— 'fi—“w -
1 I

 

MULTIPLICITY AND REPRESENTATION IHEORY
OF PURELY NONcDETERMINISTIC STQCHASTIC

PROCESSES AND ITS APPLICATIONS

Thests for Ike Deg?“ of DI». D.
MICHIGAN STATE UNIVERSITY
Vidyadhar Shantaram Mandrekar
1964

   
  

THESIS

LIBRARY L

Michigan State
University

    
   

This is to certify that the

thesis entitled

MULTIPLICITY AND REPRESENTATION THEORY
OF PURELY NON-DETERMINISTIC STOCHASTIC

PROCESSES AND ITS APPLICATIONS
presented by

Vidyadhar Shantaram Mandrekar

has been accepted towards fulfillment
of the requirements for

Ph .1). degree in Statistics

Uj/Z/AI‘A/ “7:27; /
C

‘ in: L}:.,
Major firofessor/ ,

Date November 30, 1964

0-169

,1—

ABSTRACT

MULTIPLICITY AND REPRESENTATION THEORY
OF PURELY NON-DETERMINISTIC STOCHASTIC
PROCESSES AND ITS APPLICATIONS

by Vidyadhar S. Mandrekar

The study of the representation arises in the investigation

of linear prediction problem for multivariate stochastic
processes. Using an extension of the method of Banner from

the point of view of the multiplicity theory (See A. I. Plessner
and V. A. Rohlin, Uspehi Mat Nauk l9h6; G. Kallinapur and

V. Mandrekar, Tech. Report 49, University of Minn.), representa-
tions for multi-dimensional (including infinite-dimensional)
processes are obtained. The concept of multiplicity arising
here is shown to coincide with the rank introduced by Gladyshev
for continuous parameter multivariate (finite-dimensional)
processes. In Chapter II, explicit form of the kernel is
obtained for continuous parameter Markov and N-ple Markov

processes.

MULTIPLICITY AND REPRESENTATION THEORY OF PURELY
NON-DETERMINISTIC STOCHASTIC PROCESSES
AND ITS APPLICATIONS

BY

Vidyadhar Shantaram.Mandrekar

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Statistics

196M

Again, the pursit of knowledge has its
own pleasure, distincc from the pleasures of
knowledge, as it is distinct from that of
consciously possessing it. This will be evident
at once if we consider what a vacuity and
depression of mind sometimes comes upon us on
the termination of an inquiry however success-
fully terminated, compared with the interest
and spirit with which we carried it on. The
pleasure of search like that of a hunt lies in
the searching and ends at the point at which
the pleasure of certitude begins.

John Henry Cardinal Newman,
A Grammar of Assent, 1870.

ii

ACKNOWLEDGMENTS

I am deeply indebted to Professor G. Kallianpur who
introduced me to the field of representation theory and whose
valuable suggestions have gone a long way in shaping this work.
I am also grateful to him for the guidance I received from him
during my entire Doctoral degree program. Further, I would like
to express my thanks to Miss Kathy Smith and Mrs. Judy Sjodin
for doing an excellent job of typing the manuscript. Thanks
are also due to the U.S. Army Research Office'(Durham) and to

Michigan State University for partial support during the prepar-

ation of this work.

iii

ChApter

II

TABLE OF CONTENTS

ACHRNWIEDGEMENTS

JINTRODUCTION

MULTIPLICITY AND REPRESENTATION THEORY OF PURELY
NON-DETERMINISTIC STOCHASTIC PROCESSES

1.
2.
3.
h.
5.
6.
7.
8.
9.

Second Order Processes on ¢.

Representations of stochastic processes on ¢.
Canonical and proper canonical representations.
Discrete parameter processes.

Continuous parameter weakly stationary processes.
Multiplicity as a generalization of rank.
Preliminaries.

Stochastic Pettis integrals.

Representation theorems for purely non-deterministic
Hilbert-space-valued processes.

APPLICATIONS TO WIDE_SENSE FINITE DIMENSIONAL N-PLE MARKOV

PROCESSES.

1. Preliminaries and notation.

2. Characterizations of the wide-sense Markov processes.

3.’ Vector-valued stochastic integrals.

h. Purely non-deterministic wide-sense Markov processes.

5. ”An analytical characterization of a proper canonical
representation. 3

6. Finite-dimensional wide-sense N-ple.Markov processes.

7. Stationary wide-sense N-ple Markov processes.

REFERENCES

Page

10
16.
19
35
#1

MB

52
5h

59
61

6h
66

75
76

Introduction: The main object of this thesis is to study the multiplicity

 

theory of a wide class of purely non-deterministic weakly stationary
processes and to show how this theory provides a natural means of obtaining
representations of continuous parameter processes that are extensions of
the well known result due to K. Karhunen [1.10]. Karhunen obtained his
representation of purely non-deterministic weakly stationary (univariate)
processes using spectral methods. Our work can be described as a unified
time domain analysis that applies equally to finite dimensional and certain
class of infinite dimensional stationary processes. The earliest time
domain analysis of a (univariate) continuous parameter weakly stationary
process was made by 0. Hanner in giving an alternative derivation of
Karhunen's result [1.6]. More recently, in the light of the extensive
deve10pment of multidimensional stationary processes it has appeared
desirable to separate time domain studies from the Spectral and consequently
interest in the former has revived. As an example we mention the paper

of P. Masani and J. Robertson [1.11] whose approach makes essential use of
the Cayley transform associated with the unitary group of the process.

The extension of this method to finite-dimensional stationary processes

has been carried out by J. Robertson in his thesis [1.14]. The earlier
work of E. G. Gladyshev [5] also belongs to the same order of ideas.
Hanner's paper, nevertheIess, has remained an isolated piece of work and
his method has apparently given the impression of being §d_hgg. In reality,
however, as shown by G. Kallianpur and the author [1.9], Hanner's work

is intimately related to multiplicity arguments. Thus the generalization
of Hanner's approach to multidimensional (including infinite dimensional)
processes is to be sought in the development of the multiplicity theory of

the process, i.e., in the study of the self-adjoint Operator A of the

process (see Section 1.2) and its spectral types. This is one of the
central problems studied here and its discussion is presented in Sections
A, 5 and 6 of Chapter I.
In recent years a theory of representation of purely non-deterministic
processes has been introduced by H. Cramer and also by T. Hida ([1.l],
[1.2], [1.3], [I.7]). In Sections 1.2 and 1.3, following the technique
of the latter author, we obtain an extension of the basic theorem of his
paper [1.7] to the processes considered by us. Our purpose in doing so is
to compare the representation of the Hida-Cramer theory (Theorem 1.2.2)
with the result of Section 1.5 which is independent of Sections 2 and 3.
The extension of Hanner's method leads to a definition of multiplicity
which is seen to be identical with the concept of multiplicity introduced
by Hida. Section 1.6 brings to light the natural role of multiplicity as
a generalization of the rank of a stationary finite dimensional process.
In the concluding sections 7, 8 and 9 of Chapter I we consider in greater
detail Hilbert—space-valued processes. Strengthened versions (involving
random Hilbert-space-valued integrals of the representation theorems of
Section 1.2 and 1.5 are stated in Section 1.9. The material in Chapter I
is the joint work of the author with Professor G. Kallianpur [see G. Kallianpur
and V. Mandrekar, "Multiplicity and representation theory of purely non-
deterministic stochastic processes," Tech. Report 21, University of Minnesota].
As an application of the theory deve10ped in Chapter I we study in detail
the representation of vector-valued wide-sense Markov and N-ple Markov
processes. This part of our work is presented in Chapter 11 and can be
regarded as a generalization of T. Hida's work on univariate processes of
multiplicity one. As a consequence of the representation of the wide-sense

Markov processes (Theorem 11.2.1) we derive a more precise form of J. L. Doob's

well known characterization of continuous parameter multivariate stationary
Gaussian Markov processes [11.2]. Continuous parameter q—dimensional wide-
sense N-ple Markov processes are defined and their representations are
studied as an application of Sections 1.2 and 1.3. The kernel of the rep-
resentation of such a process is a matrix analogue of the Goursat kernel of
order N. The last section of Chapter II (Section 7) discusses the question

of determining this kernel in the stationary case.

CHAPTER I. MULTIPLICITY AND REPRESENTATION THEORY OF PURELY NON-DETERMINISTIC
STOCHASTIC PROCESSES

1. Second order processes 22_C, We consider stochastic processes of the

 

following kind.
Let C be a Hausdorff space satisfying the second countability axiom but
otherwise arbitrary. We shall say that fit (-w < t < m) is a stochastic

on C if for each m in C, §t(m) is a complex-valued random variable with mean

zero and allxt(m)l2 finite. The process {§t] (-w < t < w) on C is called

a

weakly stationary (or briefly, stationary) if for all @:W in C and arbitrary

real numbers 3, t and T we have

aet+T<cpmg+T<w1 = gate) 5ng

The covariance function. é [xt(m) ESZWII of the process depends on t-S, m and W .

a

The definition of a discrete parameter process {En} is similarly given. It

should be noted that the stationarity considered here is a temporal one and does
not involve C . Nevertheless, it is sufficiently general and useful for our
purpose since it includes as Special cases many stationary random processes of
practical interest. For instance, if C is a q-dimensional euclidean (or

unitary) space and xt(m) is linear with respect to m for each t, then the 5%-

process can be regarded as a q-vector stationary process (see [15]*); if C

is an infinite dimensional locally convex linear Space and §t(m) is again
supposed linear in m (with probability one), then fit is a weak stochastic

process on C. On the other hand, stationary processes 5t as defined above

include those that are not linear in m (indeed C itself need not be a linear

 

* O I
The number here refers to the number in references to thlS chapter. References
for each chapter are given separately.

Space). Such processes can serve as useful models for certain problems in
meteorology (e.g., see [8]).
Associated with thS gt process (not assumed to be stationary) are the
following spaces:
(a) the (Hilbert) space of the process HCx), defined to beg: [§t(m),
t e T, m e C], the subSpace of L2(J}, P) generated by the family of
random variables §t(m) as t and m vary respectively over T and C;

(b) the subspace H(x3 t) of H(§) given by ch; t) =C§i§t(m), T é t, and m e C]

for every real t. “
we say that 5t on C is purely non-deterministic if H(x5-w) , the
intersection of the subspaces H(x; t) for all t e T is trivial.

The process 5% is said to be deterministic if for each t H(§; t) a H(§3-w).

THE HIDA-CRAMER THEORY.

2. Representations of stochastic processes on Q .

Although our main interest will be in the study of continuous para-
meter weakly stationary processes we begin by considering representations
of arbitrary second order purely non-deterministic processes 3%(o) on C.
It can be easily seen that the results stated in this section contain as
special cases those of H. Cramér [2] and of T. Hida [7] (if Gaussian assump-
tions are made). They will, however, be stated without proof since they are
proved by following essentially the method of the latter author. Our only
reason for including them here is for the purpose of relating the repre-
sentation and the definition of multiplicity given in this section with
similar concepts for stationary processes obtained in Sections 5 and 6.
For the sake of completeness we begin with the following "Wold decomposition"
of 3% .

Ergposition 2‘1; If {zt’ t e T} is a stochastic process on C ,
then

5%(C) : Eél)(C) + z£2)(C) for each C e C where

(i) {zél)] is a deterministic and (xé2)} , a purely non-deterministic

process on C 3 and
(ii) H(z]l)) is orthogonal to H(x‘2)) .
Observe that the topological assumptions concerning C in no way
enter into the proof of this result.
Writing J 2 T ;¢C , a 2 (t, e) , B 2 (s, C) (s,B C J) define
Ms, B) 2? EARN?) my Then, clearly, K is a covariance function
on J x J . Let us denote by H(K) , the reproducing kernel Hilbert-space

of functions defined on J whose reproducing kernel is K. Let

_ 6 _

], i.e. the subspace of H(K) generated

5E
:72
‘0
E:
||
01
,_._,
:7:
p
v
\0
Q
l
c_.

c t
by [K(-, a) ,i c e Jt} where Jt = { (u, o) , u g_t and o e ¢>J . It
is well known that there exists an isometry, which we denote by V , from
H(K) to H(§) taking functions K(-, a) into the random variables gt(o)
‘ and such that VH(K; t) = H(;; t)
The following assumptions (A) will be basic for our purpose:
(A.l) The space H(;) is separable;
(A.2) H(gc_; -co)= {0]
Condition (A.2) is equivalent to the process { 3t} being purely non-determin-

istic, while the following lemma gives sufficient conditions on the tasks gt(o)

for (A.l) to hold.

Lemmaggfl. Suppose that for eacht;(-d)< t < qfl
(i) ;t(o) is continuous in quadratic mean relative to the topology of C ,
and
(ii) the random variables zt—O(¢) and gt+0(m) exist (in quadratic mean)
for each C e C .
Then H(;) is separable.
This result is a generalization of a lemma due to Cramer [2] and takes
as its starting point the fact, proved there, that for each m , the set

of all discontinuity points of the one-dimensional process { zt(o) , t e T}

is at most denumerable.

Proof. It suffices to prove that there exists a countable dense set H0
in {§t(o) , t e T , o a C] . Let CO = {pk} be a countable, everywhere
dense set in C . The set D; of discontinuities of the one-dimensional

Process is at most denumerable. We shall Show that HO 2

rt(vk)

{§u(¢k) Ck a CO , Ilefig IHc’ or u rational} is a dense subset of

_ 7 _
{ ;t(o) t 6 T , C e C] . Since HO has at most denumerable elements,
the proof of the lemma will be complete once we establish the preceding
assertion. For T and o fixed, consider an element ;T(o) and let
8 'be an arbitrary positive number. By (i), there exists a Ck_e CO
such that Sigtkp) - _x_T(<pk )I 2 < 8/2 . If r is a discontinuity point
of the one-dimensional process (§t(¢k )} (t e T), then Since 3%(ok )eaHO,
the proof will be complete. On the other hand, if T is not a discontinuity
point of [zt(¢k )} then there exists a rational number r such that
8.370% ) - £1]qu )I2 < 8/2 . This implies that 8|;T(cp) - 3T(cpk )I2< 28,

and since 3&(ok ) 6 HO the proof is complete.

It might be remarked in passing that if 3(t) = [xl(t), ..., xq(t)]
is a q—dimensional process such that the random variables xi(t-O) ,
xi(t+0) exist for i = l, ..., q , the conditions of Lemma 2.1 are ful-
filled if we take C to be q-dimensional Euclidean Space and define ;t(o) =
.%1 xi(t) oi , o being the vector (o1, ..., mq) . In other words,
LSmma l of [2] is a special case of Lemma 2.1. In view of the isometry
, V between H(K) and H(;) , the assumptions (A) are equivalent to cor-
responding assumptions concerning the spaces H(K) and H(K; -afi . Let
us introduce the spaces H*(K; t) = {0% H(K; t+-r:-L1-) . We then have
H*(K; -oo)= {O I and H(K) = H*(K; 00;]. the smallest subspace containing
all the H*(K; t) .

The Spaces H*(;5 t) are similarly introduced. Let E(t) denote
the projection operator from H(K) onto H*(K; t) and E(t) the pro-
jection from H(g) onto H*(;5 t) . It then follows easily that the
families {E(t) , -a3< t < co} and {E(t),'ana< t <1») are right continuous

resolutions of the identity in the respective Hilbert spaces H(K) and H(§).

- g -
The two results which follow are proved.as in [7]. We omit the proof,
which is essentially based on the Hellinger-Hahn decomposition of the self-
adjoint operators E and A defined respectively on H(K) and H(;) by
the resolutions of the identity introduced above. Observe that while the
parameter set T of the process is always either the real line or the set
of all integers, the resolution of the identity {E(t)} determined by the

process is defined for all real t .

