THE EQUWALENCE OF
GAUSSEAN STOCHASTIC PROQESSES

Thesis for the Degree of DH. D.
MICHIGAN STATE UNEVERSITY

Hiroshi Oodaira
1963

THESIS

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thesis entitled
The Equivalence of

Gaussian Stochastic Processes

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Hiroshi Oodaira

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Ph.D degree in Statistics

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ABSTRACT

THE EQUIVALENCE OF
GAUSSIAN STOCHASTIC PROCESSES

by Hiroshi Oodaira

This thesis is concerned with the problem of the
equivalence or singularity of two probability measures induced
by two Gaussian stochastic processes. First, we consider the
general case and obtain a set of necessary and sufficient
conditions for equivalence in terms of mean functions and
covariance functions of the processes. A proof of the
equivalence—or—singularity dichotomy is obtained simultane-
ously. The method and techniques used in the present thesis
are that of reproducing kernel Hilbert spaces. We derive
several equivalent forms of necessary and sufficient con-
ditions and show the equivalence of our results and other
criteria obtained by E. Parzen and by J. Feldman. Gaussian
measures in abstract Hilbert space are also considered. Next,
we apply our conditions in the general case to special cases,
and obtain some generalizations of A. V. Skorokhod' result
in the additive case and of D. E. Varbergis result in the

case of Gaussian processes with covariance kernels of

triangular form. We state conditions for the equivalence of
stationary Gaussian processes in terms of their spectral dis-
tribution functions and, finally, consider a particular case
of the equivalence problem of stationary Gaussian processes

on finite intervals.

THE EQUIVALENCE OF

GAUSSIAN STOCHASTIC PROCESSES

BY

Hiroshi Oodaira

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Statistics

1963

ACKNOWLEDGMENT

The author wishes to express his sincere thanks and

rappreciation to Professor Gopinath Kallianpur for his

guidance and encouragement.

He also wishes to acknowledge with gratitude the
financial support provided by the National Science Foundation,
Grant No. G 18976, which made it possible to obtain the
results_presented in Sections 1.2 and 1.5 (also in [14]),

and that of the U.S. Army Research Office, Grant No.

CRD—AA—L-3782, to complete this work.

ii

INTRODUCTION . . . . . . . . . . . . . . . . . .
Chapter
I NECESSARY AND SUFFICIENT (N.S.) CONDITIONS

TABLE OF CONTENTS

FOR EQUIVALENCE IN THE ABSTRACT FORMULATION

OF REPRODUCING KERNEL HILBERT SPACES .

Preliminaries

Main Theorems

Equivalent forms of n.s. conditions, I
Equivalent forms of n.s. conditions, II
Gaussian measures in abstract Hilbert
space . . . . . . . . . . . . . . . . .

Comparison of various methods . . . . .

APPLICATIONS TO SPECIAL CASES . . . . .
Gaussian processes with independent

increments . . . . . . . . . . . . . .

iii

Page

16
28

36

43

49

55

55

Chapter Page

2.2 Gaussian processes with covariance kernels
of triangular form . . . . . . . . . . . . . . 60

2.3 Stationary Gaussian processes . . . . . . . . . 7O

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 79

iv

INTRODUCTION

Suppose that two measures P and Q are defined on a
measurable space (f1,£}t). P is called absolutely continuous
with respect to Q if P (A) = O for every A36} for which Q (A)
= 0. If P and Q are absolutely continuous with respect
to one another, then they are called equivalent. If there
is a set Beg? such that P (B) = O and Q(£Z-B) = 0, then P and
Q are said to be singular (or orthogonal or perpendicular).
The equivalence and singularity of two measures represent two
opposite extreme cases.

The problem of the equivalence or singularity of two
.probability measures induced by two Gaussian stochastic
processes has recently received considerable attention, be-
cause of its importance in statistical inference theory as
well as in structural problems of stochastic processes. It
has been proved, by many authors in varying degrees of
generality, that two Gaussian probability measures are either
equivalent or singular. The proofs of the existence of such
a dichotomy in the general case has been given by J. Héjek

[11] and, independently, by J. Feldman [6], [7].

Since Gaussian probability measures are completely
determined by the mean functions and covariance functions of
the processes, it should be expected that necessary and suf-
ficient conditions for equivalence are stated directly in
terms of mean functions and covariance functions. From this
-point of view it seems that reproducing kernel Hilbert space
is the most natural setup to formulate these conditions. The
purpose of this thesis is twofold: (l) to obtain sets of
necessary and sufficient conditions for equivalence, using
the method and techniques of reproducing kernel Hilbert
.spaces; (2) applying these conditions, to give a unified
treatment of results obtained by various authors, concerning
additive, Markov, or stationary Gaussian Processes.

The contents of this thesis are as follows. Chapter I
deals with necessary and sufficient conditions for equivalence
in the general case. In Section 1.1 we summarize known
properties of reproducing kernels which we require and also
give several lemmas. The statements and proofs of the main
theorems are given in Section 1.2. An alternative proof of
the dichotomy is also obtained. Several other equivalent
forms of n.s. conditions are derived from the main theorems
in Section 1.3 and 1.4. In particular, we derive E. Parzen's
criterion [19] in Section 1.3 and J. Feldman's result I 6] in

Section 1.4. In Section 1.5 Gaussian measures in abstract

Hilbert space are considered. In Section 1.6 the relationship
between our results and other recent work is discussed and
various available methods are compared with each other.
Chapter II is devoted to the study of the equivalence problem
for important classes of Gaussian processes. In Sections 2.1
and 2.2 we specialize our general theorems to the cases of
Gaussian additive processes and of Gaussian processes with
covariance kernels of triangular form and obtain some
generalizations of known results due, respectively, to A. V.
Skorokhod [25] and D. E. Varberg [27], [28]. In Section 2.3
we discuss the stationary case. A particular case of Slepian-

Feldman's result [8] is considered.

CHAPTER I

NECESSARY AND SUFFICIENT CONDITIONS FOR
EQUIVALENCE IN THE ABSTRACT FORMULATION
OF REPRODUCING KERNEL HILBERT SPACES

1.1 Preliminaries

Let (£2,53) be a measurable space, where 3“ is the 0
field generated by a class of random variables {X(t), teT),
and let P and Q be two Gaussian measures on (D, 34 ), i.e.,
_probability measures such that {X(t), teT, P} and [X(t), teT, 03
are Gaussian processes. Throughout this paper we shall assume
that the index set T is either countable or a separable metric
space and, in the latter case, both processes are continuous
in quadratic mean. Without any loss of generality, we may
assume that the mean function of the process {X(t), teT, Q} is
zero. The mean function of‘{X(t), teT, PS'will be denoted by
m(t), and the covariance functions of both_processes will be

denoted by'r;(s,t) and rb(s,t) respectively, i.e.,

m(t) = EpX(t) = X(t) dP,
.Q,

[‘p<s,t)

EpX(s) X (t) - m(s)m(t)

(
] (X(s) .- m(s)>(X(t) - m(t))dp,
41
EQX(S) X (t) = j X(S)X(t)dQ.

J2

PQ(S.t)

We also write

/\(s,t)

EpX(s)X(t) = ‘S XKs)X(t)dP, and
41

M(s,t) m(s)m(t).

As mentioned in Introduction, our principal technique
is the theory of reproducing kernels. In this section we list
several propositions which will be used constantly in the
present thesis. For the details of this theory we refer to
N. Aronszajn's papers [2] and [3].

Let R(-, -) be a nonnegative definite kernel.

The reproducing kernel (r.k.) Hilbert space H(R) with
reproducing kernel (r.k.) R(-, °) is a (real) Hilbert space,
consisting of a class of (real valued) functions defined on
a certain index set T, with the following properties:

(1) for every teT, R(-, t)€H(R),

(2) (The reproducing property of R) for every t€T and

every f€H(R),

f(t) =< f(-), R(-, t) >.

where < ',- > denotes the inner product in H(R).
To construct H(R), consider the class of functions f

of the form

f(-) = a R(', t.), t.€ T,

1 l i 1 1

HIV] :3

and define the norm of f by
, n
2
I“ f HI = Z aiajR(ti,t.).
i,j=l 3

H(R) is obtained by completing the class of functions of the
above form with respect to the given norm. As easily seen
from the construction, H(R) is spanned by functions
{R(-, ti), ti€T\ and it is unique up to congruence. We shall
find it convenient to denote by F the class of all elements
in H(R), without topology, and to call R the r.k. of the
class F.

We shall consider several different r.k. Hilbert
spaces in this thesis, such as H(FD)' H(r;)' etc. Their

products and norms will be denoted by <°’4>Q’ [I] ° [llQ
<.,.>p, H] - |||p, etc.

For any two kernels R and R we shall write Rl‘<<§ R

l 2 2

if RZ-Rl is nonnegative definite.

