£3)gEG is Jn-regular.
Proof (a). The proof that (258)8EG is not interpolable with

reSpect to J + g for all g E G follows immediately. As in the

proof of Theorem 4.28 (a), we can Show that Jn-regularity implies

a .. ,OSiS, G.
(I) 711x 60181,,8 “86>

In the proof of Main Leanna II (5.23), we saw that

a
Pi§'#, on the carrier of E

if = g (I)§(d)‘)x where g. =-

-gi £* 1 -O 1 Q , on the carrier of is,
n

P = 2 I R x O S iSn. B stationarit it follows

_10.) k-Omk, )(1i’1k), y y

that

(II) 5:81., = j*(g.x)giu)§_(di)x . o s I s n. g e c .
G
Using the same techniques as in Theorem 4.28(a), by (I) and (II) we
*

obtain [8 (G ) = 0 which implies that E is absolutely continuous.
b let 2 . If we can show that z a for all
( ) _ 6 Z73x _ 1 1143

i E {0,1,...,n} and all g E G implies that g = Q, then we will

ha show that “ OS'Sn G= .Bt
V8 n 5032148: 1 :86 ) 711x U

M; = 6 5‘: 0 S i S n G and hence = 0 which
‘pn,x (-gi-|-g’ ’ g E ) 74'9““ {.1

shows that (358)866 is Jn-regular.

By lemma 5.28 and the fact that rank _P = q on a set of

positive Haar measure, we get rank P = q a.e. m. Since 2g = _1:

a.e. m, it follows that rank 3' = q a.e. m and, hence,

82

- *
rank PE' 13 = q a.e. m. Then, by lemma 5.7,

(I) I}. _"12* dm 5‘ 9 -
G

Lat 1 be the element in ZIJ x corresponding to 2 (cf. lemma

5.23). By (I), y is non-zero. Then, similar to the proof of

Lamma 5.6, we can Show that

1)
(II) (_Jg (x) - j*(g.I)g(I)§(d).)so
G
where
1'1 . a
P F_ , on the carrier of E
g = S
Q , on the carrier of _P_ .

Let _z_ = j*‘£.(1)§(d1)2£0 6 mx such that (LESS?) = 9, 0 S 1 S n,
g E G. I? then follows that (£,U_g z) = Q for all g E G. But

(III) 9_ = (EJU x) =I 1 dE(-g,)\)g* for all g E G .
-g 9:
C
BY (11) and (III), we get

(IV) 9 = f*i(I)g*(I)(-g.nm(di) for an e e o .
G

*
Hence, :2 - 0 a.e. m. Since 2 has full rank a.e. m, we conclude

that i=9 a.e. m. Now, since F is absolutely continuous, it

follows that

(V) (e.y=fidII*-jii'i*dm=9-
G G

Hence, 5 = Q. Q.E.D.

 

1) I O
U is the shift rou of
(_g)86G s p

83

Results similar to Corollary 4.30 and Remark 4.31 hold for
the multivariate case, but will not be stated.

The following shows a relation between the concept of 4%-
singularity and the notion of interpolability.

5.30 Remark. Let (x8)gEG be a qdvariate WSRF over G,
a discrete LCAG. Let J = {g0,...,gn} be a fixed subset of G.
Then (£8)86G is 4k-singular if, and only if, for all g E 6,.
em = (9.).

Now, we will give a characterization for an.flg-valued WSRF
over a discrete LCAG which is neither 4k-Singular nor 4k-regular in
terms of its spectral distribution.

5.31 Theorem. let (xg) be a q-variate WSRF over G,

866
a discrete LCAG and E be its Spectral distribution. let
J = {g0,g1,...,gn} be a fixed subset of G and 4n = {JC + g, g e G}.

(a) If QJ 1‘ {0} and IS ,I 0, then {0} 3719“,), Em“.
(b) (i) If mJn’x 3474‘, then QJ-l-g #9 {Q} for all g E G.

(ii) let G be ordered. If {9} 94 721,0 x and there exists a polynomial
n,
.Pin.Qh such that rank 2 = q on a set of positive Haar measure then F§¥QJ
Proof (a). The proof is analogous to the proof of 4.33(a).
. _ . a
(b) (i) If mJn,x #171,“ then (gig)gEG is not "an Singul r

and, hence, by Remark 5.30 and stationarity, ¥ [9} for all g E G.

em
(ii) If Zuh,x i {Q}, then (x3)86G is not 4h-regular.
Hence, by 5.29(b), either F_ is not absolutely continuous or else
for all P E 23 rank _P< q a.e. m. But the latter is not true by
assumption. Hence, F? # Q, Q.E.D.