Theorem 2.1. Let assumptions (A) be satisfied. Then each element K(°,<2)

(a in J) of H(K) has the following representation
M M
J

o t
K(°, a) = 2 Gn(c, u) dE(u) f(n) + Z 2 a.£(a) g.L
n=1 -oo tjgt 71:1 J J.
where the symbols introduced have the following meaning:

(a) If“I

prOperties:

I is a sequence of elements in H(K) with the following

(i) The inner product (E(A )f(n), E(A2)f(m)) = 0 whenever Al and A2

1

are disjoint intervals or m f n ;

(ii) For each n , G (0, °) e L (p ) where p (n) = \|E(A)f(n)n 2 ,
M n 2 n n
o
2 f G (a u)|2 d (u) < a) and >> >> etc
:1 | n , pn pl p2 ... .

(b) For each j = l, 2, ... the sequence {83L} (L =1, ..., Mj)
are the eigenvectors of the self-adjoint Operator If corresponding to the

eigenvalue t and such that

J
(”Mi ()2 2
2 2 a a | ||g ll < a) .
J21 :1 l J JC

The elements [83%] further, form a complete orthonormal system in the

subspace [E(tj) - E(tJ-O)] H(K) with

(gJL, gim) = o if i 2 j .

For 0 = (t. a) writing Fn(¢; t. u) = Gn(a. u) and bj,(C: t) =

ajt(a) we obtain the following representation for the process ' 5t on C .

'I.

HpTheoiem 2.2. If conditions (A) hold we have the following representation

for Lt . For each t and C , with probability one

( ) ( ) M0 fil't ( ) ( ) Mj ( )
2.1 x C = 2 I‘ C; t, u d z u -+ 2 E b C; t E ,
% n21 'CD n n Lj<t ('21 if J;

where

(a) zn(u) (-d>< u < ad for each n , is an orthogonal random function

with the further property that g[zm(u) 4n(v)]= O for m # n and

éi|zn(A)[2 = pn(A) . Further, the functions Pn and pn satisfy the

conditions stated in the preceding theorem;

(b) The random variables E. (.L= l, ..., M and j = l, 2, ...)

Jo J
are mutually orthogonal with

M
on J
2 2 2 2
2 2 b. - t f' it. h o = .
1215:1013’ I JW’ )' m 9’ were 3: “hi

Definition. The cardinal number M a max [M0, sup R5] is called the mul-
J

 

tiplicity of the stochastic process 3t on C .
It is to be noted that M can be infinite, in which case of course
M is aleph null. The corresponding series that occur in our work are then
to be treated as infinite series.
' If T is the set of integers it is easy to see that Mo is neces-

sarily zero and t = j .

J

_10_
3. Canonical and proper canonical representations.
The representation obtained in Theorem 2.2 has the following property.

For 3 < t ,
M M

o J
(3.1) E()x =2 st.)d()+22b(<p; t”.
S >115 “(w u znu t<s~=l if 5’"

J
A representation satisfying (3.1) will be called canonical. From the
form of (2.1), it follows that H(x )Cgil-Kg; t)u HG; t)] where
t) = g 9:38, 9.: l, 2, ..., Mj’ tJ g t] . For applications of the

theory, however, it is more useful to consider canonical representations for

which,

(3.2) (:6;th t)U H(§_; t)] = H(_Jg; t) for all t

Following Hida, we refer to a representation with property (3.2) as proper
canonical. In [7], Hida was concerned with prOper canonical representations
of multiplicity one. In order to be able to discuss the multiplicity theory
of the more general processes considered by us it is necessary to establish
the existence of a proper canonical representation of arbitrary multiplicity
equivalent to the one given by Theorem 2.2. This we do in Theorem 3.1.

For the representation of Theorem 2.2 define the processes Bn(u) as

follows:
(i) If both Mo and Sup MJ are infinite, then
.1
Bn(u) = zn(u) + 2 Ejn for n = l, 2, ... ad inf .
tjgu

let

(ii) If M is finite and M S,Sup M ,
o o J 3

B (u) = z (u) + 2 {in for n = l, 2, ... MO

n t gu

J
n for M0 < n g_83p MJ .

_11.—

(iii) In the remaining cases define

Bn(u) z (u) + 2 E

n (n = l, 2, ..., Sup M3)

jn
t3.<.u J

= zn(u) Sup MJ g.n g_MO .
J .
With the above notation we rewrite (2.1) as

M t
(3.3) 3%(o) = 2 Jr Gn(¢; t, u) dBn(u), where

n=l -co

M = max(Sup MJ’ Mo) . What the functions Gn stand for is clear from the

J
context.

Also, H03; t) =§[H(_g; t)U H(l;; t)] . A representation of the
J M

form (3.3) will be denoted by [Gn’ Bn]
1
M
Theorem 3.1. Let (Gn , En} be a canonical representation. Then there
1 M
exists a proper canonical representation [Gn , Bn ] such that for every
1
(e, t) , (o)e:2 th Gn (e; t, u) dBn(u) with probability one.
n 11 -q)

Em. Let tiHM) 2’ 8.311(4))2 . For each (p and t , and every measur-

('q), t] , define the measure HEt)®)(S) :

(n)

u) . Then for each n , the measure u given

able subset S of

Jr'Gn( (w, t9 u) )|2 dp

by “(n )(S) e V HE:)w)(S) (see [7]) is absolutely continuous with
’ (n )

respect to pn . Let NH 2 (ul g§E-_(u u) > 0) and Bg(s) be the random
n~

set function with variance function pn and defined by the stochastic

fl<

integral Bn( —=j‘ I,N (u) dB n(u) . Further, set E;(e; t, u) z Gn(e;1t,u)
M t ~

for all e, t and 3u and consider the sum, yt(e) : E J( §g(e:t,u)d8n(u) .
1 -a3

If M is infinite, the right hand side series is easily seen to be conver-

gent in quadratic mean. From the fact that

d#(n) . d (n) 2
a“ n) < ) d;;_ <') 2 ‘Gn(Ti t; 3))" for each t, e and n , it is

easy te deduce that

-12-

t
2 . 2 _
j [l-IN (u)] [Gn(cp, t, u)| dpn(u) - O .
-00 n
Thus for all t, o,

2
I

t 2 2
(3.4) mete) - its) f \1 - IN (u)| I3n(cp; t. u>l apnm = o .
n

n=l -co

From (3.4) we find that for every t and (p

(3.5) 3%(Q) = xt(¢) with probability one

and that
(3.6) H(3g; s) = HQ; 8) for all s e T .

A similar argument also yeilds that for every measurable subset S

or ('33, t] .9

l
IIMZ

M ~
(3.7) 2 flen<cpzt.u)|2dpn(u)- f(en(cp;t,u)\2d6n(u) .
n= S S

n l

O for A f A' or n f m , we have

Since a [§n(A) §m(n')]
' M

H(§} t) =12€B H(§h; t) .
1

Therefore, to establish that (5;, 3;) is proper canonical, it suffices
to show that H(§n; t)C H(;; t) for all n and t . Now suppose that

there is a t and an n , such that

H(§n: t) ¢ Hus; t) .

Then we can find a non-zero element 2 a H(§;; t) which is orthogonal
to H(x; t) . Let s" e T be arbitrary and s g s' g t . By the canonical

property of (On, En} , (3.5), (3.6) and (3.7), the projection of 3%"(o)

M 8'
onto HQ; 5') is given by 2 f Gn(cp; a", u) dBn(u) . But zJ_H(x;t)
1 -oo

_ -13-
t
and z = h(u) dBn(u) with h 8 L2(fin) (see [A], pp. 426-28). Hence
-oo
8'
(3.8) jr Gn(¢; s", u) h(u) dfih(u) = O for all s", m
-a>
Using a similar argument with s we obtain
8' e”
(3.9) I. Gn(¢; s", u) h(u) dfih(u) = O for all s" and e .
s

Proceeding as in Theorem 1.2 of [7], it can be shown that (3.9)

implies
pn[N(h) nNn)= o where N(h) = (u) h(u) 7! 0}

Hence,

t t
and: f |h<u>|2dsn<u>= I IN (u)‘h(U)|2dPn(u)=f\h(u)|2dpn(U)=0:
—oo -oo n Nn(\N(h)

contradicting the assumption that z # O .

Remarks. (i) The relation obtained in (3.5) is an equivalence relation. Hence

M
we shall refer to (En, Bn] as a proper canonical representation equivalent
M l
to (Gn, Bn} 1 . d5
(ii) By definition of g; and the fact that Egfl'(u) = Ifi (u) if
n n
E'.:.O , we obtain I (u) = O a.e. p . But this will imply
n d (n) Nn n 2 dug?!) d (n)
p E&-- (u) > O = O . Hence (G ($3 t, u)[ which equals (u) -&--(u)
n 9n n d“(r0 dpn
vanishes almost everywhere [pn], i.e., for every' ¢ and t Gn(e; t, u) = O

a.e. with respect to pn , contradicting the fact that M is the multipli-
M
city of (Gm, Bn} . Thus the representation (an, Bn] also has multipli-

1
city M .

(iii) Finally, from the definition of E; we have

-1k-

is) fr<>a<>+v :
= u z u a
n SNn n tannS Jn
= z (S) + 2 E
13.53 Jn
J
say, where {jn = Ejn if tj 5 Nn , and 0 otherwise. Hence the proper

canonical representation obtained can again be put in the form of (2.1).

NEARLY STATLONARY STOCHASTIC PRQQESSES Q}! 2

We now turn to the central task of this paper, the study of the multi-
plicity theory of weakly stationary processes on ¢ . As we shall see,
this theory applies also to a class of infinite dimensional stationary
processes and shows that in the study of the latter, the idea of multiplicity
naturally supplants that of rank.

Before proceeding to the discrete parameter case whose results we
shall need in Section 6 we make the following observations concerning the
Wold decomposition of continuous parameter stationary processes on ¢ . If

for every real h , we define

Th LtW) = Lt+h<<p) ,

where t is an arbitrary real number and m 6 0 , it is easy to see that
this definition can be extended so that Th becomes a unitary operator.
Indeed, (Th) (-d3 h, +a3) is a group of unitary operators and for all

real a and h

ThE(a) = E(a+h)Th .

Using this fact and proposition 2.1 we are able to state the following pro-
position:
If"(xt] is a weakly stationary process on ¢ then there exist weakly

stationary processes on w , {x‘l)] and [5:25 such that

...,
(1) its) = 4%) + aims) for every t ,

(2) (1:1)] is deterministic (3:2)} is purely non-deterministic, and

(3) H(x‘l)) and H(xfl2)) are orthogonal.

_ 16 -
4. Discrete parameter processes. Let En (n = 0, il, ...) be a purely
non-deterministic stationary process on w . Since we want H(x) to be
separable, we shall assume that for each n , xn(-) is continuous in
quadratic mean in the G-topology. If in Theorem 2.2, T is the set of

integers then the resolution of identity of the process is given by

E :

t (pn - pn-l) where pn is the projection onto H(x5 n)

2
ngt
The self-adjoint operator A then hasa purely discrete spectrum, having
each integer as an eigenvalue and H(x_; n) eH(gc_; n-l) as the invariant
subspaces. The multiplicity M of the process is therefore given by

M = Sup [dim (HQ; n)eH(2<_; n-l)} ]
n
The following two lemmas show that dim {H(_x_; n)eH(;c_; n-l)] is inde-

pendent of n . Let gn(m)a:§n(m) - pn_lzh(m) .

Lemma kl. H(_x; n)eH(3c_;n-l) asgtgnfip), (p e e] (n = O,+ 1,....).
Lemma h.2. For arbitrary integers m and n, there exists a unitary

Operator Tm such that,

Tma [gn(cp), <9 e $1 = gems), <9 6 <19]

To prove Lemma 4.1 it is enough to Show that H(xjn) = H(x5n-l)€B
g[gn(cp), q) .3 <1? ] . But this is true from the definition of gn(cp)
For the proof of Lemma 4.2, we consider the group [ Tm) of unitary

Operators given by

Tm §n(m) = xm+n(m) for all n and p .

-17..

It can be easily verified that

TmPn_ll{n(cP) :pmi‘n-lgc'mi-nW) ' Hence, Tm €1.10?) : gm+n(<P)

and the proof is complete;

For the process Zn of this section we now have the following result.

11

Theorem 1+.1 x (cp) = 2 NZ b, (cp; mm) “(111), where
_n .= nén

(i) M: dim[H n)(eH 3;; nl— ' is the multiplicity of the process,
(11) For eachl. , {51011)} (m = o, :1, ...)

 

has stationary orthogonal

increments and afiz (m) €k(m' )I = 0 if k 7g t . Furthermore,
2 2 . . .
E 2 ‘ b m)| a IQW)‘ is finite and
L=1.m<0 _

(iii).2 €£>H(Ei[ n) = H(35 n) for all n .
i=1

Proof: From Theorems 2.2, 3.1 and the remarks preceding Lemma 4.1 about
the resolution of the identity in H(3fl , we have

M

(4.1) £1109) = >3 E b,’(<p: mm) Him) with Hug; n) =
min L=1

By Lemma 4.1 and (4.1)

g[gn(¢> ”@535 nén]:H ._3 n) - —2@H(Els

In particular

@8009): cm ®1=C§[£,'(o), l. = 1, 2, M]

Hence, if we define

§t(m) = T C'(O) , we have

gtzflm), L: 1, 2, M] =(r;[;,(m), L =1, 2, M],

'18-
since

ng[€_',,’(0),

6
ll
H

v
N
so

M J = Tm<§[ao<<v),cp ® J =C§igm(cp),q>e <9 1.

Therefore, H(x5 n) H(€i; n) and hence

I
begdtz'
ME: 6

: Z 0 '
x0e) H mg) by. m) gym) . with
M . . .
2 2
g >3 b,(cp;m)l glam)! <00
L—l mgc .
M
x (c) = ’I‘ x (cp) = >3 2 b.(<p; m) E (m+n)
-n n*-o é=u.mgc -. t
-M
Z 2 2 b.3(q); m-n) EL(m)
=1 mgn _

u h

-19-

5. Continuous parameter, weakly stationary processes. We shall give in this

 

section the generalization of what we believe to be the essence of Hanner's ideas
underlying his time domain analysis of oneodimensional stationary processes. The
desired generalization will turn out to be based on a study of the properties of
the maximal spectral type of the operator A of the process and its multiplicity,
thus effecting a unity with the work presented in Sections 2 and 3.

It is convenient to recall at this point some of the terminology of multi-
plicity theory in a separable Hilbert space H. Let A be any self adjoint oper-

ator with spectral measure function E(.). For any element f in H let pf

be the finite measure on the Borel sets of line (sometimes also called the spectral
function) given by of (A) = IIE(A)f||2. The family of all finite measures on

the line is divided into equivalence classes by the relation of equivalence be-
tween measures (equivalence here means mutual absolute continuity). If p is
used to denote the equivalence class to which the measure pf belongs, p will
be called the spectral type of f with respect to A. p is also referred to as

the spectral type belonging to A. If elements f and g are such that pf ; pg

they obviously have the same spectral type p. We shall say that the spectral
type p dominates the spectral type c (p >0 or o < p) if any (and thus every)
measure belonging to o is absolutely continuous with respect to any measure
belonging to p. p and o are said to be independent spectral types if for
any spectral type v such that v < p and v < c we have v = 0. An element
f is said to be of maximal spectral type p (with respect to A) if for every

g in H 98 << pf. The subspace G§3E(A)f,,A ranging over all finite intervals]

is called the cyclic subspace (with respect to A) generated by f. If this sub-
space coincides with H, f is called a cyclic or generating element of A and A

is called cyclic. Also if f is a generating element of A, f is of maximal
spectral type and the latter is referred to as the spectral type of the (cyclic)

Operator A. It is to be noted that if A is any self sdjoint operator (since

H {- nannrnh1n‘ rhnra n1unvn aviern n mnvimn1 annnrrnl rung hn1nnaina rn A

-20-

Any system of mutually cyclic parts of A of type p is called an orthogonal
system of type p relative to A. An orthogonal system of type p ‘which can-'
not be enlarged by adding to it more cyclic parts of A is called maximal. It
is a known result of this theory that all uaxinal systems of type p have the
same cardinal number. This uniquely determined cardinal number is defined to
be the multiplicity of the spectral type p with respect to A.