Proposition 1. If R and R1 are the r.k.‘s of the classes F

and F1 with the norms ”l- ”I, “l- ”[1 and if there is a

finite constant c such that Rl < < cR, then F1C: F (in

particular, Rl(°,t) E F) and |Il°|l|i Z'C-1)H ' H|2.
Proposition 2. If K1 is a bounded linear operator on H(R),

then there corresponds a kernel R such that Rl(°,t)€H(R) for

1

every t and

Klf(t) = < f(-), Rl(-,t)> .

_Proposition 3._ For any arbitrary symmetric kernel R1, the
necessary and sufficient condition that it correspond to a

bounded self-adjoint operator on H(R) with lower bound 2_c >- co

and upper bound g c'< + 00 is that cR < < Rl < < c'R.

Let T' = {tl,t ,...,tm( be any finite subset of T,

2

and let RTI denote the m x m positive definite matrix

(R(ti'tj))l£i,j£m = (Rij)l£i,j£m° The 1nverse matr1x of RT'

is denoted by RSI — (31])

igi,jg@r
Proposition 4. The norm of the finite dimensional r.k. Hilbert

space with r.k. R is given by

TI
m
2 . ij t -1
= = f ,
i,j-l
t
where fT, €H(RT,) and fT, — (fl,f2,...,fm) stands for the

transposed vector (i.e., row vector).

Proposition 5. ”If R is the r.k. of the class F of functions

defined on T with the norm IH 'IH , then R restricted to a

subset Tlc: T is the r.k. of the class Fl of all restriction

of functions of F to T1. For any such restriction, flEFl, the

norm H] fl M] l is the minimum of Hi f H] for all f€F

whose restriction to T1 is fl'

Let H(Rl) and H(Rz) be r.k. Hilbert spaces with norms
||l - Illl, III - |||2. The direct product H = H(Rl) ®H(R2) is
the completion of the class of functions g(-,*) of the form

. * = . *
where gl(-)€H(Rl) and g2(*)€H(R2), with respect to the norm

2 2 2
H! g HI = lllgllll - m 921” .

Proposition 6. The direct product H = H(Rl) ®H(R2) is a r.k.
Hilbert space with r.k. R(Sl'32’t1’t2) = Rl(sl’tl)R2(52't2)'
If {fk) and {91) are complete orthonormal (c.o.n.)
systems in H(Rl) and H(RZ)’ respectively, then
hk£(-,*) = fk(-)g (*) is a c.o.n. system in H(R) =
H(Rl) ®H(R2) , and any element ‘1’ in H(R) can be written in

the form

¢(-.*) =2 a h (-,*) =2 0!- f (-)g (*)
k,l k9. k9. k,I. kn k 9.

with

2!” i2< do.

k2
k,Q

and vice versa.

The following theorem will be used in the proof of

main theorem as well as in its specializations.

Congruence theorem ([16]). Let H1 and H2 be two abstract

and ('1')

Hilbert spaces with inner products (-,-)1 2,

respectively. Let {f(t), teT} be a class of elements which

span H , and, similarly, let {g(t), teT]'be a class of elements

1

which span H If, for every 5, teT,

2.
(Ha). f(t))l = (g<s). g<t>)2.

then there exists a congruence (an isometric isomorphism)

from Hl onto H2 such that, for every tsT,

sf”) = gm.

Proposition 7.. (L -representation of a r.k. Hilbert space

2
H(R)) ([16]). If there are a measure space (B,f3.u) and a
class of functions {¢(t) , t 6T I in L2 (B, {5 , U.) such that for

all s,t T

R(S.t) = J¢(S)¢(t)d H.
B

then H(R) is congruent to the Hilbert subspace L2(¢(t), téT)
of L2(B,13,u) spanned by (¢(t), teT}. That is, any element

f(-)eH(R) is represented in the form

f(t) = I ¢(t)gdU«,
B

10

for all tET, with g€L2(¢(t),t€T), and, conversely, any element
g€L2(¢(t), teT) determines an element f€H(R) by the above

relation.

Examples of L —representation will be given in §§2.l

2
and 2.2.

We shall also consider Hilbert spaces spanned by
stochastic processes. First, let.J:(X) denote the class of
all finite linear combinations of {X(t),tcT\, i.e., the class

of all random variables Y of the form

n
Y = Z a.X(t.),
1= 1 1

l
where a ,a ,...,a are real constants and t ,t ,...,t 6T.
1 2 n l 2 n
Let 1:2(X,P) denote the class of all random variables Z such

that there exists a sequence of random variables Yn in,£:(X)
converging to Z in quadratic mean with respect to P. Define
L2(X,P) to be the set of all equivalence classes of random

variables inwi;2(X,P) modulo the class of random variables

Z €JC2(X,P) with EpZ2 = 0. We denote byuz-P the equivalence

class in L2(X,P) to which Z belongs. ,}:2(X,Q), L2(X,Q),EQ

are defined in the similar manner. Clearly L2(X,P) and

L2(X,Q) are separable Hilbert spaces with norms

"P 2 _
||Z [I = Epzz, [IZQl]2= E022. Their inner products will be

d tdb .l. .’. .
eno e y ( )p and ( )Q

ll

*
We introduce another Hilbert space L2(X) in the fol-
lowing manner. Suppose that there are finite positive
constants c and c' such that for every Y€¢£;(X)

(1.1) CE Y2 3 E Y2 < C'E Y2.
Q P. ,

*
Let¢l:2(x) be the class of all random variables Z such that
there exists a sequence of random variables Yn in,l:(x)
converging to Z in quadratic mean with respect to both P and

*
Q, and let flbe the class of random variables Z in 4C2(X) with

E Z2 = 0. Note that from assumption (1.1) and the definition

Q

*
of d:2(x) it follows that EQZ2 = 0 implies EpZ2 = O and vice

* *
versa. Define L2(X) = 0C2(X)/1L , the set of all equivalence
* ._
classes modu10'fLof elements in¢K:2(X). Denote by Z the
*
equivalence class in L2(X) to which Z belongs. It may be

*
shown that L2(X) is a separable Hilbert space with inner

- —- ‘— 2 2
product (21, 22) = E02122 and norm N Z N — EQZ . The

verification does not offer any difficulty, but it is some-
what lengthy and is therefore omitted. In passing, it should

*
be observed that the class of elements in L2(X) can be en-

-— 2 2
dowed with different (but equivalent) norm " Z ”p = EpZ .

We shall write H «,H if two Hilbert spaces H

1: 2 55nd

1

H2 are congruent.

12

Lemma 1.1.1
(1.2) L2(X,P) :HLA)

(1.3) L2(X.Q) :Hfl-b) .

and, if there exist finite positive constants c and c' with

or; < <.P(< < c'r; .
then (the space L:(X) can be defined, and)
*
(1.4) L2<x> qu-‘QI g. L2<x,o>.

Proof. (1.2): Since

 

(Ms)? x<t)P)p =A<s.t) = </\.(-.s).]\(-.t)>p

and both {X(t)P} and.LA(°,t)\ span L2(X,P) and HLA). respec-
tively, the assertion (1.2) is obtained by applying the
congruence theorem. The proofs of (1.3) and 1.4) are quite

similar.

-_1 aiajm(ti)m(tj)

n 2
= (.2 a.m(t.))
i=1 1 1

2.0.
the kernel M(s,t) is nonnegative definite, i.e., O<<<< M.

Hence, O< < M=_/\_- Pp, i.e.,
Pp <</\.

l3

Assume the existence of positive finite constants c1

and c2 with
(
clrQ<<rp .</\<< CZPQ.
Then” r;(-,t)€H(ra) and, by Proposition 3, r; determines a

bounded self-adjoint linear operator S on H(rb) by the relation

Sf(t) = <[f(-), ln'p(°,t)>Q . Note that

§FQ(SIt) <PQ(.It)I l—‘p(':S)> Q

< Fp(.ls)l I-Q('.t)> Q
=[jp(s,t).

*
Our assumption enables us to define L2(X) and, by
*

Lemma 1.1.1, there is a congruence E from H(F‘Q) to L2(X).
Lemma 1.1.2 For fl,f2€H(Fb) ,

< Sfl,f2>Q = COVp(Zl'Zz) .

= _ E ,
Epzlz2 (.leHEpZZ)

where Zle§fl and Zzefifz.

Proof. If fl and f2 are of the form

14

then
< Sfl.f2>Q = < < fl(t)_.rp(t,-)>Q, f2(-)>Q

m n

= <1: aiI;(°ISi). 3:21 blip .tj)>Q
m,n

= l §;l aibj[;(s ,t )

m n
= covp (1:1 aiX(ti), jfl bjx(tj))

For any f€H(fB) there is a sequence (fn\ of elements of the
above form which converges to f in norm. This implies that

Zn€§fn converges to Zeéf in II - H‘ -norm, and hence, in-

Q

1|,r‘Hp-norm. This proves the lemma.
We need the following known fact on which the proof
of main theorem is based. (See [6]).