Our next goal will be to establish the Wold-Cramer concordance

relation with respect to 4% for a q-variate WSRF over a discrete LCAG.

84

5.32 Theorem (Wold-Cramer concordance for 4%). Let

(i) (x3)gEG be a q-variate WSRF over G, a discrete

LCAG, which is ordered; J = {g0,g1,...,gn} and 4% = {JC +Ig, g E G}.

(ii) (wg)gec and (yg)gEG be the components of (x8)86G

as occurred in the Wold decomposition theorem with respect to 4%;
(iii) F, F , and F be the Spectral distributions of
and w
(1‘s)gEG’ (5)366 (1)866

a S
(iv) E_, E_ be the absolutely continuous and singular com-

respectively;

ponents of F_ with reSpect to the Haar measure, as in the Cramer
decomposition theorem;

(v) ‘P be a polynomial in Q3 such that rank 2 = q on
a set of positive Haar measure.

Then

F
-y
Proof. By assumption (v) and the fact that G is ordered,

as in the proof of Theorem 5.29, rank Ff = q a.e. m,

,-1
I}:

90 ‘P,

*
P. dm #‘9, and if E is the element in fl_ x correSponding
’

I z = G E d x
( ) _ j*_()()_( i)_0
G
where
I' . a
_P F , on the carrier of F
9’ , on the carrier of F .

Also since A L for O S i S n it follows that z i .
’ £81 6 ZUkIX , _.€52%%}x

But in the proof of the Wold decomposition theorem, we observed that

85

= .L
my mJn’x and, hence, i 6 my.
By Theorem 5.29(3), Ey is absolutely continuous. Thus,

using the same type of proof as used in Theorem 4.13, we obtain

I = a +
(II) E F F a.e. m

dFS = dFS

-— -w

where F' and F' are the spectral densities of F and F
-y -w -y -w

respectively. Since 5.6329, again imititating the proof of Theorem

4,13, we get

III = E d .

( ) a, f*2<x>_( K>Xo
C

By (I), it follows that

(IV) (M) = Lywydmfiu = f*£(x)§"1(x)§*(i)m(dx) .
G C

By (111), we get

(V) < = ¢< >F (dx>¢*< > = P( >F"1 >F'( >F"1 )P*< > (d >
by f;m_y _ x j;i_ (xwx_ (x_xm x.
G G

Combining (IV) and (V) and rearranging, we have

-1 * - -1 *
I ( F' P - P F' 1F'F' P dm = 0 .
N) j;__ _ __ W_ _) _
G
— * - - *
By (II), E} 2 F' a.e. m and, hence, gig} 1P - P F' 1F'F' 1P 2 0 a.e. m.
‘7 ‘— "" -y" -'
- *
This fact and (VI) imply, by Lamma 5.7, that 2.2} 1P =

1'1 u t'1 * .
PF 2y: g a.e. m. Since rank§=q a.e. m and

rank E} = q a.e. m , we get F; = F' a.e. m. Q.E.D.

5.33 Remark. If e5 = {9). then. by Remark 5-30: <§g>geg

is 4k-singular, and, hence, F_= Ew' In this case, Ew could be

86

absolutely continuous with reSpect to m.

We will now Specialize our results on interpolation of q-
variate WSRF's to processes indexed by the integers. Under suitable
analytic conditions, we will make appropriate comparisons between
the subSpaces 711 and 7.7K in the time domain. First, though,

'0an Jn3x
we will need the following notation.
5.34 Notation. Let Z be the integers. Let (inf; be

q = .
an N -valued WSRF over Z. Let J {k k1,...,kn, k0 < k1 <...< kn}

0’

be a fixed set of n+1 integers and Ik = {j E Z : j S k}. Let
(1) m

J +k,x

.. = . s k .