Finally we need the notion of a uniform spectral type. The spectral type
p (+0) is said to be uniform if every non-zero type a dominated by p has the
same multiplicity as p itself. Most of the above definitions have been taken
from the article by A. I. Plessner and V. A. Rohlin [12] to which the reader is
also referred for further details.

When dealing with continuous parameter processes, we assume not only that

3t (e) is continuous in q.m. in the topology of o but that for each $6..

the complex valued univariate process [§t(¢)] (-w <It‘< + w) is continuous in
q.m. in t. We shall refer to this as condition (C). It is easy to see that if
(C) holds, the assumptions of Lemma 2.1 are valid so that the separability con-
dition (A.l) is satisfied. In addition, it follows from condition (C) that the

group [Th] introduced in Section h is strongly continuous. We recall from Section h

[(5.1) ThE(t) = E(t + h)Th

for All real.t, h. As 1n'[é], (5.1) is theV' -' - " . }
basic relation between the operator A and the unitary group of the process
which we propose to exploit in our time domain analysis. We shall prove the
central theorem on representation by means of a number of lemmas. The first
group of lemmas concerns the properties of spectral types.

Lemma 5.1 If f is any element of H(;_c) then, pf << Luthe. Lebesque measure.
£5225: ‘Let us define for every real t, and every measurable set S of the real

line pf(t) (S) = pf(S-t) = ||E(S - t)f||2. From (5.1), however, p£(t)(S)

equals ||s(s) thlleo Hence by the strong continuity of the group (rtr, ogt)(s)

converges to 0.18) as t -)O. The assertion of the lea-are now follows from a

-21-

Presult-due-to N. Wiener and R. C. Young [See Saks [16], p.91].

Let f(1) be a maximal element of A, i.e., an element of maximal spectral

type with respect to A. and u any positive number. If we define

(5.2) g: . [E(b) - E(a)] Ah/‘BThE(AO) f(1)dh, where Ab§10,u), A < a-u

B >'b and the integral is taken as in [6], we observe that 3: can be ident-
ified with Hanner's 2(Iab) with z = E(Ab)f(1) in the formula (3.2) of [6]
(p.166). We remark that g: does not depend on A and B as long as these
limits of integration satisfy the stated inequalities. We give here the proper-
ties of g: which follow from those of z(lab) [See [6], p.167]. For

a < b < c, we have

b C c
(5.3) s‘ + 8b - 8,:
b c
(5.h) 8a is orthogonal to gb,
and for arbitrary t,
b b + t

(5.5) Tt sa - 83 + t
It follows from (5.3), (5.h) and (5.5) that

(5.6) ”3:“2 a T(b-a) where T is a non-negative number that does

not depend on the interval (a,b].

Lemma 5.2. There exists a finite interval 456C(O,u] such that g; as

defined in (5.2) is different from zero.

Proof: We follow Hanner closely in proving this lemma ([6], Proposition C).
Suppose g; a 0, then for every 2' e E(x) and every %¢(O,u],a[gg?'] a O.
Hence,1f z - o(s1,t1) and 2' a “(52,t2), where uKs,t) : {E(t) - E(g)]f(1)

for s < t, then from the fact that 8J3: '5'] a O, we have

. u
(5'7) f £[Th'w(919t1)° £3???)th '2 O (O < slntltigptg 5 u)'

-22-
But for 5 such that O < 5 < éu, 8}[Thw(0,u) wi8,u-5)] = alTh w(5,u-5)|2
is a continuous function of h which converges, as h.-9O to g]w(8, u-8)|2.

Now, w(5, u-5) = 0 implies that [E(S) - E(u - 5)] f(I) = O and hence
[E(S) - E(u - 8)] f = O for all f eH(x_), giving H(_x;5)@ H(§; u - 8) = [O].

This contradicts the fact that the it -process is purely non-deterministic.

Therefore we can find a r (O < r < u) such that

 

. Y _
L = ufawn w(O,u) w(5, u - 5)] dh + 0. Let t0 = 5~<t1 <o.. é tn =.u-&
'T

be a finite subdivision of the interval (5, u - 5]. Then

n r
L=Zf gmh w(O,u).-u3(_E:1_,_—ti)] on.
1 -r

n r + (ti - ti-l)
Let M.=:§: ‘I2;[Th w(ti_1-r, ti + r) “Zt141’ ti)] dh
1 'Y'(t1‘t1—1)

n +11
=2 J‘ gmh ”(ti—1"“ t1 + Y) W1” dh which is zero from (5.7).
-u

Now |MPL| é 2u|| w(O,u)||max||w(t But
1

1—1' t1)||°

f(1) = P f(1) and ”(ti-1’ ti) is

w<ti-1’ ti) + PH(x_; t1,u) H(x; ti_1,u)

(1). f(I)|l2

orthogonal to H(§; ti’u) f Hence, llw(t1-1’ t1)||2 = IIPH(x; t1_1.U)
'IIPH(§; ti’u) f(I) ||2° Since IIPH( f(1)||2 is a continuous function

£3 tau)
of t, we make ||m(t1_l, ti) || as small as we please by taking a fine enough

subdivision. Hence M = L. But M = O and L + 0. We arrive at a contradiction,

thus proving the lemma.

-23.
Henceforth, Ab will denote a fixed subinterval of (O,u], such that

Ilgglla + 0 where in (5.2) we take (a,b] = (O,u]. B.

Suppose that O < b < u and consider 3: = [E(b) - E(O)] f T E(A

(1)
)f dh,
A. h

0

where A' < -u, 3' > b. Since the definition of is independent of this

b
8o
particular choice of A', B', we have

B
8: = [E(b) - E(O)] 8; = [E(b) - E(O)] f ThE(A0)f(1)dh, where A < - u and
A

B > u. Also from (5.3) and (5.h), g; = g; + g: with g: orthogonal to 3;.
u 2 b 2 u 2 b
Hence ||gO I] = Ilgoll + ||gb|| . If go a O, we have from (5.6) that

mu = 7(u - b) where r + O by Lemma 5.2. Since u and b are distinct pos-
itive numbers, the above relation is absurd and thus 38 + 0. 0n the other hand,

if b > u then again (5.3) and (5.h) imply that g; = 83 + g: with 3; being

b
orthogonal to 3:. Therefore “go”2 a ||ggl|2 + ”33”2 thus giving 33 # O

for all positive b. Finally if b'< 0, then from (5.5), T 3:. e 3% where

B
_ B , O _ b
5 a -b'. From previous arguments go + 0. Hence gb. + 0. Thus 80 + 0 if

b > O and 82. + 0 if b' < 0. We therefore obtain T + O in (5.6), since

d d-c
for any (c,d], T_CgC a go + 0.
Lemma 5.3. The spectral measure pg b = TuI, (I 2 (a,b]), where
a

uI(S) 2: u(IhS) for every measurable subset S of the real line.
Proof: Let [A be any finite interval. Then pgb (A) e ||E(AQg:||?. There-
a

b

fore, from (5.2). pgb (A) = UNA/“)8, ||2, which equals zero if am =¢
a

and, from (5.6), is equal to “r u (AAI) if AA 1 +¢ . The result follows

immediately from the definition of uI.

.2h.
The definition of g: can obviously be adjusted to make T = 1. From

now on we shall assume that this has been done.

Lemma 5.h. If p is the maxiual.spectral type of A, then p.3 p.

 

Proof: It suffices to prove that if f(1) is a maximal element then p (l) E.“-
f
' <1) _ . I
From the maximality of f and the fact, shown in Lemma 5.3, that p b = u
8
a

for an arbitrary interval I = (a,b], it follows that u << p (I). An appeal
f
to Lemma 5.1 completes the proof.

We next define a complex-valued process £1 (a) for all real a, as follows:
' o

g1(a) = ~ga if a < o
§1(O) = O

a
§1(a) = go if a > 0.

If we set §1(I) = gib) - §1(a) for every interval I = (a,b], it follows from

(5.3) and (5.u) that

(5.8) elm-:3" .

a
It is easy to see that [51(t)] (-w < t <I+ w ) is a stochastic preocess with
stationary orthogonal increments and €151 (A)I2 = h(A). Let us write 'H(§1) =

g{§1(A). A ranging over all finite :subinterv'als of ”real? line] and
H(§1;t) = gglfls), A ranging over all finite intervals contained in (-oo, t]] .

Then by (5.5) it follows that for every real t, T ) = P If we

t PH(§1 H(:1)Tt°

now define

1)
(5.9) §t( (v) = PH(§1) §t(¢).

’ B
then the 5K1) -process is stationary and Tt §(:$)=‘§(s)

t + t(cp) for all s and

m. Furthermore, since g1 is a process with orthogonal increments, we have

H(g1) = H(§13t) @35{§1(A)0 AC<t9 + “0) = H(§1;t)@3[8:, t< a 5 b < + 00]

-25-
from (508). But, by definition of g2, §t(cp)J_C§[g:.)t < a E b < on] so that
x (o) = 91ml“) _t(cp) for all t, cp. since from (5.8) and (5. 2),; 1(A) e H(§;t)

for every finite interval A lying in (-oo, t], we have H(x(1); t) CH(x; t).

Hence the it“) -process is purely non-deterministic.
1 t
Lemma 5.5 For every real t and cp in G, 5t( )(cp) = f f1(cp; u - t)d§1(u)

1
-OO

0
where f [F1(cp; u)|2 du(u) < co.
.m

Proof; Since 350(1)(cp) e H(§1; 0), it has the stochastic integral representation
(1) O O
350 (CD) = f F1(q);u) d§1(u) with f | Fl(cp;u)|2du(u) finite (See [LL], pp. hes—28).
..a

The §t(I)-process is stationary and It §1(A) '= §](A,+ t) from (5.5) and (5.8);

hence

353%) = T,_01<q>> =fo1 F (at) W“ + t) = fem; u t)del(u)

For every Cp 6 (D and t real, set yt(1)(CP) = itW) " §t(1)(cp). Then
Tt y.(l)(q>) = yfiifip) and H(y_(1);t)C_H(x;t). Hence the ltd) -process is

also weakly stationary and purely non-deterministic. From (5.9) we have

1th) = 5t”) - PH<§1) xt(cp) which implies that for all t, (p. zt(1)(<P).LH(g1).

Since H(:_t_(1))C H(§1) it follows that for every t and s

(5.10) 1191(1)"); Heme)
Lemma 5.6 H(_x_;t) -.- H(§(l);t)@H(z(1);t) for each t.

Proof: Since H(§(1);t)9H(_y_(U;t) fiH(x;t), we need to show only that

H(x_(1);t)®H(y(1);t)c_H(x;t). But this follows from the fact that for

(p e O. 357W) = §T(1)(Cp) + yT(1)(cp) which belongs to H(x(1);(yt)®H (1)3t) for

for £2 1'.

-26..

.Lemma 5.7 Let a and b be arbitrary real numbers. If we write

 

H(§_(i);a,b) = H(x_(1);b) eH(x(I);a) than
(5.11) H(§_(i);a.b> = gag. a < a .s. a s b] = gums) - E(a))g: a~5< masses)

Proof: The second half of relation (5.11) is obvious since [E(B) - E(a)]g: = 33

 

for a < 0: § 6 § b. To prove the first part we proceed as follows: For

"(.CP) : P ( 1;Xt) t (cp)P 1,1193%:

a<t§b and cpe¢,_}§ ,‘(cp) -PH( (1),a)xt

From Lemma 5.6 and (5.10),
(l l ':
it )(q’) ‘ PH(§(1);a)5t( )(q’) = PH(§1;t) it“? " PH(35;a) PH(§13t) 35th” Further"

more, for 8. § t, writing H(§_1;a,t) = H(§1;t)eH(§1;a)

(5.12) H(E1;t) = H(e1;a)®H(§1;a.t) and §T(cp)JJ(§1;a.t).
The latter assertion follows from (5.8) and the definition of g2. Thus, we have

PH(§;a) PH(§1;t) = PH(_}£;a) [PH(§1;a) TF'IPH H(§1;a t)]: P‘H(x;a) PH(§1;a a) Further.

since H(§ 1:;a)€H(_1_i_;a), we have

ad) (1)
¢) ’ PH'(§(1);3) fit (CD) : PH(E1;t) _t(cP) " PH(§1;a) §t(¢>s Hence
H615”); a, b)CH(5 :1;a a,b) which from (5.8) is the same as
Egg, a < C1 é B .5 b]. To complete the proof we have only to observe, be-

8

cause of Lemma 5.5, that for a < oz g B E b, ga is in H(_x_;a,b) and is ortho-

gonal to H(y(1);a ,b).

Let )5 8(1)) ( xt1~~8cp) — PH(X(1);a)xt(1)(cp). From Lemma 5.7,it follows that

a<t§éb and (pad).

. :1" ‘ ‘ b
(5.13) g ('1) =J F(cp; t,u)d E(u)g: where f |F(cp;t,u)|2dp(u) is finite.
a

-27-

We are now in a position to prove the following result

Lemma 5.8 The operator A is reduced by H(§(1);a,b).

Proof: It suffices to prove that for a < t é b and m e ¢,.QEL(1)(m) e H(§(1);a,b)

 

since H(x_(1);a,b) = §{§t(1)(¢)i CP 6 <9 a < t éjb]. From (513)

, t
~’1' Aga)(cp) = fu F(cp; t,u)d E(u)g: where F(cp; t,u) e L2(p.I). Hence

a

A? me) e gums) «soon: a < a < a e b] u F(cp;t.U) e Leaf).
From the preceding lemma it now follows that Agt(1)(m) e H(x(1);a,b).

'Lemma 5.9 H(§KI)) reduces the operator A.

Proof: From the properties of the resolution of the identity corresponding to A,
‘we have
(5.11;) E(A)A .—. AE(A)

for every finite subinterval A.= (a,b]. If w is any element belonging to

d5: (\HKEK1)) (which is non-empty) where 49A is the domain of A, then from

Lemma 5.8 we have

s H(_x_(1);a,b).

E(A)Aw = A E(A)w = A PH(x(l);a’b) w
Now letting a = n -l, b = n and An '= (n - l, n] we obtain
00 m
3 5‘ '
Aw == :E: EKAE)AM' €‘ 2:.‘$> H(§fil); n l, n) = H(x(1)).
n=-oo n=-oo

Leif Au) be the reduction of A to H(x_(1)). Then (Lemma 5.8)c1early

i) (1)

A( is reduced by H(x(i); a,b). We denote this operator on H(§ ;a,b) by

AI(1) (I = (a,b].). An immediate implication of Lemma 5.6 is that AI“) is a

cyclic operator with generating element g2. We-recallwfrom Lemma 5.3 that thel

spectral function of g: is given by p b = H ,

ga

Now let I, = (a,,b,] (j = 1,2,...) be disjoint intervals whose union

in

is the real line. If pj denotes the spectral type of the operator Aj
. (1) . . . .
(which we write here for AI ) then it is easy to verify that the pj s are
1
independent spectral types. For let j and m be arbitrary (j + m) and suppose

that o is a measure whose spectral type is dominated by both pj and pm. For

all k + j since qu(Ik) = O we have o(Ik) = 0. But 0(Ij) is also equal

I
to zero since H m(Ij) = 0. Hence a = O. Summarizing all the above facts

1
‘we find that we have a representation of A( I as the orthogonal sum of cyclic
,1
operators A(I) whose corresponding spectral types pj are independent.
1
It then follows that ([12] p. 152), A(}I itself is cyclic and since the spectral

I
function u j belongs to the type pj for each 3 we can conclude moreover

1
that the spectral type of A( )is equivalent to p. From Lemma 5.h it follows

i
that the spectral type of A( ) is equal to p, the maximal spectral type of A.