Lemma 1.1.3 Let 2* s be random variables that are independent,

normally distributed with mean m and variance vk'>0 with

k

respect to P and independent, normally distributed with mean 0
and variance 1 with respect to Q. Let CZn denote the o-field

generated by Z ,2 ,...,Zn and CLthe minimal<¥fie1d containing

1 2

. . 00
the un1on of all a‘n’ 1.e.. a. - >11 an'

15

If
0° 2 0° 2
Z (l-vk) <00 and 21 Ink <oo ,
k=1 k=1

then P and Q are equivalent on Cl. If

00 2 oo
2 (1-v ) = 00 or 2 m = a) ,

k=1 k=1
then P and Q are singular on 51.

The lemma can be proven by applying the following S.
Kakutani's effective criterion for the equivalence of infinite
product measures.

Let u and Q‘be two probability measures on a measurable
space (9.5% ), and let T be a measure which dominates both u

and 9. Consider

a” 1/2 dv 1/2
P(H.9) = ('3? ) ('3; ) d T,
11

It may be shown that p(u,\)) is independent of the choice of T.

Let )Qh"j;n) ‘be a sequence of measurable spaces,
and assume that, for each n, there exists two equivalent
probability measures fin and V; on (Ch,.3*n). Consider infinite

product measures ii: ® % and ‘9 = ®Qn defined on i

(mean. 3~=®3« n).

Theorem ([13]) u and 9 are either equivalent or singular

16

00
according as I].P (Mn,VB) is > O or = 0. Moreover
n
00
9(11'19) =1];- pULnIVn) -

1.2 Main Theorems
Theorem 1.2.1 If P and Q are not singular, then the following

conditions are necessary.

(1) m(')eH(rb).

(2) There exist finite positive constants c1 and c2

w1th elf; < < r; < < Czra.
(3) S has a pure point spectrum.

(4) The eigenvalues {Ak} of S satisfy the relation
00
2
ii (l-xk) <00.
k=1

Proof. First we prove (2): It suffices to prove the first
part, ClrQ < <f‘b, since the other part can be proved
analogously. Suppose that there is no such a positive

constant c . Then, for each n, there are a sequence of

l
vectors an = (a2, a:,...,a: ) and a sequence of finite sets
n
n n n n
T — (tl,t2,...,tk ) c: T such that
n
k

n n n n n 1 kn
2 a.a. t.,t. I; Z
- l JEP( l J) < -

a2a¥[3(t:,tg)
1,j=1 3:1 3 J

17

Put
kn
-1'2
b: = a? ( Z alagra(tn,tg)) / -
1,j=1 J J
k
n
Then Yn = 2 b: X(t§) is normally distributed with mean
i=1
k k
n n n 2 n n n n
m = Z 'b.m(t.) and variance C = Z b?b.[fi(t.,t.)< l/n with
n i=1 1 1 n i j=l 1 j p 1 ]

respect to P and normally distributed with mean 0 and variance

k
n
l = Z ‘b¥b§[‘(tg,tg) with respect to Q.
1,j=1 1 3 Q 1 3

Let C = lb): |Y (w) -m I < (fl/2). Then we have
n n n n

2 ,2
" (t-mn) /2 tin

P(Cn) =-———————- e dt
277-Gn ltrm VQ11/2
n n
1 -82/2
= --—-—- e ds -—9 l,
sz -l/2
|S|<16n
as n -—> co, and
2
1 _
Q(Cn) =---—- e t /2 dt -—9 0.

42” 1/2
It-mn|< on

as n -9:a3. This implies that P and Q are singular.

l8

(1): It is sufficient to show the existence of a
finite constant c with M < < ch° For, if M < < CrQ' it
implies M(-,t) = m(-)m(t)eH(rb), and hence, m(-) =

O for all

m(t)-1M(-,t)€H(rb) for t with m(t) + 0. If m(t)
t, m(') = OeH([5).
Now, assume that there is no finite constant c with

M < < CrD' that is, for every n there exists a sequence of

vectors an = (a:,a:,...,a: ) and a sequence of finite sets
n
n n n n ‘
T — (tl,t2,...,tk ) C: T such that
n
k k
n n
n22 ananr<tn,t‘.‘)< :1 a?a’?M<t§‘,t’.‘).
J 1 J
1,j=1 i,]=1
Put
kn
- n n l 2 n
b? — n 1/2 ( Z a a.l‘(tn.t )) /
1 Q 1
1,j=1
k k
n n

Let Y = 2 b?X(t?). We may assume m = E Y ==Z b?m(t?):> O,
n 1 1 n p n 1 1

i=1 i=1
since, if necessary, we may take -Yn instead of Yn. Yn is normally
distributed with mean 0 and variance l/n with respect to Q,

. . . . 2 .
and normally d1str1buted Wlth mean Inn and variance on With

2
/2 and On<_ c2/n, where c is

respeCt to P. Note that mn:> n1 2

19

the constant in (2). Letting C ={po: IY {wI-m \ < 01/2),
n n' n n

we have, as n -—5 a),

. 2. 2
1 -(t-mn) /20n
P(Cn) = —-———-' e dt --9 l ,
WC
2 n ]t-m |< 01/2
n n
and
n1/2 _nt2/2
Q(Cn) = -—-—-—- e dt
42 V / 1/2

t- \ o
l mnl‘ n

 

2,
= l J. e"5 /2 ds "5 O.
W
J2 $2 nl/2mn_nl/4C21/4

This is a contradiction.

(3): In View of (1), if P and Q are not singular, we can de-

fine the Operator S corresponding to[jp, and, since

0 < < M < < CPO (the proof of (2)), we have clrQ <(A<< c'r'Q

[taking c' = c2+ c. Hence, we can define the space L:(S), and,

by Lemma 1.1.1, there is a congruence § from H(F'Q) onto L:(X).
The spectrum of S is a nonempty bounded closed set,

since S is bounded and self-adjoint. By a limit point of the

spectrum of S we mean a point of the continuous spectrum of

S, a limit of eigenvalues of S or an eigenvalue of S of

infinite multiplicity ([22]). Denote by-{EX} (),real) the

resolution of the identity determined by S. We write

E(A) = Eb_0- Ea 1f’A = (a,b).

20

First we show that any point a # 1 cannot be a limit
point of the spectrum of S. If the contrary is true, then
there is a monotone decreasing sequences of intervals

A = ° ' (I, '
k (ak'bk) containing W1th ak —9 0L and bk —9 (1 such

that we can choose nk from E(Ak)H(r;) with.<nk,q}>o = Skj

( Skj lS Kronecker s delta) and < qu,ni:>Q = o for k i 3.

Then we have

A
k
Let Zk€§rlk’ k = 1,2, ..., Then Zis are independent, normally
distributed W1th mean mk = Eka and variance vk =‘<Snkxnk:>o

with respect to P (Lemma 1.1.2) and independent, normally
distributed with mean 0 and variance 1 with respect to Q.

Since a and b tend to Clas k.-—§ 00, v goes to CIwhich is

k k k
. . oo 2
not 1 by asSumption. Hence 2 (l-vk) = 00, which shows that
k=1

P and Q are singular (Lemma 1.1.3).

Thus, if 1 is not a point of the spectrum of S, the
spectrum of S consists of a finite number of eigenvalues of
finite multiplicity. On the other hand, if 1 is a point of
the spectrum of S, then it is either an eigenvalue of S or a
limit of eigenvalues of S. For, if 1 is not a limit of

eigenvalues, there exists an interval (1-6; 1+<Q such that

21

: 7 - . -_- U .
El-€ = El+eand every p01nt in the set (1 e,l) (1,1+€)

belongs to the resolvent set of S, since any point different
from 1 is not a limit point of the spectrum. This implies

El * El-O' i.e., 1 is an eigenvalue of S. Hence, the spectrum
of S consists of either eigenvalues or eigenvalues and their
limit points (in fact, 1 is the only possible limit of
eigenvalues), i.e., it is a pure point spectrum.