=6(}_<_j,j¥ki+k,05i$n);

m
C H m =
18" -oo Jc+k,x 'an’x

theorems, we will examine conditions under which

. In the following

CD
Ob ' 1 =
Vious y, @Jp’x 10 ml

721 '14
.0an Jn ,X

C
or 4 .
mefl‘ 22141.1(
5.35 Theorem. Let (info be a q-variate WSRF over Z.
—— -m
Suppose there exists a polynomial g in QJ such that rank I: = q

on a set of positive Haar measure. Then 711.0 x = 711.0 x'
p, n,

S

a
Proof. By Theorem 5.32, we see that F = F ; F = F
-y _' “W "
Let Fu and Ev be the spectral distributions of the components
co co ,
(En)-m and (zn)_m given by the usual Wold decomposition theorem
with respect to the family {Ik]fm. It can be Shown without much
difficulty that F s F and, hence, rank F' 5 rank F' a.e. m.
y -u -y -u
Since rank 2 = q a.e. m and F = Fa, it follows that F' has
-y -' -u
full rank a.e. m. Therefore, by Lemma 5.1, [19], we have pl

0
Hence, by Theorem 3.17, Eu = [a ; Ev = [8. Now, using a similar

.1=q°

87

proof to the one used in Theorem 4.16, we get 74.0 ,x = 721 .
P

5.36 Theorem. Let (En)fm be as above. Let

* * <;

I = = 0
log det z E£1(G ,6 ,m) and a, {9- Then 7.72., ,X my“,x 71g,

Proof. If QJ = {9}, then, by Remark 5.30, (in); is
J -singular,and, hence, 7.7.! =77) . It is well known ([32], p. 145)
n .J ,x ~x

*

that if log det F' 6 L1(G ,B ,m), then (3%)?” is non-deterministic

and, thus, 171%,), 9 711," Q.E.D.

5.37 Remark. If we ass me = 0, then =
__ U 910,1 7.7.1", ,X flax

and, hence, J ,x =mx.
n

This concludes our discussion on the problems of interpola-
tion with reSpect to 4% = [JC + g, g E G}, where J is a fixed
set of n+1 elements of G, a discrete LCAG. We will devote the
rest of §5 to interpolation theory with reSpect to 4;, the family
of complements of finite sets of C. First, though, we will recall

some notation.

5.38 Notation. Let (x ) be an flg—valued WSRF over G,
—"‘""'_ '8 SEC
a discrete LCAG. Let Ff be the Spectral density of (5g)g€G’

,#

F'# the generalized inverse of Ed, and g_= E. F}. Let J be

any finite subset of G. Then, Similar to the notation in 5.19,

we set
(i) zsz=mic flax;
’ J ,x

ii = P: P = Z A A 'S are arbitrar x
< > 2, {_ _m gangs)? _g # y q q
* * *
complex-valued matrices; 2g = g a.e. m and g P_" g E L1(G ,B ,m)}.
We are now able to give the following definition of inter-

polability.

88

be a q-variate WSRF over G,

 

5.39 Def°niti n. Let x
1 ° (1)869

is inter olable if x
366 p ( g)gEG

is interpolable with respect to every finite set of elements of G.

a discrete LCAG. We say that (x8)

The following remark follows immediately from Corollary 5.26.
5.40 Remark. Let x be as in 5.39. Then (x )
-—-—-' (-g>gEG ‘s gEG
is interpolable if, and only if, Q5 = {Q} for any finite subset
J of G.
The following establiShes a relationship between the concept

of non-interpolability and that of 4Lfregu1arity.

5.41 Theorem. Let (5g) be a q-variate WSRF over G,

86G

a discrete LCAG, and E_ be the Spectral distribution of (Eg)g€G°

is non-trivial and is.J -regular, then
on

(a) If (5g)gec
there exists a finite Subset J in G such that Q5 ¢ {93 and
F, is absolutely continuous with respect to the Haar measure m;

(b) Let G be ordered. If F. is absolutely continuous
and for some finite set J in G there exists a polynomial

2:6 Q5 Such that the rank 2 = q on a set of positive Haar measure,

is 42 -regu lar.

the
n (£8)8€G 00

Proof (a). The proof that there exists a finite set J
in G such that Q5 # {93 is trivial. Using similar techniques
as in the proofs of Theorem 4.44(a) and Theorem 5.29(a), we can
easily Show that F. is absolutely continuous.

(b) Trivially,

(I) 7.71,,me .an={JC+g.gec1.