(I

I
Let us recall that H(x) = H(x )){$}H(y()) and the self-adjoint

l
operator A is reduced by H(x( )). Hence A can be written as the orthogonal

sum of the reduced operators, A = AH(*(1)§- + AH(y(1))

Q\
Now, A (y(1))’ a self-adjoint operator on H(y£ )) is the operator of

H- ..

l . .
the weakly stationary non-deterministic process [yt( )f , -m < t <'+ w)
I I 1

We may, therefore, apply the above analysis to this process replacing H(x) by
H(_y_(i)) and A by AH(y(l')) . We then have, Hula») = H(_x(g));®H(y(23»),
where the x Q?) process .3 constructed frim the yt(}) -process in the same

. '5‘)
L.

way as the §t(r) -process is obtained from the given Et -process. The XCK

-process is stationary and purely non-deterministic. We also have the orthogonal

-29-

decomposition

A = A(1) + A02) + AHQIEI) .

(i)

where A = AH(x(i)) Continuing the above procedure we arrive at the follow
ing relations,

1‘ 2‘ -' , M
(5.15) M33) = H(3§( UGBHQ 5G}. . .({)H<§( )).
(5.16) A = A“) + Am) + . . + A(M),
‘where x (1) (¢) = P x (@> and [E (u), - w < u < + m] are mutually
orthogonal processes with stationary orthogonal increments. The operators A<1>
are cyclic, all having the same spectral type 9 (the maximal spectral type of A).

Further M is a cardinal number at most equal to F(%.

Also from Lemmas 5.5, 5.6 and 5.7, we have

. t
(5.17) 551%) =f Fi(¢;u—t) dg1<u>

with K H M
(5.18) H(§;t) => G9 113(1)») = 2%) H(gi;t).
i=1 n=l

Let f(1) be the generating element of A(1). Since

[E(b) - E(a)] f(i) = PH(x(12a,b) f(i), clearly H(x(i)) is the cyclic sub-

space generated by' féyl, i.e.,

(5.19) H(§(1)) = G§§{E(AJ E(l)’ a_ ranging over all finite subintervals
of the real line]. We also have pELi).; u. From (5.15) and (5.19), we have
M
V. i, (1) .
H(X) = .. f "iiELA)f , Ll ,ranging over all finite Subintervals] and
1:1
(5-20) pf(l) .i oft!) : - ‘ ; pf(M)o

Hence, it follows that M is the multiplicity of the xt-process. (See Sec-

tion 2 where this notion is defined). Assembling all the results of this

-30-
Section together we observe that we have established the following basic re-

presentation theorem.

 

Theorem 5.1 Let xt (-w < t < + m) be a weakly stationary, purely non-

deterministic process on w satisfying (C). Then

M t

(5.21) new: f new-twee),
1:]. .00

where,
(i) M is the multiplicity of the process,
(ii) each §i(u) is a process with stationary orthogonal increments (homo-

geneous process) and the 51's are mutually orthogonal. Furthermore,

M . M O
H(x;t) = if é&)H(§i;t) for every real t, and :E: ‘jfl IFi(¢;u)I2du(u)
igl l -m

is finite.
It can be easily seen that the homogeneous processes gi,(i=1,2,...M) of
the representation (5.21) are uniquely determined upto a unitary equivalence.

The above theorem is a generalization of the Karhunen representation to

stationary stochastic processes xt on ¢. This result also generalizes the

Rozanov-Gladyshev representation for' q-dinensiona1 stationary processes as
will be seen in the next section. The reader will observe that (5.21) has been
derived essentially independently of the Hida representation (2.1) and the
latter is referred to at the end of the proof only for the purpose of identi~
fying M as the multiplicity of the process. Indeed, the whole point of the
problem is to study the maximal spectral type and to construct the homogeneous
processes gi(u). Once (5.21) has been obtained, however, it is easy to dis-
cover the special properties that the representation possesses in this case,
(i)

e.g to see that all the elements f occuring in it are equivalent, with a

common spectral type equivalent to u. Moreover, starting with the gi's

one can construct without difficulty a sequence [f(1)] for the representation

-31-

(2.1) of Section 2. This can be done as follows: It is clear that the elements

fi (i = I,...,M) occurring in the proof of Theorem 5.1.and with the property
that they have all the sameispectral type equivalent to u (see(5.l9 and (5.20))

can be chosen as the elements in the Hida representation of xt. If we now set

dpf.‘ ‘_%
§1(A) = f Mu) decori .
A. dp

it is easy to verify that the gi are mutually orthoganal random set functions

each having p as itS‘measure' function, and that (A being a finite interval)

dpf. 5
E(A)fi =[ 1 (u) d§i(u).

d
. A “

 

If we now make the appropriate substitution in (2.1) and compare it with the

representation (5.21) it follows that for each t and m

dpf -%
mm...) = Fin; u-t) 1‘ (u) (1:1,...M>

Ldu

 

a.e. with respect to u.
Thus, for stationary processes, the generalization of the approach of Hanner
given in Theorem.5.l leads to a deeper analysis which includes the proof of
(5.19) and (5.20) and yields directly the representation we seek. It is inter—
eating to explore further the connection between p and M. The following dis-
cussion presents another aspect of the problem and provides additional information.
Theorem 5.2? :p is a uniform.spectrai type with (unifoflm)tmultiplicity M.
.EEESE: We use the ideas of Plessner and Rohlin [12]. It will first be shown

that p has multiplicity M. Let {Aé ] be an orthogonal system of type
p and cardinality PP, , i.e., a system of orthogonal cyclic parts AB' of the
operator A, the spectral type of each cyclic operator Aé being p. According to

to the terndnology of [12] M is the multiplicity of p if we can prove that

-32-
MI 5 'M. Observe that neither M nor M? can exceed EV}, for otherwise we would
arrive at a contradiction of the fact that H(§) is separable. Furthermore, there

is obviously nothing to prove if M =?€b . Thus the only case to be considered

is when M is a finite cardinal. If possible let M'I> M. We shall show that this
leads to a contradiction. Let h. (i=l,...,M) be a generating element of the

1
subspace H(x‘1)) and hé (B = 1,...,M') be similarly a generating element of

the cyclic subspace corresponding to AB . Clearly, there is no loss of gener«

ality in supposing that all these elements have the same spectral function, say.

p'. From (5.15) and (5.19) it follows that for each B we have
M
I _ 2 g .
hB —: fFiB(u) dE(u)hi where 2 f IFiB(u)l dp (u) 18
i=1 i
finite. For every measurable set A, we obtain
‘M
U ' __ '
E [E(A)hB. h7] — f ; F15 (u)F1?,(u] dp (u).
A =

The left hand side of the above relation is zero if B +7 and equals

p'(A) if B = 7. Hence for u not belonging to a set N of zero pl—measure

57
we have M .
,4“
F. F I I = 5 .
Z 16 (u) 17 u 67,
i=1
Since M' is at most 1 1) the set N = UTN is measurable and p'(N) = 0.
8,7' B7
Choosing a fixed point uO in the complement of N we see that
‘M
(5.22) E 19150.0) ‘T-Fiy uo) - 867' for all s, 7.
i=1
If we now set aB = [ F15(uo),..., FTB(u0)}’ the relations (5.22) imply that

the a are MI orthonormal vectors in M dimensional unitary space. Hence M'

f3

-33...

cannot exceed M. In other words p has multiplicity M.
The proof that the spectral type p is uniform is achieved by a modification
of the above argument. The reader will no doubt, observe that the conclusion

about uniformity rests on the fact that the orthogonal system [A(l), i=l,...M7]

is not only maximal but that the orthogonal sums of the A(1) is equal to A (see

(5.16))-

Let a by any spectral type dominated by p. The only change we make in

the proof given above is to let [Aé] be an orthogonal system of type a and

B

cardinality M'. Let h' be a generating element of the cyclic subspace of A

B

Assuming, as we may that the hi have all the same spectral function p' and

that the h' have the same spectral function 0' we obtain the relations

B

3.4-, do' do'
(5.23) 2 F (u) F1 (67 z.- ———-—— (u) o , where u t N and

 

dp'

is the Radon-Nikodyn derivative of o' with respect to p'. Since the set

3

S = {'11: (u) > O] has positive p -measure we can choose 110 in S (\Nc

 

dp'

'.~measure. 'Substituting u for u in

when as before N is the set of zero p 0
the relations (5.23), we are again led to the conclusion that Mfg M. Thus it
has been shown that the multiplicity of any spectral type dominated by p is
equal to the multiplicity of p . Hence p is a uniform spectral type.

Remark: It follows at once from the theorem just proved that every
spectral type belonging to the operator A of the stationary process it has
multiplicity M.

To find the funtions F1 and the value of M in the representation (5.21)
in specific instances one would have to consider, individually, concrete ex-
amples of spaces o and purhaps have to assume additional properties of the

process it such as linearity in m. The study of some of these questions we

postpone to a later paper. However, since it is important to relate our work

-3h-

to recent developments in the theory of multidimensional stationary processes

we consider in the next section the case when c is a q-dimensional unitary space.

-35-

6. Multiplicity gg‘a Generalization of Rank. In the theory of finite dimensional

 

weakly stationary processes the notion of rank plays a conceptually essential role.
Zasuhin, in l9hl, was the first to define the rank of a q-dimensional, discrete
parameter stationary process as the rank of the (q X q) ”error matrix" (See [18]).
More recently, the definition of rank for a continuous parameter process has been
given by Gladyshev [5] to be the rank of the discrete parameter process associated
with the process. This point of view has been further explored in the recent
thesis of Robertson [1h]. It is also well known in the literature that the rank
of the process is equal to the rank of the spectral density matrix. (See [15]
where the rank is defined this way and [1h].)

we shall show in this section that the multiplicity M occurring in the
representation given in Theorem 5.1 constitutes a generalization of rank in the

following sense: If x_ is a weakly stationary process on 0 where c may be

t
infinite dimensional (and xt(m) itself may or may not be linear in.m) then M.is
equal to the multiplicity of the associated discrete process (Theorem 6.1). In
the case where o is a q-dimensional unitary space and §t(m) is linear in m, so
that we are dealing with a q-dimensional stationary process, it is shown in
Theorem 6.2 that the multiplicity equals the rank of the process and the
representation of Theorem 5.1 coincides with that obtained in [5] and [1h].

The connection between multiplicity and spectral theory for infinite dimen-
sional stationary processes‘xt will be considered in a later paper.

If [gt] (-w < t < +im) is a given stationary stochastic process on 0

satisfying condition (C), then for each m, the one dimensional weakly stationary

+ao
process {35t(q))] is continuous in q.m. and hence for fixed cp, xt(cp) =ߣ eitAdXG'hBOQp)

where [C(x), -m < A < + m] is a resolution of the identity of the unitary group

[Th] of the‘ggt process.

With the process [§t(¢)] (for fixed m) is associated a discrete parameter

process,

-36-

~ 7’ in}, 1 -1
(6.1) ens) = Ie d,G(-o:,,~t an A) eon), (n = o, : 1....) [[4], [11]].
JH'

Let us now write for each m and t, H¢(x;t) =(EixT(m), T é t] and
Hq)(§;m) = @gnfip), 11 § m} (m any integer). We have for all cp, Hq)(x;+oo) =.Hcp(’x';+oo)

and Hcpfaégsf. - Hqfim) (See [4], [11]). Therefore,

(6-2) H(§.; +°° ) = H("’; +°°) and E(ziso) = MEG) -

From stationarfly and (6.2), the following lemma is immediate.

Lemma 6.1: {gt - w < t < +'w] is deterministic if and only if fifi, n = O,‘:.l,...]

’

 

~is deterministic.

we recall here two lemmas from [5] which will be frequently used in what follows.
It should be observed that in Lemma (G2) stated below the process can be infinite-
dimensional. Its proof, however, involves no change and is an easy consequence
of (6.2).

Lemma (G1). If [nt] is a one-dimensional weakly stationary, continuous in

q.m., purely non-deterministic process, then the‘fin- process is purely non-deter-

ministic .
Lemma (G2). If [qt] and [gt] are stationary processes on o satisfying

condition (C) and such that H(rl;t)C H(§;t) for all t, then H(?]’;m)c H(z;m) for

every m and conversely.
we shall now obtain from Theorem 5.1, a representation for the EL- process.

The notation will be that of Section 5. Let us define for each i = 1,2...M,

t
(6.3) 'xé1)(m) = [Fi(¢;u-t)d§i(u), where the right hand side expression is the

-oo

term appearing in the representation (5.21) of xt(m). Consider now the process
12
h(1)(t) = fee-tdgi(s) (—w < t < +-m). Then [h(i)(t)] is a one dimensional
.m

stationary stochastic process with Tth(i)(0) = h(i)(t). Furthermore, since

-37-

51(t) ' 51(5) = {h(i)(t) - h(i)(8)] + fth(i)(u)du (s < t), it follows that

for all t
(6.h) H(513t) = H(h(1);t) (i = 1,2,....M).
The hél)- process which is obviously continuous :Lq,m., is also purely non-deter-

ministic, since from (6.h), {] H(h ;t) = f] H(gi;t)c:if\ H(x:t). The discrete

parameter process {h(1)(m)} is thus purely non-deterministic and therefore has a

moving average representation given by

(6.5) EUR...) =Zbi(i)ui(m-£) ,
i=0 ' '
where:
(6.6) H(h(i);m) =@ui(m-£), o s a < + co] and [ui(m)] (for fixed 1) is a

process with stationary orthogonal increments. From (6.2), (6.h), (6.6) and the

mutual orthogonality of {§i(n)], it follows that the processes [ui(n)} (i=l,2....M)

are mutually orthogonal. Also from (6.3) and (6.4), H(x‘i);t)C: H(h(i);t) for

each t. But from Lemma (G2) and (6.6), Hcg‘i);m) is a subspace of

figiui(m-L), L=O,l,2....]. Hence

(6.7) grin“) -Zc1(cp;e>u,<m-z)-

i=0
(i) . . . . .
From (6.3) the [xt (m)] process is stationary and continuous in q.m. Wlth

. +“.
Tt§§1)(¢) = xgig (¢). Hence xé1)(¢) = feltAdAG(A)§éi)(m). Furthermore,

IN
(6-8) 35th) = 293:1)(cp) for every t;
1 M
where the (possibly) infinite series converges in q.m., since 2 glgéihcpnz

l
is finite. Also,

-38-

"(i) 1” ink 1 '1
(6-9) 5,, (Cp) = fr 6 duds; an Meow) ~
7’ i), -1
Since S = f e de(%fit an A) is a bounded linear (in fact, unitary) operator
-Tr

on H(x) from (6.8) [with t=0], (6.9) and (6.1), we have

M s
(6.10) am) =‘ gym).

1 M n
From (6.7) and (6.10) g;(¢) = :E: :E: Ci(m;n-L)ui(£) . From Theorem 5.1 and (6.h)
M M i=1 6:430 - -
H(ggt) : :@ H(gi;t) =Z$ H(h(i);'t). In other words
i=1 "=1

(6.11) H(xgt) =g{h(i)('r), ‘r é t, i = 1,2,....M]

From Lemma (G2), (6.11) and (6.6) we have
M M

(6-12) H(gsm) -'-' Z®H(z(l);m) =Z®§iui(m-£), L = 0,1,2...) .
i=1 74:1 . .

(6.11) and (6.12) imply (see Theorem 4.1) that

(6.13) M = dim{H(g;n)eH(g;n-l)} .

We summarize the above results.

 

Theorem 6.1. Let xt(-w < t <'+.m) be a stationary, purely non-deterministic

process satisfying condition (C). Then its multiplicity is equal to the common
dimension of the subspaces H(gm) e H(ggn-l).

The above discussion pertaining to multiplicity is very general since we have
been dealing with weakly stationary processes on an arbitrary Hausdorff space,
satisfying the second countability axiom. It is instructive to consider the case
when 0 is a finite dimensional unitary space and the process‘ggt is linear on 0-

we have referred to the fact that some recent work of H. Cramer [2] can be

regarded as a special case of the results of Section 2. In [2], Cramer also

-39-
includes a brief discussion of the stationary case and shows that the multi-
plicity of the q-dimensional process does not exceed q. We shall now deduce
from Theorem 6.1 that the multiplicity is actually equal to the rank of the
process. This corollary (Theorem 6.2), incidentally, provides an alternative

proof of a theorem due to Gladyshev (Theorem 1, [5]).