(4): Let {AkS be the set of all eigenvalues of S and let

{Wk} be corresponding orthonormal eigenvectors, i.e.,

<nk7n5> Q = Skj' Note that if h = A = ... = Ak+Q—l' where

l is the multiplicity of the eigenvalue A, then

nk'nk+l"°"nk+1-l

span the subspace (El -E3-O)H(rQ) and {nk\
is a c.o.n. system in H(FD)° Let Zkéimk/ k = 1,2,... The

' = ( , , = A , ' th t
relationS‘<nk,nj:>Q Skj and \ SnknJ > k8¥3 imply a
Zk's are independent, normally distributed with mean m =

k

Eka and variance hk with respect to P (Lemma 1.1.2) and
independent, normally distributed with mean 0 and variance 1

with respect to Q. The assertion then follows from Lemma

1.1.3. This completes the proof of the theorem.

Theorem 1.2.2. If conditions (1)—(4) of Theorem 1.2.1 are
fulfilled, then P and Q are equivalent.
Proof. By condition (1), M(-,t) = m(.)m(t)€H(Fb) for every

t€T. Then the relation< f(-),M(-,t)>Q= <f(-),m(-)>Qm(t)

22

defines a linear operator K on H(Fb), that is, Kf(-) =

< f(°),m(°)>Qm(-). K is bounded and self-adjoint, since

f>=

2 ( 2 4 ‘/
III kf Ill Q ... Ill f III Q III m ill Q and. Kfl. 2 Q

a \= \, (z 2
< fl'm>Q< f2,m/Q <f1,kf2,Q. Hence 0 < < M . \ |H ml” Ora.

Sinceufik==FL + M, this relation and condition (2) imply

(2.1) clrb < <dflt< < c‘rb ,

where c' = c2 + I“ m Hi This enables us to define the

2
Q’
*
space L2(X). By Lemma 1.1.1, there is a congruence E from
*
H(Fb) onto L2(S).

Conditions (2) makes it possible to define the
operator S on H(Fb). Conditions (3) then implies that the
normalized eigenvectors {qkk of S is a c.o.n. system in H(Fb).
Let zke'éik, k=l,2,... Let an be the o-field generated by

Z Z , ...,Z , and let 52. = V (1 . Consider an arbitrary
n n

1 2 n=1
*
element Y in a<:2(X) and define

n

n
_ z — — =
Yn _ (Y’Zk)QZk z (Y,§nk)ozk .

k=1 k=1
Yn's are (Zn-measurable and, a fortiori, a-measurable. Since
{73‘} is a c.o.n. system, so is [QTE‘L and we have

n

"ii. "Q = H Y - Z (iénkfink ||Q ——§ 0 ,
k=1

23

as n —-) 00, and, by (2.1),
|| Y-Ynllp ——> o
as n ——) 00. Since l| Yn-lelQ -) O as n, m —> 00, there is

a subsequence )Yn) converging to an a-measurable function
k

1':

Y on the set D6 a, where D =[o.): 11m Yn (on) = lim Yn (cm).
k k

 

*
Y (on) = lim Yr1 {to} on D. O elsewhere, and mm = 1.
k

... -*'
Observe that I] Yn- Y ll 0-90 as n —1> 00. Again by (2.1),

there is a subsequence {Ym i of the sequence {Yn} which

3
tit-'4:
converges to an Ci -measurable function Y on the set

__ ink
D ={uo: 11m Ym (w) = lim Ym ('10)) of P-measure l and Y (to) =
j 3'

lim Ym (w) on D and O elsewhere. Since {Ym) is a subsequence

j j

_ _ *‘k ‘k

of {Ynfi , D C D, and, hence, Q(D)= l and Y (on) = Y (00) on D.

k
Therefore,

__ _** 2 * 2 _ _* 2
]| Yn-Y I] Q- JD (Yn-Y)dQ— ]|Yn-Yl]Q—-)o.

as n —--9 00. Furthermore, by (2.1),

_**

“Yn-Y Ilpg|IYn-ij]|p+]lymj-Y HID—90
asn——)m. Hence
_** _ _**
(2.2) “Y-Y HQ=||Y-Y ]]p=0.

24

'1’
Define CL =[AUN: ACa, N634, P(N) = Q(N) e o}. (2.2)
' , * *
shows that every random variable Y in 0C2(X) is a. -measurable
it
and, hence, .ijCZ .
a *
Since Zk€dC2(X), there is a sequence of random

variables {YE} in 4C(X), that is, of the form

Q(k,n)
Yk = Zak X (tk._\,
n . n1 n1
i=1
converging to Zk in quadratic mean With respect to both P and Q.
L(k.n)
Noting that Q-lsz = Z ak I"! ,t .) converges to §—lE = r ,
‘n i-l n1 Q n1 k k

we have

. k
mk — Eka lfim EpYn

l(k,n) Q(k.n)
= lim Zak. E X (tk) = lim Zak.m(tk.)
n1 p n1 n1 n1
n . n .
1:1 1:1
1(k.n)
. k k
= 11m 2 ani< m(').l:)( 'tni)>Q
n .
1:1
Q(k,n)
. k k
= lrllm (m(-). 2: aniIQ(°’tni)>Q
i=1

<m(->, 73((-)>Q.

25

Since m(°)GH(rb) (condition (1)),

M8

. . 2 . . 2
<mi~i,nk(-)> = Mimi-Mu Q (00.

(2.3) 2 m Q

Zk's are independent, normally distributed with mean mk and

variance )k with respect to P (Lemma 1.1.2), and independent,
normally distributed with mean 0 and variance 1 with respect

to Q. The positivity of M follows from condition (2).

'k
Condition (3), the relation (2.3) and Lemma 1.1.3 together
. _ *
imply that P and Q are equivalent on El. If E = ALJNEZCL
and P(E) = 0, then Q(E)_§ Q(A) + QiN) = 0 Since P(A) = 0.
Similarly Q(E) = 0 implies P(E) = 0. Hence P and Q are
* ..
equivalent on 51 , and, hence, on {p . This completes the

proof.

Summing up, we obtain the following main theorem.

Theorem 1.2.3 P and Q are either equivalent or singular. For

 

the equivalence of P and Q it is necessary and sufficient that

(l) In(-)6H(Eb),

(2) there exist finite positive constants c1 and c2

such that

Cl Q < <r'p << CZFQ’

(3) S has a pure point spectrum, and

(4) the eigenvalues {3k5<of S satisfy

0° 2
Z (l—ik) <too.
k=1 '

26

Remark. Conditions (3) and (4) may be replaced by

(3') I - S is a Hilbert—Schmidt operator.

Theorem 1.2.4 The following condition is equivalent to
(2)-(4) of Theorem 1.2.3.

F; has a representation of the form

00
. _ , \
Q(Srt) — Z H-qu\s gk(t)r
k=1
with
00
Z (l-Uk)2 < a3 and Uk 2 O for all k,
k=1

where {9k} is a c.o.n. system in H [D)°

Proof. Assume (2)-(4) of Theorem 1.2.3. The second half of
(2) implies r;(-,t)€H(Ig; for every tCT. Let (HRI be normal—

ized eigenvectors corresponding to the eigenvalues {bk} of S.

It forms a c.o.n. system in Hira) by (3]. Then
CD
F‘p(-.t) = 2 <[;(.,t),nk(.)>an(.)
k=1
(I)
=2 <P >, («2 <
k=1 Q nk 09k
=2 5 i), Q( t) ()
k=1 “k a").
CI) (13
=kZl 1 k<nk(.)'g U>Q nk(°) =k2§ Aknk(t)qk( )

27

for every t€T. Since norm convergence in H(Fb) implies point-

wise convergence,
oo

r;(s,t) = kélyknk(s)nk(t)

for every 5, t€T. The requirements on constants “k =1Ak are
fulfilled by the first half of (2) and (4).

Assume the condition of theorem. Noting that

CD
. . 2; . .
conditions that Z (l-nk) Q 00 and Uk ) O for all k imply the
k=1
existence of finite constants cland c2 with c1I§u k.£ c2 for

all k, define on H(Fb) an operator S by
00

S: ZIJ'P,
k=1 k k

where Pk is the projection on the one-dimensional subspace

spanned by gk. S is a bounded, self—adjoint linear operator

with upper bound c and lower bound c , and has a pure point

2 l
spectrum (Uk's are the eigenvalue of S). Since
00
sfb(-,t) = z ukgk(-)gk(t)
k=1
for all t€T,
oo
SP(s.t) = Z LL 9 (sh; (t),
Q k-l k k k

which equals r'(s,t) for all s, tET. Hence
P
.l = S .lt-‘H I
Fp( t) PQ( ). (Po)

and

28

Sf(t)

< Sf(-).PQ(-.t>>Q

. I .It 0
< f( ) Pp( >>Q
Therefore, ClrQ < <Cr; < < czrg, concluding the proof.

1.3 Equivalent forms of n.s. conditions, I.
Since I - S is Hilbert-Schmidt, it is expected to re-
place (3) and (4) of Theorem 1.2.3 by a condition stated in

the direct product of two r.k. Hilbert spaces.

Theorem 1.3.1 Conditions
(Q) There are finite positive constants c1 and c2

such that
Cer<<Pp<< C2 0
(B) [‘0 - Pp e H(r‘p) ®H<iy
are equivalent to (2)-(4) of Theorem 1.2.3.
Proof. Assume (a) and (5). From (a) it follows that
Hence, there is a bounded, self-adjoint linear operator K on

H(Fb) corresponding tolfiQ— FL, and K = I - S.
Let ifkl and {gt} be c.o.n. systems in H(Fb) and

H(Fb), respectively. Then by (B),

["Q(-.*)-l"l')(-,*) = kzl a k"fk(-)g,;(*)

with

29
2a - t
Note that l, kak( killrg) and, by (C0, €H(rb), for every k.