..anJ’

Therefore, our assumptions satisfy those of Theorem 5.29(b) and,

th S,
u (§g)g€G

is 4B-regular. Hence, in view of (I), (Eg)g€G is

89

.0 -regular. Q.E.D.
on

In the following remark, we will state a characterization
of Joe-Singularity for an Yq-valued WSRF over a discrete LCAG.

5,42 Remark. Let be as in 5.40. Then x
__ (:18) (_g)gEG

EEG
is Jab-singularif, and only if, QJ = {Q} for any finite subset
J of G.

Next, we will give a characterization for a q-variate WSRF
over a discrete LCAG which is neither 4200-3 ingular nor Jim-regular.
Its proof is Similar to that of Theorem 5.31 and thus is not given.

5.43 Theorem. Let (53) be a q-variate WSRF over G,

gEG
a discrete LCAG and F be its Spectral distribution.
(a) If QJ 1‘ {Q} for some finite subset J of G and
S C C
3‘ .
z #9. then {9} my“ m,

(b) (i) If ,x #mx, then QJ #4 {Q} for some finite

74900
subset J of G.
(ii) Let G be ordered. If 9 #mJ x fland there exists a poly-
’
:n

nomial _P_ in QJ for some finite subset J of G such that

3
rank P = q on a set of positive Haar measure, then F_ 5‘ Q.

We will now State the Wold-Cramer concordance theorem for
the multivariate case with reSpect to the family Ja' Its proof
follows from the proof of Theorem 5.32 .in the same way that the proof
of Theorem 4.49 follows from the proof of Theorem 4.35, and, hence,
is omitted.

5.44 Theorem (Wold-Cramer concordance for .9”). Let

(1) (353)ch be a q-variate WSRF over a discrete LCAG

G, which is ordered; 42‘” = family of complements of finite sets of G;

90

ii w and be the com onents of
( ) (_g)86G (xg)gEG p
(lg-g)86G as occurred in the Wold decomposition theorem with reSpect
t0 .9;
G3
(iii) Ed Ey’ and Ew the Spectral distributions of
x and w res ctivel ;
(—g)gEG’ (Lg)g€G’ (1)gEG P9 y
(iv) Ff, E? the absolutely continuous and singular com-
ponents of E. with reSpect to m, as in the Cramer decomposition
theorem;
(v) For some finite subset J, 65 contain a polynomial

‘P such that rank 2 = q on a set of positive Haar measure.

Then

=21; =53-

F F
-y -w

5.45 Remark. If Q5 = {93 for every finite subset J of

G, then, by Remark 5.41, is 4L:singu1ar and, hence, E.= Ew'

x
(‘8)86G
In this case, Ew may be absolutely continuous with reSpect to m.

In Specializing our results for 4;, to the case when G = Z,
the integers, we will simply State the results comparing 2%? x and

9
2U? x’ since their proofs follow closely the corresponding proofs
’

for 4%.

5.46 Theorem. Let (5n)0° be a q-variate WSRF over Z.

—— -oo
If there exists a polynomial 2’6 Q3 for some finite set J of integers
such that rank P = a.e. m then = .
._ q , me’x me ,x

5.47 Theorem. Let (511)”0° be as in 5.45. Let

* *
log det I: E L1(G ,B ,m). Suppose QJ = {Q} for every finite set

f’t . h 9 = .
J o in egers Ten me,x me,x %

5.48 Remark. If 91

91

O,

l

= 0, then

711

JP

,X = EJGSX

Z; .

6. SOME EXAMPLES AND FURTHER REMARKS ON FINITE
AND INFINITE DIMENSIONAL STATIONARY RANDOM FIELDS

AS we pointed our earlier, this section will be devoted to
the construction of some examples and to the discussion of some open
problems on q-variate WSRF'S over LCAG'S. We will also remark briefly
on the problems of minimality and interpolation of infinite dimen-
sional WSRF's over LCAG'S.

Our discussions in the preceding sections have been mainly
on processes over discrete LCAG'S. Concrete examples of such groups
are as follows:

6.1 Examples. (i) G = Z, the set of all integers;

(ii) G = R, the set of all real numbefis;

(iii) G R“, n-dimensional Euclidean Space;

(iv) G 2“, the set of all lattice points in n-dimen-
sional Euclidean Spaces.

The following discrete LCAG Should be of interest in the
study of WSRF'S. AS far as we knOW, in connection with stochastic
processes, this group has not been considered.