Suppose [ei] (i:l,2,....q) is an orthongrmalbasis in 6. If {5t} is a weakly
q

l-

stationary process linear in m then, if m =:E:aie. , —t(¢) :::E:aixi(t) where
1:1 i=1

xi(t) ='§t(ei)' Now, (x1(t), x2(t),....xq(t)) is a q-dimensional process which

is weakly stationary. Since {fit} satisfies condition (C), {xi(t)} (i=l,2...q)

are continuous in q.m. Also, if (x1(t), x2(t),...,xq(t)) is a q-dimensional

weakly stationary process continuous in q.m. then there corresponds a stationary
process [xt ] on the q- -dimensional unitary space 0 which is linear in m and

satisfies condition (C), [viz ,x (m) =:E:ai x 1(t) if m is the vector (ai,a2,...,aq)].

=1
Furthermore, H(,_)_c_;t) =Cv=§[xi(.u), u é t,1 i = 1,,2 ...,q].
Theorem 6,21 Let (xl(t),x2(t),...,xq(t)) be a continuous in q.m., purely

non-deterministic, weakly stationary process. Then

x (t) a: [Fm (u no, (.1)

”Xi-5:1 -00
where the gi-processes and the number M are as introduced in Theorem 5.1,
‘M
@[xi(u), u é t, i--;'1,2,....,.q]. 4.: Z®Hfgi;t) and M is the rank of the process.
i=1

Proof: All the assertions of the theorem follow immediately upon setting<p= ei

in the representation obtained in Theorem 5.1. It remains only to show that M
is the rank of the process. From Theorem 6.1 and Lemma 4.1 it follows that
= dim[.H(E;n) Griffin-1.)] é dim (61am), owl)- Writing Ci =<'s'[3<'i(m). m é n.

. ~ 4' ,cv . «I .
integer, i=l,2,...,q} and g£n) = xi(n) - Eéfi xi(n) we find that

,5 .
' I
V,

q ’1“:-
8n(tp) = Zaigi(n). Therefore, M = dim(§[gi(n), i=1,2,...,q]. But the latter
i=1 '

quantity is the rank of the q x q "error matrix" with elements 8§;(o)§3(o),
(i,j = 1,2,...,q), i.e., the rank of the process (x1(n),xé(n),...;§;(n)) [[18]].

Hence the multiplicity M of xt-process (Theorem 5.1) equals its rank.

Theorems n.1, 5.1 and 6.1, apply to weakly stationary processes‘xt on a
Hausdorff space 0. The only assumptions on the process is that it satisfies
condition (C) and is purely non-deterministic, while no condition is imposed on Q
other than that its topology satisfy the second countability axiom. If, in part-
icular, ¢ is a locally convex linear space (e.g. if ¢ is an infinite-dimensional
separable Hilbert space) with a countable basis {e1} and if §t(¢) is linear in
m (e.g.‘xt is a‘wggk_process on 0) then we may consider the‘xt-process as having
an infinite number of components xéi) =.xt(ei) (i a 1,2,...,) . Thus we may
conclude from these results and Theorem 6.2 that for infinite-dimensional processes
the representation given in Theorem 5.1 is a generalization of the Karhunen-Gladyshev

representation and that the multiplicity is the appropriate generalization of rank.

HILBERT-SPACE VALUED PROCESSES

7. Preliminaries. In Theorem 2.2, and for the stationary case in Theorem 5.1
‘we obtained a representation of the purely non deterministic process on an
arbitrary Hausdorff space ¢. Suppose now that o is a locally convex linear
space and that for each t"§t is a random variable taking values in ¢', the

dual space of ¢3 i.e., for each t, there exists a mapping x fromsn to ¢' such

12
that (i) < xt,¢2> D< m,¢'2> denotes the value of the functional @' at m] is

a random variable on Q, and (2) for all qgo,‘xt(m) [w] = < xt(w),q>>' with

probability one. As is well-known these assumptions are stronger than the ones
made in the concluding paragraph of Section 6 dealing with weak processes. we
shall call [gt] defined as above a process in ®'- The definitions of deter-
ministic and purely non-deterministic processes in ¢' are the same as the ones
given in the Introduction.

By a representation of a purely non-deterministic process [gt] in ¢', we
mean a process {It} in ¢' such that,‘§t ='Xt with probability one for each t

and 1t represents a "moving average" over the present and past of xt-process

analogous to what was obtained in Theorem 2.2. In this section we confine our
attention to the case in which ¢ is a real separable Hilbert Space and refer
to [gt] as a process in ¢- Although this is the only case studied in detail
here, we feel that a similar theory can be developed to cover more general
situations, e.g., where ¢ is a separable, reflexive Banach space or a nuclear
spaCe. The last mentioned problem could well have points of contact with
recent work of K. Urbanik and others on the representation of purely non-
deterministic homogeneous generalized random fields ([17]).

We shall also make the stronger assumption that 8| lxtl I2 is finite for

each t, with the help of which we are able to prove a strengthened form of the

-hg-

Wold decomposition stated in Section 2.

.EEERgiltion 7.1. Let [xt] be,a process in o withgllxtHa < m , for each
t. Then, with probability one we have fit = x (1) + §t(2)and -§t(l) i=1,2 which are

defined except possibly for an w-set of probability zero, have the following

properties:

i
(1) {Eta-)1} and {§t(2)] are processes in d5 with 8°H§t ”2 < go (i = 1,2);

(2) H(§(1)) is orthogonal to H(x(2)), and

(3) [§t(1)} is deterministic and [§t(2)] is purely non-deterministic.
Proof: The process 3;(¢) = ‘<'§t’ m > is a stochastic process on ¢- Hence
Proposition 2.1 gives us xt(¢)=:’t(1)(¢) + x t(2)(¢). It suffices to show that

¥(1)(¢) =‘<x (11), m > (i = 1,2) where {x£1)} are processes in o with the above
mentioned properties. This is achieved by means of the following lemma.

Lemma 7.1. Let Lit} be a process in w and let P be a projection operator onto

 

an arbitrary subspace M of H(xgt). Then there exists an almost everywhere weakly

measurable mapping-xt p from n to @ such that with probability one < 5E P’ ¢ > =
5 3

P <Et’ m >’ for every m 5 ¢

Proof: Let t be fixed. It is well-known that our assumptions on x imply that

—t
for all cpl, (pa ino 8[<xt,cp1> <xt, cp2 >] = <Btcp1, (pg), whereB
is an S-operator (see [13]). Choosing a complete orthonormal (C.0.N.) system of
eigenelements corresponding to the eigenvalues {in} of Bt and observing that Bt

, W
has finite trace, we obtain 2 [P < 35th»), cpn >12 <.oo. This implies-.that-~there
1

is an w--set N of zero probability such that

(7.1) 2 [P < xt(w), cpn >]2 is finite, if m ¢ N.
1

For every m 6 ¢ and w t N, define

r

-h3-

G3

(7.2) 1%,,wa = Z < cp, cpn > [P < its). q», > 1

n=l

Then “t p is an a.e. weakly measurable, bounded linear functional on ¢. Hence,
3

fl(¢)[w] = < qt 6w), m > for de. Clearly, for each m,
t.p ’

8,[P < xt(w),cp> - < ”t p(w),cp >]2 = O and from (7.1), ||q(w)| I2 is finite.
’ t.p

If {xm1 is any other C.O.N. system then following the above argument we obtain

an a.e. weakly measurable function gt p from Q to o such that
3

.< gt,p(w),¢ > = P < xt(w),¢ > , Ilgt,p(w)||a < m and

£[P < xt(w),cp > - < gt p(on), q) >]“ = O for every cp. Thus we have

(7'3) Hump“) - gc.1300)“2 = Zfik TIrma”) ' g12,13(m)’ Xm>J 2 = O
1

since for every (p, 8[ < ”t p(m),cp > - < gt p(m),cp >1Z = O. Let£2(Q,P) be

the space of weakly measurable functions g from Q to d), satisfying gl |g(w)| l2 < co
(strictly speaking, equivalence classes of functions, see Section 8). From (7.3)

we see that q and g are elements of the same equivalence class, say, x
t,P t,P -t:P

belonging too(;(Q,P). Identifying 5t p with any of its elements we have
<xt,p,cp> =3 P<xt,cp>.

~ 1 «.2
Since 5: )(m) 2 PH(§3-m) <xt,¢> and x: )(m) < xt,¢ >,,

PH(rs t )A H129 -°°)

it follows from the lemma that there exist processes (5:1)], {xé2)} in d, defined
. _ ~(1) _ (i)
for each t, except poss1bly on a null w set such that xt (m) _‘< xt ,¢ >

for i = 1,2. Obviously, [xéi)] satisfy all the other desired properties.

Before proving the representation theorem for purely non-deterministic pro-

cesses 3t in 0, we need to introduce stochastic integrals taking values in 0, which

we shall call Stochastic Pettis integrals.

-uu-

8. Stochastic Pettis integrals. Let (A, 0‘. , u) be an arbitrary a -finite

 

measure space and o£2(A,p) be the set of all weakly measurable functions g
from A to ¢ such that jA||g(a)||2dp(a) is finite. It is well known that upon
identifying functions which are equal almost everywhere [u] (i.e., setting f = g
if f||f(a) -g(a)| |2du(a) = O), £2(A,p.) becomes a Hilbert space with inner

product given by

(g1, 32) £20,,“ = f<gl(a). 82(3) > du(a).

The norm of g w111 be denoted by llgll <0£Q(A’")° It is easy to show that

c(;(A,p) is separable if the Hilbert space L2(A,u) of real functions square
integrable with respect to p is separable. In particular, if A = T, the real ‘

line and u is a a -finite measure on Borel sets then the Hilbert space a£é(T,p)
is separable. In what follows we write o[é(p) for q[E(Tm).

Lemma 8.1 Let 2 be a real orthogonal random set function with. &[2(A)]2= 0(a).
If g e ‘uC2(p), then there exists an a.e. [p] weakly measurable mapping J(g)
from Q to ¢ with the following properties:
(8.1) J<g> e $40.2);

if g1, g2 are any elements of.d[2(p) and c c2 are real numbers than

1,
(8.2) J(c1g1 + c2g2) = C1 J(81) + 32 J(82),
the equality holding in the sense of .Cé(Q;P);

for every m 6 ¢,

(8.3) '< J(g),¢ > = ji< g(t),¢ > dz(t) with probability one, where the

right hand side integral is an ordinary stochastic integral.

-h5
The element J(g) is called the Stochastic Pettis Integral of g(t)

with respect to z and is written fg(t)dz(t). We also have

(8.1+) &<[g,(t)dz<t). fg2(t)dz(t) >1: f< gin). gem > do(t).
Proof: Let [mk} be a C.O.N. system in ¢ and let g be any element of.£%(p).

Strictly speaking, each g represents an equivalence class belonging toeCé(p)

and it is clear that elements of this equivalence class give rise to the same

stochastic integral ‘/]<g(t),mk’>vdz(t) since the latter is itself defined up

to an equivalence. Denoting it (more precisely, a random variable belonging

to the equivalence class) by L(g, mk) we have

2 & [L(g’cpk) ]2 = Z I < g (t),Q)k >2dp(t) < oo, 80 that
k=1 k=l

co

2 [L(g .opknw]

k=1

2
] < 00 except possibly when w in a set N of

zero p-measure. If, for any m, we now set
w
L(s .cp)[w] = Z <cp.<pk> L(g, pk) [w]. (w t N). it follows that
k=1
L(g, ;)[w] is a bounded linear functional on @. Hence we obtain
L(8.CP)[w] = < J1(8)[w] . (13>.
where J1(g)[w] e @. It is further easy to see that J1(g) [.] is a.e. weakly
measurable and that g! |J1(g)[w]| [2 is finite. It is evident that we have
relied on the choice of a particular C.O.N. system in our definition of J1(g).
However, if [mm] is any other C.O.N system in ¢ and J2(g) [.] is the cor-

responding a.e. weakly measurable mapping, then we have

-h6-

2
I I

EHJ1(g)[w] " J2(g)[w] = O, 1.8., l.|.J1(g) - J2(g‘)||°('2(0'£) : O.

In other words, J1(g) and J2(g) belong to the same equivalence class, say

J(g), of 062(Q.E). Thus, the equivalence class J(g) in .fi,2(Q,E) is unambiguously

defined for each g in. 0L2(p) and further llg||°( (p) = ||J(g)||(; (Q R)’
2 2

For every g e o£2(p), the corresponding element J(g) of oCé(Q,P) will be

called the stochastic Pettis integral of g with respect to the orthogonal

process 2 and will be denoted by /. g(t)dz(t). The assertions (8.2)-(8.h)

of the lemma are easy to verify.

If 2 z are orthogonal random set functions with measure functions p1 and

1’ 2

respectively and are further mutually orthogonal then it can be shown that
EB [81(t)dzl(t) . [82(t)d22(t) >3= O for 81 e 0C2(01) and g2 6 {2(02).

The proof follows by the definition of the Pettis integral.

The following result will be useful in the next section.

Lemma 8.2. Let 2 k = 1,2...) be mutually orthogonal processes with ortho-

k(

gonal increments and with respective measure functions pk. If gk 6:1;2(pk)

 

are such that

(8.h) 2E: ‘l-llgk(t)||2 dpk(t) is finite, then
k=1
:5: 'l’gk(t)dzk(t) is an element of (J;2(QflP) (the series of
k=1

Stochastic Pettis integrals converging in the OL (HIP) sense), and for every

¢ 6 w,

(8.5) _ <2 I gk(t)dzk(t),cp>= Z f< gk(t),cp>dzk(t) with
k=1 k=1

-h7-
Probability one.

Proof: It is clear from the definition of ‘fgk(t)dzk(t) that [Cm] where

m
gm = Z! gk(t)dzk(t) is a Cauchy sequence of elements in 449791)} since
1
(m' > m), m'
2 2
Hcm. - cmll 42mm = E fllgk(t)llrdok(t) —> o
m

by (8.1+). Hence the limit (an £50,?) sense) of Cm exists which we denote by

as

Z I gk(t)dzk(t). The other conclusions of the lemma are similarly proved.
l

-h8

9. Representation Theorems For Purely Non-deterministic Hilbert Space-valued

 

Processes. In this section we consider a purely non-deterministic process

{5t} in o , with €_|Ixt||2 finite. As in Section 2 we confine ourselves to

the continuous parameter case. The representation we seek for xt is obtained in

terms of Stochastic Pettis integrals. Since

2 2 2 .
8f<xt,qi> - < it”? ] é allxtll IIcp-WII , 1t follows that it
~process is continuous in the topology of ¢. Hence, from Lemma 2.1, the space
H(x) is separable provided the limits it O((p) and §t+0 (m) exist for each

(9 e 0. We shall refer to this condition as assumption (B).

 

Theorem 9.1. Let {xt} be a purely non-deterministic process in o with

a I last] I2 finite and satisfying assumption (B). Then for each t, with pro-
bability one

M

Mb j
‘- ft < >d <> ’- b <)
(9.1) x: Ft,uzu+‘ :.te.
—t A 360 n n A 3% -_]L
1 t3 é t L21
‘where Mb, th the processes 2n and the random variables Ejt have the same
meaning as in Theorem 2.2.
Furthermore, for each t,
(9.2) Fn(t’°) e c£;2(pn), on being the measure funct1on of zn, and
bj,(t) e s for every j, L;
‘ Mb A t
2 .
(9.3) 2 J IIFn(c.u)|I dpn(U)<°°.
n=l .00
M.
w J
(9.1+) Z Z ||bj,(c)||2g(§§e) <m; and
j=1 L=1

(9.5) H(x;t) = @[H(3;t)U H(igtH for every t, where

-kg-

H(zgt) =g[zn(u) I u 2’ t, n=l,..,MO] and H(fit) =g[§jb | I..—.l,..,Mj ,tj gt].
Proof: Since <§t,m > is a S.P. on ¢ Theorem 2.2 applies without any change
to it. Furthermore, it has been shown in Section 3 that the representation for
‘<.§t’¢ > can be chosen to be proper canonical without changing the numbers
Mb and Mj and hence without affecting the multiplicity M of the process.
This accounts for the conclusion (9.5) of the theorem. In order to prove the
remaining assertions we need to use the additional hypothesis in the present

case, viz., that alll‘il |2 < co.