Since

2 m Kg (3?) III 2 2 m<g <->.r‘ (*,-)-P(*,-)’> m 2
k Q R Q p Q Q

k k
= z m m Mfkm m3
k L
-1 2
= c 2 "I z a f (*)|"
2 k L kQ k p
= 031 >3 2|ak |2
k i Q
<<D,

K = I - S is Hilbert-Schmidt. This implies (3) and (4). (2)
is the same as (0).

Conversely, assume (2)—(4). Then I-S is Hilbert-
Schmidt and [b— r; is the corresponding kernel. Let Niki be
the eigenvalues of I—S and let {9k} its corresponding eigen-
vectors forming a c.o.n. system in H(r‘). Noting that

Q
9k€H(r;). by (2), define

Kr.w = in <g.f> t(wg(*b
j’k k k 3 p 3 k

where {fj3 is a c.o.n. system in H(rg). Denote by l“ - "I

19810

the norm in H(F'p) a H(F'Q).

2
. = ' 2
m,n .
=1 3:].

m

2 2
2

< 9k,fj> P

30

n
'. 2 2
' Z
n .
k 3

OO
2 2
< z Ink] lllgklllp

(D
-1 2
$01 Zlul
k k

(00.

Hence K(-,*)€H(["‘p) ®H(|“Q). Also

,* = U ,f. . *
K(t ) 323‘ k <gk 3>pf3(t)gk( ) 6-: H(PQ)

for every t€T. For,

2
2
ZIP] 2< J.) f.(t)
k k lk 9k Jpjl

2

2 Eu < ,.>f.t
1"]. k 9kgnp3‘)‘

Z Ilpklzigkun 2
k

szngm' Q(iit)>o|2

2
5:. [pk]. [‘Q(t,t)

<00.

31

Since, for every gk,

<gk(°).K(t. ->>Q = ; uk<gk,fj>pfj(t)

= ngk(t)

(I—S)gk(t)

= <gk<->,r5<t..>—r;<t,->)Q,
K(t.-) =PQ(t,-)-]"p(t.°).

Hence, K(s,t) =[~Q(s,t)-]fip(s,t) for all s, t€T. Therefore,

FQ('.*) -l;('.*)€I-I(["p) ®H(["Q), which is ([3). (0L) is identical

with (2). This concludes the proof.

Using similar arguments we obtain

Theorem 1.3.2 Conditions (3) and (4) can be replaced by the

condition
(B') [“6 - FEeH(rb)<8iH([6)
or
(a > [“0 - Ppemrp) snap.
Let T' be any finite subset of T, and letlfi and

PT'

IWQT' denote the covariance matrices, i.e.,

rPT' = (rp(ti’tj))ti,tjeT'

rhT' = (rQ(ti'tj))ti,tj€T'.

32

Assume that FPT' and rQT' are non—singular for any finit sub-

set T'(: T, i.e., their inverse matriceslfigé, andI‘QT,

exist.

Under this assumption we prove the following

Theorem 1.3.3 ([19]). P and Q are equivalent if and only if
(1) m(')<-:H(PQ),
and one of the following three conditions holds.

(a) P0 - PP€H(F‘P) <3 H(F'Q)
(b) [“0 - PP€H(I"Q) ®H(I"Q)
(c) (“Q - FPeHWP) @mr‘P).

Lemma 1.3.1 Condition (a) implies the existence of finite

p031tive constants cl and c2 W1th clrQ < < PP < < CZPQ.

Proof. Letc= “ll—'0- PP mP®Q' LetT ={tl,t2,...,tm§

u I I .
be a finite subset of T and let II "I P ® Q,T' denote the

norm in the finite dimensional r.k. Hilbert space obtained

by restricting functions in H(r‘P) ® H(PQ) to T' x T' . Then,
i

by Propositions 4, 5 and 6,

c2>= lllFQ- Pp III2

PQQ, T'

{r' ixTi- {IT.i+

Pp'i'l

Traceif‘l where IT' is the m x m identity matrix.

QT PM)

33

There is a non-singular matrix U which transformsr'P and

T' T'

FQT' into diagonal matrices, i.e., such that

 

 

 

 

 

t
UT'FPT'UT' “ IT'
t \
UT'r‘QT'UT' = DTI = rdt i
l 0
d
t2
0 '.
dt
K mJ I
where tUT, denotes the transposed matrix of UT" dt '5 are
i
the roots of detIr‘ rrl - xI I: O and they are positive.
QT' PT' T'
Since the transformation by UT' does not change traces,
m ,
<1-d )2
2 ti
c ‘2 2
i=1 dt.
- 1
2
(1-dt )
Z 1 for i = l,2,...,m.
dt.
1
Therefore, for every dt ,
i
2 2 2 1/2 2 2 2 1/2
- - 2 -
(3 l) O < (2+c ) (;2+c)pu4) pp < d < (2+c )+(( +c ) 4) (am

'- t,'- 2
1

34

Suppose now that there is no finite positive constant

c2 with I; < < CZrQ’ Then, for any n, there exist a finite

subset T' = [t ,t ,...,t }and an m -dimensiona1 vector a
n n1 n2 nmn n n
such that
t t
n anrQT' anS aanT' an .
n n
-1
Let b = U , a . Then
n T n
n
t
b
ntanT, n < bbn,
n ._
i.e.,
m m
n n
2
Z 2
b ni Z n dtnibtni
i=1 i=1
m
n

"V
N

n~(min dt ) 2 b .
i ni . ni

Hence min dt ‘g 1/n. This contradicts (3.1).
i ni

The existence of cl can be proved analogously.
Proof of Theorem 1.3.3 If (a) is assumed, then conditions (a)

and (B) of Theorem 1.3.1 are satisfied in View of Lemma 1.3.1.

Converse is clear. It suffices to_prove the equivalence of

(a) and (b), since the proof of that of (a) and (c) is similar.

Assume (a). Let {fj),{gk\ be c.o.n. systems in

H(PP) , H(F‘Q). Then PQ - PP is represented in the form

35

l-‘Q(°z*)- FP(.I*)

Z 0. f.(-)g (*)
j.k 3 k

with

|2<oo.

Z [or
3k

j,k

Noting that PP < < CZPQ, we have

llll'-P "[2 = 2: mengc'lzla |2<co,
a I
Q p Q®Q j,k| . l O 2 ',k 3k
where I“ ' lIIQ Q 18 the norm in H(F‘Q) ®H(I"b). Hence

Po _ PP€H(FQ) ®H(PQ), i.e., (b) holds.

A ow b . L t ' = - d l t

ssume n ( ) e c I" PQ ["P "I Q Q Q an e
I“ . “IQ ® Q,T' denote the norm of the r.k. Hilbert space
restricted to T' x T' of H(F‘Q) ® H(PQ) . Then, for any finite

subset T' C T,

azngQ-PPmQ®QIT

Trace(IT, )-Trace(FPT,P;,]I’H )-Trace(P;,]I',,r'PT, )

-1 -1
+Trace(PPT'rQT'PQT'FPT' )

= z (1-at)2
t€T'
2
>_ (l-dt) for every t€T'.

From this one can conclude that there is a positive constant

36
c" with c"r; < < r;. Hence, if g€H(rb), then g€H(r;) and

2 II-1 2 ' '
HI 9 l" P.S C H] g I|IQ If {gk‘ is a c.o.n. system in

H(r‘Q),
FQ(-,*)- PP(-.*) = z Bklgk(-)gl(*)
k,Q
with
Z ‘5 [2 < oo .
k.Q k‘
and
2 2 2 ”-1 2
[II FQ.[;[|[P ®Q -_- 33.33“] m gklll P go 12le 5kg) <oo.

This shows thatl; - PP€H([—£3) ® H(PQ) , concluding the proof.

1.4 [Equivalent froms of n.s. conditions, II

In this section we derive from Theorem 1.2.3 other
equivalent conditions given by J. Feldman [6]. We assume
that m(-)= O, i.e., both processes have zero mean functions.
Let V* denote the adjoint operator of a linear operator V.

Definition ([6]) A linear operator V from a Hilbert space

 

to a Hilbert space is called an equivalence Operator if V is
one-to-one, onto, bounded, invertible and I-V*V is Hilbert—
Schmidt.

'Pheorem 1.4.1 P and Q are equivalent if and only if the

correspondence

37

m n

2 . .
airb( ,ti) __; 2 aiF;( ,ti)

1 i

is induced by an equivalence operator from H(rb) to H(rk).

Proof. We prove the equivalence of the above condition and

(2)-(4) of Theorem 1.2.3.