6.2 Example. Let T denote the unit circle. Let T0°
stand for the infinite (countable) Cartesian product of T with
itself. Since T is compact, Tan is also compact under the usual

product topology. Let 2: denOta the set of all infinite (countable)

sequences of integers only finitely many of which are different from

92

93

zero. It is clear that 2: is a discrete LCAG.

By Theorem 2.2.3, [26], it follows that the dual of z:
is Tm. Since 2: is a discrete LCAG and Tan is compact, the usual
Bochner theorem, 3.4, holds. Hence, our results in §4 and §5 on
minimality and interpolation of WSRF'S indexed by elements of z:
are valid.

The following is a counterexample to L. Bruckner's claim,
Theorem 4.1 of [I], that a process must be either 4b-Singular or
4b-regular. Because of Theorem 4.11, it suffices to find a process
whose spectral distribution F has the properties that
l/f E L1(G*JB*,m) and FS # 0. The example is as follows:

6.3 Example. Let G = Z, the integers. Then 6* = [0,2n].
Define dF in the following manner:

(i) f = 1 on [0,2n];

(ii) p be the singleton measure with mass 1 at n;

(iii) dF = f dx + du.

Clearly, FS # o and j‘ 1/f(x)dx = 211 < ...
*
G
Any WSRF over the integers with Spectral distribution F

will constitute a counterexample to L. Bruckner's claim.
Our next example will be to construct a process over the
* *
integers whose spectral density f is such that log f E L1(G 46 ,m),
. * 2 * * _
but, for any polynomial P on G , ‘P‘ /f i L1(G ,6 ,m). This
example will Show, among other things, that the assumptions in

Theorems 4.17, 4.39, and 4.52 are not vacuous; i.e., there do

c:
exist rocesses such that # = = = .
p ”Ups ”has ”Cans ”90.x ”S.

94

on

6.4 Example. Let G = Z, the integers. let (xn) be

-co
any WSRF over Z, whose Spectral distribution is absolutely con-
1M).

tinuous and whose Spectral density is given by f(x) - e

= _ , * * . 1m.= °° 1
Then log f UV), 6 L1(G ,8 ,m). Since e “:0 W,

‘bysimple manipulations one can Show that \PIZ/f é L1(G*,5r,m)
for any non-zero polynomial P.

Our last example will Show that if G is not an ordered
group, then it is possible to construct a non-zero polynomial P
on 6* such that P = 0 on a set of positive Haar measure, as
Remark 4.31 claims.

6.5 Example. Let G = {0,1} and its binary operation

"+" be defined in the following way:
0+O=0;0+1=1+0=1;1+1=0o

It is easy to see that G cannot be ordered compatible with its

*
structure and that G contains only two elements, 1 and K2:

1

defined in the following manner:
i1<0> = 1; 11(1) = 1; i2(0> = 1; i2<1> = -1 .

Define P by P(x) = (0,x) + (1,k)- Then P(x1) = A1(0) + x1(l) = 2;
P(x2) = x2(0) + x2(1) = 0. Note that P = 0 -on the set {x2}
where m({x2}) = 1/2 and P # 0 on the set {A1} where
m({x1}) = 1/2.
Next, we will mention some open problems that arose from our

study of q-variate WSRF'S over LCAG'S.

95

6.6 Open problems.

)Q be a q-variate WSRF over 2, the integers.

(I) Let (an _m

Wiener and Masani, [32], showed that log det E} E L1([0,2n]45*,1eb.)
if, and only if, the rank of the process with respect to the past is
full. Later, Wiener and Masani, [34], extended this result to cover
bivariate processes, not necessarily of full rank. The most inter-
esting result, in connection with this area, is due to Matveev, [16].
He gave a necessary and Sufficient condition in terms of the spectral
density of the process for the process to have any rank between zero
and q.

In Theorem 5.8, a characterization for full rank of a WSRF
over a discrete LCAG with respect to the "past & future" was given
in terms of the Spectral density. It would be very interesting to
extend this result, in the same spirit that Natveev extended Wiener
and Masani's result, and obtain a characterization in terms of the
spectral density for the rank of a WSRF over a LCAG with reSpect to
the "past & future" to assume any value between zero and q.