From Theorem 2.2, we obtain

M
o no t

(9-6) 2 Z I Fn2(q>k;t,u)dpn(U) i E, II ztl I‘2 < co. where
=1 k=1 -00

[pk] is a C.O.N. system in o. A fortiori, there exists a set An of pn - measure

zero such that for u t An,

00

(9.7) 2 Fn2 (cpk3t,u) < co.
k=1

2 . th t. '

For m e o setting ck ‘< m, mk >, we obtain from (9.7) a for u k An,:E:can(mk,t,u)
k‘

converges and is in fact, equal to Fn(m;t,u) a.e. [on]. Hence Fn(m;t,u) is a

bounded linear functional on o for u * An' We may therefore write

Fn(¢3t.09 =‘< Fn(t,u),m >3 where Fn(t,u) is an element of o and moreover, Fn(t,o)

is an element of J82(pn). From (9.6) we have

Mb ‘%n
(9.8) E f I] Fn(t,u)||2dpn(u) <00.
n=l -w

Since al Ixtl I2 is finite it follows that for all j and L there exists a

bounded linear functional bjL(t) such that for each t,

-50-

(9.9) bjbwst) =<bj,,(t).<p> with Z Ilbj.,(t)||"-‘aj,2 <oo.
Lt
By (9.8), Lemma 8.2 and (9.9), we have
MO t M.
it = f Fn(t,u)d zn(u) + Z : bj%(t)§j%
n=l ~00 tj g t. (:1

The corresponding results for weakly stationary (see Introduction for de—
finition of stationarity) db -valued processes are stated below without proof.
Theorem 9.2. A discrete parameter weakly stationary, purely non deter-

ministic, process in d), with 8| lit-t] I2 < co, has the following representation.

n M
in = Z b£’(n’m) £310“)-
m=-oo L=1 - -

Here M is the multiplicity of {in}
(i) the discrete parameter processes [§£(m)} (£=l,..,M) have orthogonal

increments and: .are mutually orthogonal;

M
(ii) H(xm) = z $H(§i;n) for each n,
i=1

0 M
(111) bL(n-m) e o with E Ilb:(m)||2 fi[5§(m)] <0. .
m=-oo £=1

The number M is the multiplicity associated with the Stochastic process.

Theorem 9.3 Let [it] be a continuous parameter weakly stationary pro-
cess with values in 0 satisfying the assumptions of Theorem 9.1 and condition

(C). Then for each t with probability one,

M t

(9.12) _x_t=Z f Fn(u-t)d§n(u).
n=l 4:6

In this representation

(1) the gn's are mutually orthogonal and each 5n is a homogeneous orthogonal

-51-
randOm’settfunction (with Lebesque measure p for its measure function),

M
(11) H(x;t) = Z ®H(gi;t) for every t,

i=1

(iii) M is the multiplicity of the process, and

(iv) Fn(u-t) e ch2(u) (n=l,...,M) such that

M 0
Z I lan<U>ll2du(u) <..
n=l -m,

CHAPTER I; APPLICATIONS TO N-PLE MARKOV PROCESSES

Wide—Sense Markov Processes

1. Preliminaries and notation. Throughout this chapter a q-dimensional
second order stochastic process will be denoted by [5%] (-m < t < m)

where for each t,'x is a column vector (x1(t),...,xq(t))*. Associated

t

with [x will be the following spaces:

t}

(i) The space of the process up to t, L .xgt) is the subspace€§{xi(T), T é t]

2(

of L2(Q) generated by the random variables (xi(T)} (T 5 t, l = 1,2,...,q)
L2(§; 4n) the intersection of L2(x:t) for all real t and L2(x) is the

smallest subspace of L2(Q) containing all L2(x;t) for each t.
(ii) For the processes with mutually orthogonal increments or those which are
wide-sense martingales the notation H( ; ) of Chapter I will be used.

(iii) will denote the projection onto.A( .

PM
Definitions of deterministic and purely non-deterministic processes are the
same as in Chapter I. The following definition of a q-dimensional wide-sense
Markov process is due to F. J. Beutler ([1]).

Definition 1.1. A q-dimensional process {xt} (-m < t < + w) is wide-sense
(t),

Markov if for each i (i = 1,2,...,q) P xi(t) = P

was) {x1<s>,...,xq<s>}"i
(s < t).

For our purpose we need the following definition of a q-dimensional wide-

sense martingale. The notion of a wide-sense martingale for q = l is due to Doob

(81.12.1620.

Definition 1.2. ut-process is called a wide-sense martingale if for each k,

 

= so. = . . . é a
(k 1,2, ,q) PH(g;s) uk(t) uk(s) w1th probab111ty one for s t
The assumption (D) given below will be used throughout this chapter.
(D.l) 'xt-process is continuous in q.m.; i.e., each component process {xi(t)] is

continuous in q.m.

-53-
(D.2) For all t,s real the covariance matrix function F(t,s) is non-singular.
The assumption (D.2) and the definition of wide-sense Markov process imply

q
('1
that PL (x;s)xi(t) =
2-
J=1

aij(t,s)xj(s), where the matrix A(t,s) = (aij(t,s))

-1
is given by A(t,s) = P(t,s) P (s,s) for 3 § t. It is easily verified that A(t,s)

is non-singular for each s,t (s g t). The function A(t,s) is called a transition
matrix function and is defined only for s g t. Beutler [1] has the following
theorem which furnishes an operative criterion for verifying the wide-sense Markov
property.

Theorem B ([1] Theorem 2). The following statements are equivalent

(1) (fit is wide-sense Markov

(2) For 3 § t é u A(u,s) = A(u,t) A(t,s)
(3) With A(t,s) = F(t,s) P- (s,s) for 8 § t é u A(s,u) = A(s,t)A(t,u).

In the case of stationary processes A(t,s) = B(t-s) (3 § t). Hence B(')
can be considered as a function on non-negative real numbers. As will be shown

in Theorem 2.2, one can easily characterize wide-sense Markov processes in terms

of the transition matrix function B(-). we remark that (t a O) B(t) = A(t) =

-1
P(t,O) r (0,0).

-54-
2. Characterizations g£_the wide-sense Markov processes. We first consider the

non-stationary processes.

Theorem 2.1. If 'Et (-m < t < + w) is q-dimensional stochastic process

 

satisfying (D) then it is wide-sense Markov if and only if xt =~§(t)u_t with

probability one, where for every t, E(t) is a non-singular q X q matrix and 2t

process is a q-dimensional wide-sense martingale with H(g3t) = L2(x;t).

Further for all s,t the matrix J(t,s) = Qui(t) uj[s)) is non—singular.

Proof. Sufficien_y. Let fit = E(t)ut where E(t) and (gt) are as described

above. Then for s g t 1f we donate by PL2(§:S)§t the column vector
. * o s o s - s -
(PL2(§53)§j(t),o..’PL2(§55)§q(t))_ we have by def1n1tion of a Wide sense mart1n

gale, with probability one,

PL2(£;S) it = PL2(§;S) E(t)}it = PH(_U.;s) 14th,: = 31,1th
Since Es = Efl(s)§s with probability one, we obtain that the transition matrix

function A(t,s) = E(t) E}(s). The proof of sufficiency is now complete by
appealing to Theorem B, (2).

Necessity. Let xt-process be wide-sense Markov. Then denoting by A(t,s) the

 

transition matrix function we recall that for 3 § t

(2.1) EL2(x;s)§t = A(t,s)xs with probability one and for 3 § t g u

(2.2) A(u,s) = A(u,t)A(t,s).

Following Hida, we now define for every real t the function

E(t) = A(t,so) if s _s_ t
. _1 .
nmA (so,t) if t < s

where so is a fixed real number. we shall show that for all s,t (s < t) real

(2.3) A(t,s) ‘= nails) .

-55..

First of all if s < 30 g t then (2.3) is a restatement of (2.2) i.e.,

A(t,s) = A(t,so) A(so,s). Secondly, if 50 § 3 < t, from (2.2) we have

. -1
A(t,s) A(s,so) = A(t,so) i.e. A(t,s) = A(t,s) A (s,so)
giving A(t,s) = §(t)@-1(s). Finally, if s < t < 30 we get A(so,s) = A(so,t)A(t,;

And hence A(t,s) = i(t)§-1(s). The fact that E(t) is non-singular follows from non-
singularity of A(t,s) and the definition of g(t). Therefore from (2.1) and (2.3),

for s < t

(2.#) Pi (X'S)£t ="‘Ir(t)\F-1(s)>_cs with probability one.
2-’ ‘ “

If we define 2t =---'T).r"1(t))_c_t , then
(2.5) L2(x:t) = H(E5t) for every t.

Thus from (2.h) and (2.5) we get

(2.6) §H(u's)'3t = gs(with probability one).

Since F(t,s) = fi(t)J(t,s)§§(s) and 'fixt) is non-singular for every t, we have
J(t,s) non-singular.

Corollary. If the continuous parameter process x; is continuous in q.m. then

I:

so is 2. and E(t) is a continuous function in the sense that each element ofjfi(t)

t
is continuous.

Proof. If rij(t,s) denote the elementseof-F(t,s) then by the continuity in q.m.

of the process {x.(t)} we get for every fixed: 3 lim r,.(t,s) = r,,(t ,s);
1 t—at 1j 13 0'
o

i.e., lim F(t,s) = F(to,s) . But by Theorem 2.1, F(t,s) = E(t)J(s,s)§*(s)
t—at
o

(for s'< t). Hence ‘Ext) = P(t,s) [J(s,s) Eé(s)]-1 as a function of t is
continuous (note that s is fixed). To prove continuity in q.m. of 2t; consider

111(t,s);glui(t) - ui(to)|z = jii(t’t) - Jii(to’to) (to g t). Now

-56-

J(t,t) = {)'-l(t)l‘(t,t)[1T/'E(t)]-l and hence we get lim J(t,t) = J(to’to)° We
” "' t—> t
0

therefore have linlajui(t) - ui(to)|a = O. A similar argument gives
tit
o

limEZIu,(t) - u,(t )l2 = 0, thus completing the proof.
t1~t 1 1 o
0

We now study stationary wide-sense Markov processes. In this case

(& [xi(t + h)- szt)]) for any h is a function of h. We denote it by R(h). By

Theorem 2.1 and properties of wide-sense martingale it is easy to see that for

every h g 0 and t real

(2.7) R(h) = g(t + h) J(t,t) g(t) .

Let h = 0, we get

(2.8) 12(0) = g(t) J(t,t) 31*(0 .

With t = 0 in (2.7), one has

(2.9) R(h) = 101) J(O.o) 1*(0).

Relations (2.7) and (2.9) imply for h e 0 and t e 0
(2.10) R(h) = R(t + h) [J(0,0) Won'l J(t,t) We) .
From (2.10), (2.9) and (2.8) for t, h e 0,

(2.11) R(h) .... R(t + h) R'1(t) R(O)

With R1(t) = R(t) R'1(0) (2.11) reduces to

(2.12) R1(t + h) = R1(t) R1(h) .

we prove the following theorem.

Theorem 2.2. If {gt} (- m < t < +»w) is a q-dimensional stationary process

 

satisfying assumption (D) then it is wide-sense Markov if and only if the trans-
ition matrix function B(t) = etQ for every t e 0 where Q is a uniquely determined
constant q x q matrix, none of whose eigenvalues has positive real parts.

Proof Necessity. we have already shown that for R1(t) = R(t) R-1(O), equation (2.12)

 

holds. Further from (D.l) it follows that R1(t) is a continuous function and

-57-
therefore R1(t) = etQ (t a O) is the solution of (2.12), where Q is a q x q
constant matrix. The assumption (D.2) in addition implies that R1(t) is non-
singular and hence Q is uniquely determined by Rl(t). We recall that
B(t) = R(t) R—1(O) for t e 0. Hence B(t) = etQ (t e O). The statement about
the eigenvalues will now be proved. Observe that for any non-negative integer n
B(n) = [B(1)]n. Q has an eigenvalue with positive real parts if and only if
eQ (=B(1)) has an eigenvalue K with |x|> 1. Suppose that there is an eigenvalue A

with [AI >'l. Then

(2.13) lim Sup|x(t)| :1» where x(t) is an eigenvalue of B(t) corresponding to
t—)oo

the eigenvalue k of B(l). But

IIA

moi e tr<B<t>B*(c)>

tr(R'1(0)[R’1(0)] trgR(t)R*(t))

tr<n'1(o>[n'1(o>1* (Z lxi<o)|2)2.
1

Il/\

Therefore for all t |x(t)| is bounded contradicting (2.13).

Sufficiency. Clearly A(t,s) = B(t -s) = e(t-S)Q(s g t) satisfies Theorem (B) (2).

 

The proof is now complete.

Theorem 2.2 is proved by Doob in his important paper [2] on elementary Gaussian
processes. One of the central problems of his paper is to characterize purely
non-deterministic stationary Gaussian Markov processes. we shall give an alter-
native proof of this result (in our notation) based on Theorems 1.5.1 and 2.1.
First, we state Doob's theorem in its original form for the sake of comparison
with our derivation given in Theorem 4.2.

Theorem D (Theorem 9.3 [22]). If‘xt is a continuous parameter non-degenerate,

continuous in q.m., purely non-deterministic, Gaussian Markov process then

1’.
(2.1h) IEt = ”f e(t-u)Q Sd§(u) where (i) Q, a q X q matrix, having no positive

-®

-58-
real parts is uniquely determined by R(t) (ii) {i(u)} is a Gaussian fi-processes
(see [2] p. 263) with covariance matrix lu-vl U where U is a diagonal matrix
zero and 1 over diagonal (iv) R(t) = 28QR(6) for t e O R(-t) = R(OSeFQf
(v) the matrix Q furnished a solution of the prediction problem (vi) the matrix
S is uniquely determined and measures the dispersion of xt-process from its

. . U .
predicted value i.e., the variance matrix of §u+t - e git 1s equalto

R(O) - duQR(O)euQ* ._ u82 as u —9w .

Clearly the assertions (iv) (v) (vi) of Theorem D follow from (2.1%). Hence
it suffices to obtain the representation (2.14) by means of our method.

In concluding this section we point out that the vector-valued stochastic
integral ‘[‘F(u)d§(u) where E(u) is a q-dimensional Efprocess is defined by
Doob ([2], p. 263) for continuous matrix-valued functions F. A complete and
rigorous definition of vector-valued stochastic integral is to be found in the
recent paper of M. Rosenberg [7]. This definition together with an explanation

of the notation employed is given in the next section.

-59-

3. Vector Valued Stochastic integrals. If H is a Hilbert-space then H(q)

 

denotes the space of all q x l vectors h with hi 6 H. In H(Q) is intro-

, q
duced norm |||h| l I2 = Z llhil |2H and an inner produce given by the Gramian

1:1
(q)

matrix [1132'] for any h, _h_' e H(q). A linear manifold in H is a non-void

subset M“ of H(q) such that if h, h' e c/“then All + B_h_' ea“ for all q x q

matrices A,B. A subspace of H(q) is a linear manifold closed under the top-
ology III III. For properties of the Gramian and further structural questions
we refer the reader to N. Wiener and P. Masani [19].