Assume (2)-(4) of Theorem 1.2.3. Since

n n
2
“I ,ZaiPP(.’ti) I” P = z.aiajPP(ti'tj)
1 1!]
and
n
, 2
‘H ; airb("ti) ‘” Q = .2. a'ajro(ti'tj)’
l 1,]
it follows from (2) that
n n
2 2
(M) C, III >i3air'Q(-.ti) m Q: Ill 3 aiFP(-.ti) III P:
. n
’ z . 2
C2 “I i aiPQ( Iti) "l Q .
n(k)
Hence, if k f a]: PQ(-,t}i<)} k is a “I ° “‘0 -Cauchy
n(k)

sequence, then. ‘{ Z a: r;(-,t:)} is a “I ~||lP-Cauchy
i

k
sequence, and vice versa. For any element f€H(rb) there is a

n(k)
|u . _ i 2 k . k .
I" Q Cauchy sequence i ai PQ( ,ti) k converging to

38

f, and, hence, there exists an element 9 in H(r;) to which the

n(k)

corresponding sequence {V Z at'r;f-,t:)} converges in
i

k
"I ° "‘P —norm. It may be shown that g is unique, i.e.,

independent of the choice of Cauchy sequences. Therefore we
can define an operator ® from H(F‘Q) to H(Pp) by

GDf = 9.

From (4.1) we obtain

(4.2) cl m :- \uggm @f lugs czlll f mg.

Hence ® is linear, one-to-one, onto, bounded and has a bounded

inverse. Now,

m n
(3:: aiPQ(-,si). ijjT"Q(-,tj)>Q

m,n

= i jZaibj <f;(~.si>,l‘o(.,tj)> Q
111,11

= Z a.b.r1(t.,s.)
i,j 1' J p 3 1
m,n

= 3:3- aibj§g"'si"r£>("tj’ >p

m n

= Z . Z .

< i aiPp< ,si). jbjfy ,tj)>p.

That is, for functions f, g of the form

39

m n
= 2 0 = Z . 'Ito I
f i air;( ’Si) . g i bjrb( J)

< Sf,g)Q=<@f,®g> .
P
By the continuity of inner products and (4.2), we have, for

any f.9‘€H( PC) I

< Sf,g>Q

<@ f: ®g>p =(®*®f.9>o .
Hence
® *®= s .
and I - ®*®= I—S is Hilbert-Schmidt, by (3) and (4).
To prove the other direction, suppose that ® is an

equivalence operator from H(Fb) to H(FL) and

n n
® :12 aiPQ("ti) = :3 airp(°’ti)‘

Since ® is bounded and invertible, there are finite positive

constants c1 and c2 such that

2 2 2
(4.3) elm f “‘05 Ill @: mp < czm f mQ

for all f€H(rb). In particular, for any f of the form

n
f = ZaiPQ(.pti)I
i

40

n n
2 , 2 < 2
cl Ill 1 aiPQ( .ti) Ill Q_ III ® i. aiPQ(-.ti) "I p
n
2

n
5 c2 In zaiPQ<-,ti> m g,
i.e.,
(4.4) elf; <‘<)j§‘<‘< czrg.
which is (2). The second half of (4.4) implies r;(-,t)eH(rb)
for every t€T. For any f,g€H(rb) there exist sequences of
m

n
functions {- Z air'Q(',si) “ , i Z bjPQ(',tj)5 converging to
i j

f,g, respectively.

m n
< * z -, . , >3 -
® ® 1 aiPQ( s1) j bjPQ( .tj)> Q
m n
= <® i aiQ(.,si),@§ bjPQ(',tj)>
m n
=< 2L aiPp( ,si), 2 bfl,‘ .t )>p
3
m,n
= Z a.b.P (t.,s.)
1,3 1 J p 3 1
m,n
= z aibj<r;)('ISi)I|-.Q( It )> Q

41

m n
=< Z ai‘;('.si), Z bjr-|Q(°Itj)>Q
l J
m n
= < f ai <f;(*.-).fi5<*,si) >0. i bjr5(-.tj>>

m

1 Q 3 Q

Q

n
=<< 2 a-P(*:Si):rp(*l°)> Q! i b-P(°Itj)> Q.

.1

Using the continuity of inner product, we can conclude that
* .
69 69in ) = < f<*),!Ԥ<*. )> Q
* . * .
Hence, by definition of S, ® @= S. Since I- ® ® is
Hilbert—Schmidt, so is I—S, from which (3) and (4) follow.

This concludes the proof.

Theorem 1.4.2 ([6]). P and Q are equivalent if and only if,

for any 26 aC(X), EP 20

(set-theoretically) and the cor-

respondence ZQ —4§ EP is induced by an equivalence operator
from L2(X,Q) to L2(X,P).

Proof. Necessity. By Lemma 1.1.1, H(r‘p) 3L2(X,P) and
H(Fb) 2’ L2(X,Q). Let §p denote the congruence from H(F‘p) to

L2(X,P) , and, similarly, let i0 denote the congruence from

H(PQ) to L2(X.Q). Note that

 

n n
P
§Ep f aiF;(-.ti) — E ai X(ti)

42

and
n n
——-—Q
3.20 :3 aiPQ(°’ti) _ i5; ai X(ti) .

r—u

Define an Operator :4 =§p® 32561 , where ® is the equivalence

operator in Theorem 1.4.1. Since §p and §Q are congruences,

'27 is an equivalence operator from L2(X,Q) to L2(X,P).

From (4.3) we have

—Q 2 73—0 2 —Q 2
(4.5) clll Z ||Q_<_l|.__.Z “p éczu Z “O

for all EQ€LZ(X.Q). Let Ze {2(X,P)f\&.2(x,0) and let YEEQ.

Then, since P and Q are equivalent,

( SlledP)l/2§_ ( f|z|2dP)1/2 +( [IY-ZIZdP)l/2
J1 ~0- .fl.

= ( J |z|2dP)1/2 + ( f lY—ZIZdQ)l/2
.n. .n.

=( [|z|2dp)l/2 < oo.
.1).

Hence YeoC2(X,P)noC2(X,Q). By (4.5),

—P —P 2 r—v—Q —Q 2 —Q—Q 2

Y — z = —- Y— z Y-Z I = .

\I "p H, ( p<_c2|| lo 0
. -P .. . —P —Q

This shows YeZ . Similarly, if YGZ , then YGZ . In

(particular, this is true for any ZeaC(X). Therefore, for all

Z€ 0C(X), the P-equivalence class EP and the Q—equivalence

_ _P .
class ZQ are the same set and the correspondence 20 a Z is

' - 3—1
induced by an equivalence operator -- .

43
Sufficiency; let.EZ be an equivalence operator from
J: rfi-Q -P
L2(X,Q) to L2(X,P) such that, for all 26 (X),;:,Z = Z .

-lfi a I
= - . l o erator from
Let ® §P jig Then ® is an equiva ence p

H(rb) to H(rE), and

n n
_ -l'-—' 0
® .Zairo("ti) ‘EP :2 (gait-1d 'ti)
1 l
n
=§—l E Z a X(t )Q
P .
l
n
-1 p
l
n
= . aiF‘P( .t > .
1

Hence, by Theorem 1.4.1, P and Q are equivalent.

1.5 Gaussian measures in abstract Hilbert space.

In this section we consider the equivalence problem
of Gaussian measures in Hilbert space.

Let H be a separable real Hilbert space with inner
product (-,-), let {5 be the 0-field of subsets of H generated
by all continuous linear functionals on H, and let P and Q
be two Gaussian measures on CH,13 ). We identify, as usual,
the conjugate space of H with H. Then any element f in H

may be considered as a random variable, since (f,x) is

44

6 -measurable. For the sake of simplicity, we assume that

EP(f,x)

J{ (f,X)dP(X) = O
H

and

E (f,x) J. (f,X)dQ(x) = 0
Q H

for all f€H. The operators A and B on H, defined by

(Af.g)

EP<f.x)(g.x)

(Bf.g)

E f, . .
Q( X) (g X)
are known to be bounded, self—adjoint, and positive ([21]).

We shall write V .S V

1 for any two self-adjoint operators if

2

V2 - V1 is a positive operator.

Define nonnegative definite kernels)“P andlfiQ by
F'P(f,g) = (Af.g)
F'Q(f.g) = (Big).
Letfl = H, j = 3 , and T = H. Then a version of Theorem

1.2.3 may be stated as follows.

Theorem 1.5.1 Necessary and sufficient conditions for the
equivalence of P and Q are that

(1) there exist finite positive constants c1 and c2
with

CB_<_A_<_c

1 B'

2
(2) the operator S on H(rb) corresponding tol"P has

a pure point spectrum, and

45

(3) the eigenvalues of S satisfy the relation

00

Z (143‘)2 <oo.

k=1
Proof. We need only to show that (l) is equivalent to (2) of

Theorem 1.2.3. If clr'Q < < PP < < CZPQ’ then, for every feH,
clPQ(f,f) _<_ PP(f,f) S czr'Q(f,f), i.e.,
(Cle,f) _<_ (Af,f) _<_(c23f,f),

which is (1), by definition.