(II) For the integers, Robertson (Theorem 3.20) gave a
complete characterization for concordance between the Wold decomposi-
tion with respect to the past and the Cramer decompasition in terms
of the rank of the spectral density. For a.WSRF over any discrete
LCAG, we believe a characterization for concordance between the
Wold decomposition with respect to the "past & future" and the Cramer
decomposition in terms of the Spectral density is possible. In fact,
Theorem 5.14 tells us that our conjecture is true when we assume
that the rank of the Spectral density is full a.e. m and its inverse

* *
is in L1(G 45 ,m). Similarly, in general, the concordance between

96

the Wold decomposition with reSpect to 4% and 4; and the Cramer
decomposition remain open.

(III) In the univariate case, for a non-trivia1.WSRF over
a discrete LCAG, we saw, by Corollary 4.9, that 4b-regularity implies
that 1/f E L1(G*43*,m). In the multivariate case, for a non-trivial
WSRF over a discrete LCAG, Theorem 5.9(a) implies that E. is
absolutely continuous. If the rank (with reSpect to the "past &
future") of the WSRF is full, then, by Theorem 5.8, Ef-1 exists
a.e. m and is in 'L1(G*,Br,m). In general, when the rank is not
full, it seems reasonable to assume that perhaps a similar implica-

tion holds; i.e., if

F .#

(xg)86G is 4b-regular, then maybe
6 L1(G* ,B* ,m) .

(IV) In the statements of some of our results in both the
univariate and multivariate cases involving 4%- and 4L-regu1arity;
e.g., Theorem 4.28(b), Theorem 4.35, Theorem 5.29(b), Theorem 5.32,
and Theorem 5.44, we assumed that the group G was endowed with an
order relation compatible with its structure. We feel that one
Should be able to dispense with this aSSumption to carry out the work.

(V) In several of our theorems in the multivariate case,
such as 5.9(b), 5.17 and 5.35, we have assumed that certain matrices
have full rank. It may be possible to obtain these results under
weaker assumptions.

We now direct our attention to a short discussion on in-
finite dimensional stationary random fields.

6.7 Remark. Based on the isomorphism, Theorem 3.7, between
, a q-variate (q finite)

6G

WSRF over a LCAG G, we were able to obtain analytic characterizations

the time and Spectral domain of (5g)g

97

of the notions of minimality and interpolation for (Eg)g€G' This

work enabled us to establish various interesting results concerning

the time and Spectral domain of (£8) as presented in sections

gEG’
4 and 5.

Recently, V. Mandrekar and H. Salehi [9] have studied the
structure of the Space of square-integrable operator-valued functions
with respect to a non-negative operator-valued measure. They
established [11] an isomorphism theoremibetween the time and spectral
domains of a WSRF over a LCAG. Based on this, they settled some
questions on subordination of an infinite-dimensional WSRF with reSpect
to another infinite-dimensional.WSRF [11]. They also used this
isomorphism in connection with infinite-dimensional linear differential
Systems drived by white noise [10].

The same way that LQ(G*,6r,§) was used to study various
results on minimality and interpolation of a q-variate WSRF over a
discrete LCAG, one can use the Space of square-integrable operator-
valued functions with respect to a non-negative operator-valued
measure (cf. [9]) to Study the problems of minimality and inter-
polation for infinite-dimensional WSRF'S over a discrete LCAG. These
questions are under study by us and the results will be announced

elsewhere.

 

REFERENCES

[1]
[2]

[3]
[4]

[5]
[6]
[7]

[8]

[9]

[10]

[11]

[12]

REFERENCES

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Cramer, H., On the theory of stationary random processes,
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Doob, J.L., Stochastic Processes, Wiley, New York, 1953.

Grenander, U. and SzegB, G., Toeplitz Forms and Their
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Halmos, P.R., Measure Theory, D. Van Nostrand Company, Inc.,
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Hille, E. and Phillips, R., Functional Analysis and Semigroups,
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Khintchine, A.Y., Korrelationstheorie der stationare Stochastischen
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Kolmogorov, A.N., Stationary sequences in Hilbert Space, Bull.
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Mandrekar, V. and Salehi, H., The square-integrability of
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Mandrekar, V. and Salehi, H., Operator-valued wide-sense Markov
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Mandrekar, V. and Salehi, H., Subordination of infinite-dimen-
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[13] ‘Masani, P., The prediction theory of multivariate stochastic
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