Let P,Q be any q x M matrix valued functions. Then we say that (P,Q) is
integrable with respect to an M x M hermitian matrix valued measure I? if

the matrix function PF'Q'X' is integrable with respect to the tri . We then

define I pde* =fP f'Q-X'dtr

P is said to be square integrable [f'] if tr (defp-X') is finite. If we denote
by (£é(F) the class of all measurable P which are square integrable with re-

spect tofwhere functions P,Q with [P(u) - Q(u)) F'(u) = O a.e. [trf] are

L—

identified. 060(3) has the‘ norm HPI loC ( \ 2 tr IPd P*" and gramian

[P, 01.0)] ( )v'e [Pd (2*, for 2112.0 6 OC( 3,. , (1 ‘1,

f) .
‘— (-

We shall call _§_ an orthogonally scattered random vector valued measure

of dimension M on the real line if for each BéCBJgB) e Lémm) and for A,B 563

[_§_ (AL: (B)] = P(A(\B) wheref is a hermition matrix valued measure and

B the class of Borel sets on'the real line . With this set up Rosenberg de-

-60-
defines fP(u)d$_(u) for P e¢£2(f’) in the same way as Doob does for M = q
0’
(See J. L. Doob [3] p. 596). Further if one denotes by t[2(§) the subspace of

of Léq)(fl) generated by [5(B),B e B} with q x M matrices as coefficients then

we have the following [See [7] Theorem 11.6]

Theorem R. The correspondence P -) I Pd _§_ is an isomorphism from

42(2) to 052(1).

Remark In the above discussion q and M are fixed positive integeaQwith

 

(M é q) and the space °C2(F) is complete in the norm defined.
a!

-61-

h. Purely nonedeterministic wide-sense Markov processes. We first prove a

 

representation for the non—stationary case.

Theorem h.l If xt is a continuous parameter purely non-deterministic

 

process statisfying assumption (D) then it is wide-sense Markov if and only if

q M t
xi(t) =2 2 flak“) hkj(u)dzj'(U) where {iik(t)}(i,k=l,...q) are
k=l j=1 4»

elements of a non-singular q x q matrix E (t), hkj(') for each j belong to

to L2(f3) with zi’Fj having the same meaning as in Theorem 1.2.2, M is the

M t
multiplicity and for ever k,:§: J) Ihkj(u)|2de(u) is finite. Also H(E:t)=H(x:t)
j=1 ...00

for t.

 

Proof, Necessity. As stated in Theorem 2.1 it = E(t):t with L0(x;t) = H(E;t).

Also from Theorems 1.2.2 and 1.3.1 we have a representation for xt-process with

L2(x}t) = H(33t). Since 3t is a wide-sense martingate and H(E;t) = H(ugt)

M t
we have uk(t) =2 f hk.(u)dzj(u). The result now follows, since
J
q j=1 -00
xi(t) = :E: Ԥ;k(t)uk(t) for all t and E(t) is a non-singular q x q matrix.
k=1

M t
Sufficiency. Define uk(t) :2 )[ hkj(u)dzi(u). Clearly xt=_11_{(t)_1_1_t.
1 -w

Therefore, to complete the proof it suffices to show that fit is a widetsense
martingale. We note that since E(t) is non-singular L2(x;t) = H(Ejt). As we

are given that L2(x:t) = L2(§;t), we get L2(u;t) = L2(£;t) for every t. Con-

sider now for s < t,

M t
PH(u;s) (uk(t) - uk(8)) = PH(z;S) [:E: )’ hkj(u)dzi(u)] = O,
_ » — 1 s

-62-

where the last inequality follows because 2 's are mutually orthogonal processes

J

with orthogonal increments. The proof is now complete.
For stationary purely non-deterministic processes we recall that M, the
multiplicity of the process does not exceed q [See Theorem 1.6.2]. Also from

Theorem 1.6.2 and the definition of vector valued stochastic integrals we have

t

(11.1) 2, = fut-mags

-00
where F(t-u) is a q X M matrix-valued function and E(u) is an M-dimensional
orthogonally scattered measure. Also we have L2(x;t) = L2(§;t) for each t.
Using representations of Theorem 2.1 and an argument similar to that of Theorem

4.1 (Necessity), we obtain that 2t = / H(u)d§(u) where H(u) is a q X‘M matrix

-00

function and hence

t
(11.2) 2, = fps) H(u) dgu) ,

with L2(x;t) = H(ggt) for each t. We have the following theorem:

 

Theorem 4.2. Let xt(-m < t < + m) be a stationary q-dimensional process
satisfying assumption (D). Then x{ is wide-sense Markov and purely non-determin-

istic if and only if

t
(4.3) xt =‘/. e(t-u)QCd§(u) , where

-oo
(i) Q is a q X q constant matrix with properties described in Theorem 2.2
(ii) C is a q X M constant matrix where M equals the rank of the process

(iii) gt is an orthogonally scattered random measure such that

[g(B), g(B')] = p(B(\B')I where B, B' are real Borel sets, p Lebesgue measure

and I is an M X M identity matrix. Further L2(x3t) = L2(§;t).

Proof. Necessity. From (4.2) and stationarity we have a q X M measurable

-63-
matrix function G(t-u) such that for u é t
G(t-u) = §(t) H(u)
Since H(u) is given almost everywhere, if it is not defined at the origin,

completing its definition at zero we obtain for t a O

G(t) =jf(t) H(O)

However since R(t) = etQR(O) from (2.8) and (2.9) we get for t e O
(4.1) G(t) = etQ w(0) H(O) i.e. G(t-u) = e(t‘“)Q C(u s t)

where C = E(O) H(O). Hence from (4.2) and (4.4),

... jte(t"”)Q Cd_§(u) with L2(_x;t) = H(Efll)

-CD

it
t

Sufficiency: If we denote by 2t = J'euQ d§(u). Then obviously'ut is a q-dimen-
-oo

sional wide-sense martingale and therefore from Theorem 2.1 it follows that};t

is wide-sense Markov since etQ = R(t)R-1(O) is invertible. The proof is complete
if we show that fit is purely non-deterministic. But this is obvious from the

fact

ll

[VP/1:

nH(§;t) mugs) =/\ ®H(gi;t) = (0)
t t t
1

which follows because {§i(t)] (-m < t < + m) (i = 1,2,...,M)

are mutually orthogonal processes with stationary orthogonal increments.

Since the Gaussian wide-sense Markov processes are Markov processes, Theorem
4.2 reduces to Theorem D. The Et-process occuring in the expression (4.3) is an
M-dimensional orthogonally scattered measure where M is the rank of 5t as defined
by E. G. Gladyshev [4]. Its covariance matrix function,A(u,v) is of the form
Iu—VII where I is the M X M identity matrix. Therefore Theorem 4.2 renders a

more precise form of Theorem D (ii).

-64-
N-PLE MARKOV PROCESSES
In the study of representations of N-ple Markov processes we require
analytic conditions for prOper canonical property.

5. ‘Ag analytical characterization 2£.2 proper canonical representation.

 

Henceforth we shall assume Mié q. Further we denote by 0C2(§;t) [a (.) is
a q-dimensional orthogonally scattered vector measure] the subspace of L(q)(Q)

generated by [g(B), B a Borel subset of (-w, t]} with coefficients q X M

matrices.

 

Lemma 5.1. H(q)(£;t) =og(_z_;t) (-oo < t < + co).
Proof. A typical element of H(q)(§5t) is a column vector (yl,...,yq)* =

(y1,0,...,0)* + (0,y20,0,...,0)* + ... + (0,...,yq)* where yje H(ggt).

It suffices therefore to prove that for each i, the vector (0,0,...,zi(B),O,...,O)*

for each Borel set B belongs to (£2(55t). But this is obviously the case as is
seen by taking a diagonal q X M matrix with unity in the ith place in the diagonal
and zero everywhere else. The fact that ,Lé(§;t)c: H(q)(§;t) follows by observing
that for each Borel set B in (-w,t] and q X M matrix A .A£(B)eH(q)(§;t).

The following is a direct extension of Theorem 1.7 of Hida [5], to q-dimensional
processes with Mlé q. we shall denote a representation for such processes by
{F(t,u),d§(u)} where F(t,u) is a q X M matrix function and 5(3) is an Mkdimensional
orthogonally scattered random vector measure with components zi(B) (i = 1,2,...,M).

The notion of a proper canonical representation of arbitrary multiplicity M
has already been introduced in Chapter I. Under the assumption Mié q we give

necessary and sufficient analytical conditions for a proper canonical representation.

Theorem 5.1. A canonical representation {F(t,u),d5(u)] is proper if and

 

only if for any real tO

t o
(5.1) ij(U)d (u)F*(t,u) = O for t E to implies P(u) = O a.e. [E]

-65-

where is the hermitian M x M matrix valued measure (’(B) = [E(B)’ 3(8)]

N

and P(u) is a square integrable q'X M matrix-valued function on the real line.

Proof Sufficiency. Let (5.1) hold and let to be such that H(§;to) + L2(x;to)
and we know that L(q)(x‘t )C: H(q)(2°t ) Therefore there is a V + O in
z _’ 0 _’ 0 ° 3 _ _

H(q)(§;to) such that [ygxt] = O for t g to . Consider now H(q)(§5to) =J:2(£5to).

t
0
Then by Theorem R of Section 3 we have 2 =./ P(u)d (u) + 0 such that for all
-00
t
t(§ to), ‘/ P(u)d (u) F*(t,u) = 0. By (5.1) we get P(u) = O a.e. [P]
-00

contradicting 1+ 9.

Necessity. Suppose that H(£;t) = L2(§,t) for all t, and let tO be a real number

such that

t
(5.2) 1’ P(u)d (u) F*(t,u) = 10 for every t é to.
—00
Observe that since from the proper cannonical property Léq)(x3to) = H(q)(gito) =
t

oC)(Z;t) the vector V=f
t._ 0 _

-oo

OP(u)dg(u) belongs to Léq)(x5to). But (5.2) implies

that L!’-§t] = O for all t é to. Hence y“: 9 giving P(u) = O a.e. [P].

This proves the theorem.

The above criterion will be useful in our discussion of N-ple Markov processes.

-66-

6. Finite dimensional wide-sense N-ple Markov processes. In the definition

of vector-valued wide-sense N-ple Markov processes we require the concept of

the projection on a subSpace of Léq)(§) . We recall here a lemma due to

N. Wiener and P. Masani [10], which proves the existence of the projection

of an element h_and gives its structure. The notation used is that of Section 3.
fl.g(Lemma 5.8 [10]). (a). Ifmis a subspace of H(Q) there exists

a subspace e/(Cof H such that YRr-uflq), wheres/“((1) denotes the Cartesian

product afl®d4l69 @c’gvith q-factorsw/(C is a set of all components of all elem-

ents in]H;. (b). Iflw;is a subSpace of H(q) and heH(q), then there exists

a unique heWk such that ”h - h'HH(q) < ”h - g] [H(q) for all ge772. For

this h', h: = Puhi,,§2being as in (a). .An element hf satisfies the preceding
condition if and only if h_- hfi]&:where orthogonality is in the sense of

the Gramian. (c). IfllE,7E are subSpaces of H(q) andlflzChf, then there exists
a unique subSpace 77'CC1E such that )2: =-. ’1'7'5'6 Wt’and ha is orthogonal to’m ' .
Parts (d) and (e) of Lemma 5.8 of [10] are not given here because they will

not be referred to. Following Wiener and Masani we give

Definition 6.1. The unique element hf of Lemma WM (b) is called the ortho-

 

gonal projection of h_ontolfltand is denoted by QEDQ).

Extending usual idea of linear independence, we give following definition

of linearly independent vectors h1,,h2,...,hN€H(9).

 

Definition 6.2. The vectors hieH(q) (i = 1,2,...,N) are linearly indep-
. (q). . _
endent in H if for any q X q matrices A1,...,AN , ZAihi _'Q and Aihi
is different from the zero element of H(q) for at least one i implies that A1

are zero matrices.

Now we define a q-dimensional real continuous parameter wide-sense N-ple
Markov process. For one-dimensional continuous parameter Gaussian processes

the definition is due to Hida [5] and for discrete Gaussian processes the

-67-

definition goes back to Doob [2].
Definition 6.3, We say that a q-dimensional continuous parameter process
is wide-sense N-ple Markov if for any sequence [ti] of N-real numbers
. q .
(t1 < t2 < ... < tN) and for tO é t1, the vectors (xtilLé )(§,to)) are

linearly independent in L

(q) . ‘
2 (x,to) and the vectors (xtilLéq)(x5tO) are

linearly dependent if i = 1,2,...,N+l and tNkl>’tN°

We now proceed to the extension of Theorem 11.2 of Hida, to obtain a
representation for a q-dimensional (not necessarily stationary) wide-sense N-ple
Markov process using the theory of Chapter I.

Lemma 6.1. Let t and s (s < t) be any real numbers. If F(t,s) is

 

non-singular, then the vector (xtlLéq)(§5s)) is non-degenerate, i.e., its

covariance matrix is non-singular.

Proof. From Lemma WMwith Wk: Léq)(x;s) we get (xtlLéq)(x;s)) is the

column vector (PL (X's)xl(t)’°m PL 2(_5 s)xq (t)). First we observe that none
2_)

of the elements PL2(§5s)xi(t) (i = 1,2,...,q) can be zero; for otherWise

’ = 2 = - . = f 11 ' = 1,2,...

(11.0.8) ..(xi(t)xj<s>> tag“) P12(x;s)x.<t>> o 1 .q,

contradicting the non-singularity of F(t,s). If the vector is degenerate then

for some i, P x (t) :2 .aij PL 2(x s)xj (t). Also

L2(X;S)1j+i_]

PL w<x s)xi (t) 4 O . Hence there is at least one j such that aij + 0. Now

5(xi(t)"k(5” = ”i 5'0“ W3)? 3“” ””8 E(t’s) =11131-18PL2(?£;S)XJ'(t)xk(S)

= (t, s), (k=1, 2,. -,q)
1+3 aij jk

-68-

This contradicts the non-singularity of F(t,s) and the lemma is proved.

From the definition of wide-sense N-ple Markov processes it follows that

if {Si} (31 < 32 < ... < SN) is a given sequence and T>SN then for each

30 § 81’ there exist q X q matrices AIST; 51,...,sN) such that

(x [L(q) X' s a N - (Q) . .
"T 2 (.3 0)) kglAk(T,Sl,o.-,SN) CESkILB (x,so)). Taking a sequence

{tj] (tN>tN_1 ... > tl > SN) we have

(6.1) (§t_1L§q)e; so»
J

N
kil Ak(tj; 31"°°’SN) (ésleéq)(§3 80)) -

th

A
Denote by A(E, g) the qN X qN matrix having Ak(tj; s ,...,sN) for its (k,j)

l
(q X q) block matrix, (k,j = 1,2,...,N). Then we have the following lemma.

Lemma 6.2. If xt(-w < t < +'m) is a q-dimensional wide-sense N-ple

 

Markov process satisfying assumption (D.2) then R(E, s) is non-singular.

Proof. We first prove that for any sequence {ti} (tN > tN-l > ... >t1 > $0)
the set
(6.2) {P . x.(t.)} i . 1,2,...,q, j . 1,2,...,N
L2 X’SO) 1 j ‘
is linearly independent in L2(x). If not, then there exist aij not all zero

such that

. 13

iZJa.. yi(tj) - 0 where we write yi(tj) = PL2(x;sO)xi(tj) ,

(so being fixed throughout the argument). Since from Lemma 6.1, for no pair
1,3 yi(tj) a O, letting aij + O, we have
(6-3) y (t ) - Z*'b y (twig
i j km k -’
k,m

where 2* denotes the summation over all k,m (k = 1,...,q; m = 1,...,N)
kym

<-

-69-

such that no pair (k,m) = (i,j); though b depends on (i,j) we do not indicate

km

it here in order to keep the notation simple. Also since yi(t.) + O
J
(Lemma 6.1) there is at least one (k,m) + (i,j) such that bkm + 0. ‘We now

consider the following two possibilities.

Case I. Suppose bkj - O for all k(+ i).