Since
n n
2 = Z
. . aiajr;(fi,fj) . . a.aj(Afi,f.)
1:] 1:]
n n
= (a Z a.f., a f )
. i i i
i i
and, similarly,
n n n
E = Z
. . aiajcll-‘Q(fi,fj) (clB X aifi' . aif )
1,] i i
n n
2 . . .I - = Z - or - . i
i j alajc2r22(fl f3) (c2B i alfl .i alfl)

clrQ< < PP < < czr'o follows from (1).

Conditions (2) and (3) of Theorem 1.5.1 are stated
in terms of operator S on H(rb). One can formulate these

conditions on a different space. Let us recall the construction

46

*
of space L2(X) which was used in the proof of main theorem.
In the setup of this section aC(X) is nothing but H itself.

H may be considered as a subset of the intersection of

0C 203.3 .P) and 0C2(H. 3 ,Q), where

0C2(H, 3 ,Q) osz) = (h: J h(x)2dQ(x) < co),
H

since, for any feH, ~r- (f,x)2dP(x) and .[ (f,X)ZdQ(X) are
H H

finite.

Write, for he 0C2(P)

lthP = (h,h):/2 = [J h2<x)dP(x) ] 1/2,
H

and, for he 0C2(Q),

H h||Q= (h,h):'2/2 = [.I‘ h2(x)dQ(x) J 1/2 .
H

Observe that if fEH,

n f Hf,

.{‘ (f,X)zdP(X) (Af,f)
H

_{ (f.x)2do(x)
H

so that from condition (1) of Theorem 1.5.1,

(Bf,f).

2
Hf HQ

2 2 2
(5.1) c1 N f u 0: u f "Pic," f IIQ

*
for all feH. Just as we defined the class 42(X) , define k

47

to be the class of all real {3-measurable functions h in
0<:2(P)/“\JC2(Q) such that a sequence (fn\ exists in H with
“fn-h "P ——§ Oand Ilfn- hIIQ —-) Oasn ——> 00.
Then it follows from (5.1) that, for all h€,fic,

2 2 2
(5.2) cluhIIQS ll h "P: c2 II h HQ.
Letting ’YL be the class of functions in «with

H h ||Q = II h "P = O, we define K = 16/44., the set of

equivalence classes modulo 1L of elements in k . It may be
verified that K is a separable Hilbert space with inner

and norm H ° II . This space K corresponds

product (-,-)Q Q

* ._
to L2(X) defined earlier. Denote by h an element of K, i.e.,

the equivalence class to which h belongs. From the con-
struction it is clear that H spans K. Since, for and f, g H,

< FQ(':f)IB(°Ig)>Q = ll-‘Q(flg)

=(Bf.g)

f (f,X) (QIX)dQ(X)
H

= (f3)Q ,
H(r‘o) g. K, by the congruence theorem. Let § denote the

* _
congruence from H(F‘Q) onto K, and define S =§S§ 1. Note

that, for f,geH,

48

.§)Q =< STE—if. §_lg>Q

<l;('lf)I‘-<‘2('Ig)>o

PP(f.g)

(Af.g)

f (f,X)(g,X)dP(X).
H

for h [H

*_ _
Hence, in View of (5.2), (S hl’h2)Q, l

2GK, is the

covariance of hl and h2 with respect to P.

We have proved the following
Theorem 1.5.2 P and Q are equivalent if and only if
(1) there are finite positive constants c and c

1 2
such that

clB _< A<_ CZB,

*
(2) S on K has a pure point spectrum, and
(3) its eigenvalues (A?) satisfy
00
. 2
(l-lk) <:oo.
k=1
It might be of some interest to note that the fore-
going theorem can be proved without making use of techniques
of r.k. Hilbert space. we give the outline of the proof.

First, condition (1) is proved by the argument used

in the proof of (2) of Theorem 1.2.1. It implies (5.1) and

49

(5.2), and enables us to define the space K. For‘hl and'h'2

in K define the symetric bilinear form L(hl,h2) = CovP(hl,h2),

where
CovP(1—il,-h2) = [th(x)h2(x)dP(x) .

(Remark that, from the assumption,

EPh = .[ h(s)dP(x) = O
H

for all heK.) The definition of L is unambiguous. L is
bounded, since, by (5.2),

|L(hl,h2)| = \CovP(hl,h2)|
Slihlllp-llhzllP

*
Hence there is a bounded self-adjoint operator S on K such

*— — _ _
that (S hl'h2)Q = L(hl,h2) = COVP(h1’h2)° The remainder of
the proof, including sufficiency part, is similar to that of

Theorem 1.2.3.

1.6 Comparison of various methods

The problem of the equivalence of Gaussian stochastic
processes has been studied by many authors under various as-
sumptions (e.g., [6]. [7]. [10], [11], [12], [16], [17], [18],
[20], [25], [26], [27], L. LeCam (unpublished) and C. Stein

(unpublished)). E. Parzen ([16], [17], [18]) exploited the

50

notion of reproducing kernel Hilbert space, which was intro-
duced earlier by M. Loeve in developing the theory of second
order stochastic processes, and formulated a condition for
equivalence (in the case of same covariance functions and
different mean functions) in the form (1) of our Theorem 1.2.3.
The existence of the equivalence-or-singularity dichotomy in
the general case was established in 1958, independently, by

J. Feldman [6] and J. Hajek [11]. Their methods of proof are
entirely different. Hajek's approach is information-theoretic.
More precisely, he considers "J-divergence" defined as follows.

Let (X1,X .,Xn]be normal with respect to both P and Q, and

2,..
let p and q denote their normal densities. Then the J-
divergence of p and q is defined by

J = EPlog(p/q) - E 109(p/q).

Q

Suppose now that (xt,t€T,P)and (Xt,t€T,Q [are real Gaussian

processes. Then the J-divergence of P and Q is equal to the

supremum of the J-divergences of finite-dimensional distri-

' P ..., I
butions t ,...,t and Qt ,...,t of vectors {'Xt , Xt)
l n 1 n 1 n
i.e.,
J = sup J
T t1, ..,t €T t1, ..'tn.

His criterion is: P and Q are equivalent if and only if

JT < oo.

51

Feldman’s method is Operator-theoretic and his
conditions for the equivalence of P and Q are stated in terms
of "equivalence Operator" from a Hilbert space to another
Hilbert space (see §l.4).

Recently, efforts have been made to simplify their
proofs and to obtain more effective criteria for equivalence
([14], [19], [24], and T. S. Pitcher's unpublished work which
has been reproduced in [23]). Rozanov [24] gave a simple
proof of the dichotomy relying on properties of the information
function and obtained necessary and sufficient conditions for
equivalence which are stated by properties of an Operator de-
fined on L2(X,Q). His idea of getting conditions is to regard
r;(s,t) as a positive bilinear form L on the space L2(X,Q).
Rozanov's condition is that the operator corresponding to L

has a pure point spectrum, its eigenvalue {Kkfi satisfy the
relation Z(1-7\_k)2 < oo and the corresponding normalized eigen-

vectors {nkfi have expectations Epnk = mk with respect to P

satisfying the requirement 2m: ‘<OO. In [14] Professor G.

Kallianpur and the writer considered Gaussian measures in
Hilbert space and, using the same idea as Rozanov's, obtained
an alternative proof of the dichotomy as well as conditions

of Rozanov's form simultaneously. They also formulated

52

conditions directly in terms of mean functions and covariance
functions applying the reproducing kernel Hilbert space method
(see Theorems 1.2.3 and 1.2.4). Independently, Parzen [19]
observed that Hajek's J-divergence can be expressed as the
norm of a function in H(r‘p) ®H(I"'Q) and thus obtained con—
ditions which are also stated in reproducing kernel Hilbert
spaces (see Theorem 1.3.3).

The proof presented in § 1.2 of this thesis is es—
sentially a reproduction of that given in [14]. However,
there is a slight difference. Here we bring out the role of
r.k. Hilbert space more explicitly. Also in.§1.5 we derived
Theorem 1.5.2 from our main theorem as a corollary, which was
proved in [14] independently. The key condition in our proof
is (2) of Theorem 1.2.3, which makes it possible to define the
space L:(X) and the Operator S. As indicated in § 1.5, one
can transform conditions from H(FQ) to L;(X). One could also
state conditions in the space L2(X,Q) as Rozanov has done.
However, it seems to the writer that one needs some condition
like (2) to do so. The space L:(X) was introduced in our
previous paper [14], but the writer found later that the some—
what similar idea was used in [12], which deals with the same-

covariances-different-mean-functions case.