Then (6.3) has the form
* H
(6.h) yi(tj) = #2 bkm yk(tm) .
,m
(ml-1')
Consider now q X q matrices AL (? = 1,2,---,N) SUCh that Aj = ((

(j) ‘ (t) (4')

£1)

nP

M: $12 =1

and anp - O otherW1se; for f + 3 A1 = ((ahp)) Wlth aip = - bp£ for
£31) ' ‘N
p a 1,2,...,q and = 0 otherwise. Then from (6.4) we have 2 A. y_ = O ,
np L 1 L tL -

I
. (<19 .‘ ~
Aiztj+'g and Aj is not a zero matrix, i.e., the vectors (§E£|L2 (x,so))

Q

(L - 1,2,...,N) are linearly dependent. This contradicts the definition of the

wide-sense N-ple Markov process.

Case II. There is a non-void subset JCI[1,2,...,q} such that bkj + O keJ (i<$ J).

a;
(6'5) yi(tj) - 1363‘] bkj yk(tj)

is zero then for v = 1,2,...,q we have E>[yi(tj) yv(tj)] ==kiJ bk§{?k(tj)7i+fi y

But this contradicts Lemma 6.1. Hence the element given by (6.5) is not zero.

We now rewrite (6.3) as

* 1‘ ..
(6.6) yi(tj) - kEJ bkjyk(tj) = 15m bkmyk(tm)
(L)

Now introduce the matrices AL = ((ahp)) where

. . (1) ~ (1') (j) . .
(1) t = j, a1p a - bpj (peJ), a1].- a l and anp a O otherw1se,

-70-

.. , (a) (a)
(11) p + j alp = -bp£ (p = 1,2,...,q) and ahp a 0 otherwise.

Then (6.6) becomes

(6.7) A X :9.
g; f tz

Further Aj 1t +'g since the element in (6.5) has been shown to be non-zero.
1

As in the concluding part of Case I, these facts imply a contradiction of the
N-ple Markov property.

Thus we have established the linear independence (in L2(x)) of the set (6.2).
by a similar argument the set {yi(sk), i a 1,2,...,q, k = 1,2,...,N) is
linearly independent in L2(x). Also we can write (6.1) as

(y1(t1), y2(t1),---.yq(tl),---, YICCN),..o,yq(tN))*

A

(6.8) - *
=‘A(E’ §) (yl(sl), y2(s1),...,yq(sl),...,yl(sN),...,yq(sN))

A .
Hence A(E, g) is non-singular. This completes the proof of Lemma 6.2.
We now state the main result of this section.

Theorem 6.1. Let {gt} be a real continuous parameter purely non-deter-

 

ministic q-dimensional wide-sense N-ple Markov process with multiplicity Mlé q

and satisfying the assumption (D). Then

N t._
(6 9) at = 'é1 _f 11(t) Gi(U)d§KU)

where for each i, E;(-) is a q X q matrix-valued function such that for any N
points {ti} (t1 < t2 < ... < tN) the qN X qN matrix with (i,j)th q X q block
matrix [ii-(tin is non-singular and Gi(u) is a q X M matrix valued function
in L2(f) (:(B) -:§(B), 2(3)] LéM)C§) ). The functions {Gi(u)] are linearly

independent in L2(f;(-W,t]) i.e. for each C, and for any q X q matrices

N

A1, .‘l'AiGi(u) x o (Gi(-) restricted to (-w,t]) and AiGi(u) + o for

l

-71-
at least one i implies Ai = 0 for all i.

Proof. By Theorem I.2.2 and Theorem I.3.1, Kt has a prOper canonical repres-
entation of multiplicity M. Since Mlé q this representation can be expressed

as {F(t,u), dgj where F(t, -) is a q X M matrix-valued function in L2(V).

Let (ti) be a sequence of distinct points with tN > tN-l > ... > t1

and T'> t . Then by the wide-sense N-ple Markov property for all 0 § t

N 1

there exist q X q matrices (Aj(T;tl,... not all zero such

’tNNj =1,2,...,N

I

that

IIMZ

33c j 1AJ'(T3t1’°"’tN) 354:5 L Léq)(§;0) (a g t1)

where orthogonality is in the Gramian sense. Hence for all 0 § t1, we obtain

N

G
O I: [£13 2 A.(T;t1,...,t )étj’ 350,] =-oj;[F(T,u) -

N
-X-
A. ;t ’00.,t Ft.) (11 F ,
NJ .3 301- 1 NM, 101,01) (...)

j 1
Hence by Theorem 5.1,
N
(6.10) F(T,u) = j)ilAj('r;t1,...,tN) F (tj,u) (p;(-w, tl]]’
since the representation (F(t,u), d§(u)} is prOper canonical. (In (6.10)

”-(-m, t1]] means almost everywhere [E] on the interval (-w, t1].). If we have

\ J
.1,

A

another sequence {sk} (tl > sn > ... > s1) then from (6.10) we obtain

N —.
(6.11) F(tj’u) = k:lAk(tj;sl,...,sN) F (sk,u) E}; (—w,s ]] .

Now from the definition of Ak(tj;sl,...,s ) (k,j = 1,2,...,N) and Lemma 6.2
the matrix ‘A(E,§) defined there is non-singular. Let fi(§’5) = A-l(t,s).
From (6.10) and (6.11) we deduce

(6.12) F(T;U) = {Zkéj(T;tl"'"tN)Ak(tj;Sl"°"SN)F(Sk’u) =.iAk(T;Sl,...,sN)F(sk,u)
J:

[3; (~oo,sl)].

"72-

Now (6.12) implies that

i (iAj(T;tl’°'°’tN) Ak(tj;sl,...,sN) - Ak(T;sl,...,sN)) F(sk,u) = 0 [i}(-w,sl]] ,
which can be rewritten as (with sequence (ti) [Si] and number T fixed)

(6.13) E(sz(sk;u) a 0 [i;(-w,s )] .

a

Consider (x8 lLéq)(§551))° Since by the canonical prOperty
k
M s

q
1 (q)
P . x 2 E f.. s ,u dz. u , we et x L x;s a
L2(§581) x1(sk) i=1 j=1 _i 1J( k ) J( ) g L-sk| 2 C— l)

S
-i1F(sk,u)d§(u). Now if in (6.13) CkF(sk,u) . o q:;(-e,s1]] and ck + o

"N

then we get C (x8 lLéq)(§331)) =.g. This contradicts Lemma 6.1. Hence Ck is
k

k

a zero matrix for each k by the wide—sense N-ple Markov prOperty and (6.13).

Hence

N
(6.1h) Ak(T;Sl,...,SN) - JEilAj(11121,...,tN) Ak(tj;sl,...,sN) .
If ¢%(T;§) denotes q X qN matrix with q X q block matrices Ak(T;sl,...,sN), viz.,

gf€(T;§) a {A1(T;sl,...,s ),...,AN(T;sl,...,sN)} then (6.14) can be expressed as

(6.15) (.4 (mg) worst) Km.) .
Recalling that ‘P(§,t) = A-1(t,s) we define

(6-16) 58(1) =64..('rs$) g(sfi) ~

' ' < ... < ' < ... ...
If 31 < 52 SN 31 < SN < tl < < tN_< T then we get

£810?) = 01(133') g(g'fi) = —{ (ng') P(s’ffi) fi<§RE> since
A(t,s') - A(t,s) A(s,s') from (6.15). Hence (6.15) and (6.16) give 3;.(T) = $g(7).

Let/2 be the set of all sequences § - (Si) where 51 < 52 < ... < SN,< T ,

T being fixed throughout. For any two sequences 5, s' in 2? define the relation’fi

-73-

as follows: 8' s if s ' < s , t '3 ea t t a . . .
.. ‘R,v N l I 1 sy 0 see h t <\1s a direction on

the set/5:0f all such sequences. Further for each T the limit of the net

2; ,

(WS(T), sei} exists from the fact, proved above, that for s"<\§'i T

41—” c: ’ c:

wS(T) - ES,(T). Denoting this limit by w(T) we find from (6.16), (6.15) and

"N AI —

the non-singularity of A(t,s) that the qN X qN matrix {Ei(tj)} of the theorem
.. ' A

is non-singular where E;(T) denotes the ith block q X q matrix of E(T).

We write equation (6.10) as

l r f . ,-.
(6.17) F(T,U) =~y{(7;§) i’(t;u), [_; (-°°,t l]
where '5 (t,u) denotes the qN X M matrix (F(tl,u),...,F(tN,u))*. Let C(u,s,£)

be the qN X M matrix IL-1(§,t)f?(t,u). Then (6.17) takes the form

A A
(6.18) F(T,u) a w(T) G (u,s,t) a.e. R;(-m, t1]].
Let {ti'} (i n 1,2,...,N) and [sj'} (j - 1,2,...,N) be sequences in.§ with
s"< E) then

2. A
(6°19) F(T,u) ’ W(T)G(u,s',t') [ 5(‘m:t 1] -
Now from equations (6.18), (6.19) and the non-singularity of [§d(tj)) we obtain
E(u Ws t) = C(u,s',t'). [(; -w,t J . Hence we may set

(6'20) C(u,§j,t:) = é(u), say, for alls'1F'6 7

Hence from (6.18) and (6.20)

N
F(T:U) = Z 11(T)ci(u) [ ;(-w,t 11
1:1 -' _' .
for each t1 < T. Also lim I|F(tl,u) - F(T, u)||2 Q) = 0 .
t1-—> 7
Therefore

N
F(T,u) a z .i'i(T)Gi(u) L:;(«n,T]].

i=l‘-

_’l. ’1.

-74-

Thus (with T replaced by t) we get

N t
lit - .21 IEiCt)Gi(U)d.z_(U) .

To complete the proof we observe that for (u é t) F(tgi)==z§&(t)Gi(u)£LF(tj,u)j
are linearly independent in cC2(E) for tj >’t (j a 1,2,...,N) and that the
matrix [E:i(tj)] invertible. This implies that {Gi(u)} restricted to (-m,t]
are linearly independent inr62(i) for each t.

Remarks. 1. If we define for each i
. t . . .
(6.21) uél) = -f Gi(u)d§(u) then uél) - 2&1) i. Léq) (2‘1);s) (s < t) ,

(orthogonality again in the Gramian sense). Hence gél) is for each i is a wide-

sense q-dimensional martingale and
N .—
(6.22) 1.12.}: Ei(t)Gi(u)
1:1
q (1)
Furthermore since L2 x;t)g:G§ED]H(u_ ;t))C:_L2(§3t) n L2(x3t) from (6.21),
a:
(6.22) and the proper canonical property, we get

N (.
(623) new) =1,-.21 u at. ”an
inl ,

If N = 1, this reduces to the representation of Theorem 2.1. However, the
result here is obtained for purely non-deterministic processes.
2. The assumption M.§ q is not very restrictive since it is satisfied

for stationary processes.

-75-

7. Stationary‘Wide-sense N-ple Markov processes. From (6.22), (6.23) and
Theorem I.5.1 the corresponding representation for stationary purely non-

deterministic N-ple Markov processes satisfying (D) is given by

N t
(7.1) at - 2 f§i(t)Hi(u)dg(u>

1:1 -00

N_ N-
Here 2 W i(t)Hi(u) is a function of t-u. In fact it is .2 Yi(t-U)Hi(o) (u é t)

in]. — 1:1
- N _
where wi(-) is zero on the negative real line or Z .Ki(O)Hi(u-t) (u g t) where
.... ' =1

Hi(-) is zero on the positive real line. The further determination of the

N _
kernel 2 yi(t)Hi(u) leads under certain conditions to a vector generalization
i=1

of continuous parameter Ornstein-Uhlenbeck processes. These purely non-determin-

istic processes also have rational spectral density matrices and are of importance

in multidimensional prediction problems (see A. M. Yaglom [9]). It is proposed

to study these questions in detail at a later time.

-76-

REFERENCES

Chapter I.

[1].
[2].
[3]-

[1.].

[5]-
[6].
[7]-

[8].

[9]-

[10].

[ll].

[12].

Cramer, H. "On some classes of non-stationary processes," Proc. #th

Berkeley Sympos. Math. Stat. and Prob. II, 1961, 57-77.

Cramer, H. "0n the structure of purely non-deterministic processes,"

Arkiv For Math. 5, 2-3 1961, 2&9-266.

Cramer, H. "Stochastic processes as curves in Hilbert Space," Theoriya

Veryatnosteii i e Primeneniya IE” 2, 196A.

 

 

Doob, J. L. Stochastic Processes, New York, 1953.

Gladyshev, E. G., "On multidimensional stationary random processes,"

Theory of Probability and Its. A921. 111, 425-428.

Hanner, 0. "Deterministic and non-deterministic processes," Arkiv For

Math 1, 1950; 161-177.

Hida, T. "Canonical representations of Gaussian processes and their

applications," Mem. Coll. Sci. Kyoto, A33, 1960.

Jones, R. H. "Stochastic processes on a Sphere as applied to meteorol-
ogical 500-millibar forecasts," Proc. 2f the Sympos. 23 Time Series

Analysis, Brown University 1962 (Ed. by M. Rosenblatt, John Wiley & Sons).

Kallianpur, G. and Mandrekar, V. "0n the connection between multiplicity
theory and O. Hanner's time domain'analysis of weakly stationary

stochastic processes," Tech. Report E9, University 2f Minnesota, 196A.

Karhunen, K. "Uber lineare methoden in der Wahrscheinlichkeitsrechnung,"

Ann. Ac. Sci. Fennicae Al, 31, 1947.

 

Masani, P. and Robertson, J. "The time domain analysis of a continuous
parameter weakly stationary stéchastic process," Pac. J, 2f Math. 12,

1962.

Plessner, A. I. and Rohlin, V. A. "Spectral theory of linear operations

II," (Russian), Uspehi mat. Nauk I,‘l, 1946, 71-191.

[13]-
[14].

[15].

[16].

[17]-

[18].

-77-

Prokhorov Yu, V. "Convergence of random processes and limit theorems

 

in probability theory," Theory gprrob. and Its Appl. I, 1956.

Robertson, J. "Multivariate continuous parameter weakly stationary

stochastic processes," Thesis, Indiana University, June, 1963.

Rozanov, Yu. A. "Spectral theory of multi-dimensional stationary

random processes with discrete time," (Russian) Uspehi Mat. Nauk 13,

g 1958, 93-112.

Sak, S. Theory of the Integral, New York, 1938.

Urbanik, K. "A contribution to the theory of generalized stationary

random fields," Trans. 2: the Second Prague Conference, Prague, 1960,
667-679.

Zasuhin, V. N. "0n the theory of multidimensional stationary processes,"

(Russian) Doklady Akad. Nauk, USSR, 33, 1941, 135—137.

-78-

Chapter II.

[1].
[2].

[3].
[4].

[5].
[6].

[7]-

[8].
[9].

[10].

Beutler, F. J. "Multivariate wide-sense Markov processes and prediction

theory," Ann. Math. Stat. 3%, 1963.

Doob, J. L. "The elementary Gaussian processes," Ann. Math. Stat. 15,
1944, 229-282;

Doob, J. L. Stochastic Processes, Wiley, New York, 1953.

Gladyshev, E. G. "On multidimensional stationary random processes,"

Theory pf Probability and Its Applications III, 1958.

Hida, T. ”Canonical representations of Gaussian processes and their

applications," Mem. Coll. Sci. Kyoto A33, 1960.

Ince, E. L. Ordinary differential equations, Dover, 1956.

-

Rosenberg, M. "Square integrability of matrix-valued functions with

respect to a non-negative definite hermitiom.measure," Duke Math. J, 31,

No. 2, 1964.

Schwarz, L. Theorie des distributions 1, 2, Hermann, Paris.

Yaglom, A. M. "Effective solutions of the linear approximation problems
for multivariate stationary processes with a rational spectrum," Theory

‘gf Probability and Its Applications II, 1957.

Wiener, N. and Masani, P. "The prediction theory of multivariate

stochastic processes," Acta Mathematica 98, 1957, 111-149.

 

IIlilllwlllellfltflllfllwllzllwglljlljlfllllllH