53

In [4] G. Baxter extended a strong limit theorem for
the Wiener process, which was discovered by P. Levy and,
independently, by C. Cameron and W. T. Martin, to a wide class
of Gaussian processes. His theorem is stated as follows.

Let {X(t),0 $:twg_1) be a Gaussian process with mean
function m(t) having bounded first derivative and covariance
kernel rks,t) which is continuous and has uniformly bounded

second derivatives for s # t. Let

D+(t) = lim‘fi(t':%;rws't)

s—at+

P(tit)-rli(sltj_

D (t) = lim t—s

sab-
f(t) = D-(t) — D+(t) .

Then, with probability one,

 

 

n
2 2 1
lim 2 [X( k )- X( k'1)] = ff(t)dt .
n-)oo 2n 2r1
k=1 0

This result of Baxter yields immediately necessary conditions
for equivalence (i.e., sufficient conditions for singularity)
for a large class of Gaussian processes (e.g., Slepian (1958)
(see [23])and [2711 Gladyshev [9] and, more recently,
Aleksev [l] generalized Baxter's theorem and applied their
results to the equivalence problem for stationary Gaussian

processes on finite intervals. The writer feels, however, that

54

the proofs of sufficiency of conditions thus obtained are dif—
ficult in many cases, and, further, one needs more general
limit theorems to get conditions in the general case.

In the following chapter we specialize our general
theorems to important particular cases and get some generali-
zations of known results previously obtained by applying some
limit theorems or other techniques. Feldman [8], applying
his general condition to the stationary finite interval case,
proved the sufficiency of Slepian's result. It is also re-
ported (see [29]) that Rozanov simplified and generalized
Feldman's theorem. For other recent work in this field we

refer to A. M. Yaglom's survey paper [29].

CHAPTER II
APPLICATION TO SPECIAL CASES

2.1 Gaussian processes with independent increments
Suppose that'[x(t),t€T\ is a process with independent
Gaussian increments, where T = [a,b] or [a,ab] (a,b, are

finite). Then, m(t) = E(X(t)-X(a)) is continuous and

V(t) = Var(X(t)-X(a)) is monotone increasing, continuous on
T. Conversely, if a continuous function m(t) with m(a) = O
and a monotone increasing function V(t) with V(a) = O are

given on T, there exists a process [X(t),t T} with independent
Gaussian increments such that m(t) and V(t) are, respectively,
the mean and the variance of X(t)-X(a).

Let (X(t) ,teT,P[ and (X(t) ,teT,Q\ be processes with
independent Gaussian increments with means m(t), O and
variances VP(t), VQ(t), respectively. The following theorem
was previously obtained by A. V. Skorokhod [25]. We give an

alternative proof.

Theorem 2.1.1 For the equivalence of P and Q it is necessary
and sufficient that

(a) VP(t) = VQ(t) for all teT,

55

56

and
(B) there exist a function Y(t) such that
t
m(t) = “/r Y(t)dVQ(t)
a
with
2
j; [V(t)] dVQ(t) < co.
First we seek for a prepresentation Of H(rb). Note
that
rb(s.t) = VQ(S) for Sig t
Vb(s) for 5.2 t
=[Tx[a.81(flxla,tl (“mo”) ’
where
jzka't](T) = l for a'S TjS t

0 fort('t.
Observe also that VQ, the measure induced by function VO(t),

is nonatomic, since VQ(t) is continuous. By Proposition 7,

there is a congruence § from H(F'Q) onto L2(VQ), where
2
L2(VQ) e {ct | [ |¢(T)I dVQ(T) < co}.
T

Proof Of the theorem. From the above representation it fol-
lows immediately that condition (Q) is equivalent to saying

m(')€H(rb), which is (l) of Theorem 1.2.3. Hence it suffices

57

to prove the equivalence of condition (a) and (2)-(4) of
Theorem 1.2.3.

Assume (2)-(4) of Theorem 1.2.3. By (2), FP(.'t)€H(FQ)

for all teT, and, hence, r}(-,t) can be represented in the

form
PP(S.t) = [Txbfia] [T)Pt(T)dVQ(T) .

where pt('r) = §PP(-,t)€L2(VQ). Since [X(t),t€T,P& is also

a process with independent increments, for all s 2 t,
T T =
hum“ ”’6 )vam Pp(s,t)

= Fp(t.t>

= [Tx[a.t] [T)Pt(T)dVQ(T)

= J‘T‘xia’t] (T)pt(T')x[a,S] (T )dVQ(T) -

Hence

(1.1) pt('r) = pt('r)X[a’t] (T) a.e.[VQ] -

For all s S t,

,[Txtm S] (T)Pt(T)X[5, t] (T')dVQ(T)

= jgtamflflpt”) dVQ('r)

= r;(s,t)

58

PP(S.1)

[wa' S] (T)pl('r)dVQ(T)

[Tx[a, s] (T)P1(T)x[a, t] (1:)va(7) .

Therefore,
(1.2) pl(T)x[a,t](T) = pt(T)x[a,t](T) a.e. [VQ].
From (1.1 and 1.2) we have

pt('r) = p1(T)x[a,t](T) a.e. [V
i.e., as elements in L2(VQ). pt = plx[a,t]'

. * * ,1 *
Define the operator S on L2(VQ) by S =§S§ . S

has the properties (3) and (4) of S. In particular, if there
are eigenvalues different from 1, then they are of finite
multiplicity. Since S mapsl“Q(-,t) into r%(o,t), 8* maps
x[a,t](.) into pt(-) =pl(.)x[a,t](.)° Hence S*¢ = pl¢ for
every $6 L2(VQ). We shall prove that 8* is the identity
operator, i.e., p1 = 1.

*
Let A be an eigenvalue of S , and let

¢ ¢ ... ¢
k kl' k2' ' kmk

be corresponding normalized eigenvectors, i.e., a set of
eigenvalues which span the invariant subspace corresponding to

' Y
Wk. Since for all e L2(VQ)

(p1¢ki'Y)L2(VQ) = (s*¢ Y)

ki L2(VQ) = (kk¢ki’T)L2(VQ) '

59

. 2 2 2
(1'3) “ (p1 _ Ak)¢kil|L2(VQ) .[ [P1(T)-Ak‘ [(Dkim)I va(T)
T

= 0.
Let Ck be the union of the supports of ¢ki's (i = l,2,...,mk),
i.e.,
= T: ¢ . = I .= I I000! I
ck ( kl(T) o 1 1 2 mkx
and let

Dk = [T3 P1(T) = Ak}-

(1.3) implies that VQ(Dk (N Ck) = 0, where Dk' is the comple-

ment of Dk' and this, in turn, implies Vq(Ck r\ Cj) = O for

k,j W1th hk # kj, Since Dk F\ Dj = ¢ for Wk * Aj. Observe

also that these relations are independent of the choice of

*
¢ki's, i.e., if we choose another set of eigenvectors ¢ki'

* *

say, and if we denote by Ck the union of the supports of ¢ki's,

* * *
then VQ(Dk F\ Ck) = O for every A and VQ(Ck r\ Cj) = O for

k

-measure, otherwise ¢ .'s

Ak+1j. C must have pOSitive VQ ki

k
cannot be normalized eigenvectors. On the other hand, if

Wk * 1, Ck must be of VQ-measure zero. For, if it were not,

the invariant subspace corresponding to kk # 1 would be of

infinite dimension, since the measure VQ is nonatomic. This
*

contradicts the finite multiplicity of 1k # 1. Therefore, S

*
cannot have eigenvalues different from 1, i.e., S is the

60

the identity Operator. Condition (1) then follows immediately.
If (a) is assumed, rb(s,t) = [5(s,t) since the
processes are additive. Conditions (2)-(4) of Theorem 1.2.3

are trivially satisfied, concluding the proof.

Remark. In case of separable Hilbert space valued random
variables (see § 1.5), condition (a) may be stated in the
form

A=B.

This generalizes Skorokhod's result.

2.2 Gaussian_processes with covariance kernels of triangular
form.

 

Suppose that T = [a,b], a finite interval, and
{X(t),t€T,P} and [X(t),teT,Q] are Gaussian processes with

mean functions m(t), 0 and covariance kernels Pp(s,t),Fb(s,t)

of the following form (i.e., triangular form)

r;(s,t) = 9(s)¢(t) for s.g t
[ 6(t)¢(s) for 5‘2 t,
(A)
[E(SIt) = { u(s)v(t) for s.§ t
u(t)v(s) for 5‘} t,
where

¢(t) > O and V(t) > 0 for teT.
It is known that if a Gaussian process has zero mean

function and covariance kernel of triangular form, it is

Markov process.

61
It may be easily shown ([15]) that if rb(s,t) is of

the above form, then W(T)=u(T)/V(T) is nonnegative and non-
decreasing. Let w denote the measure on T induced by function
w. Define the measure H on T by
u(T) = u(a)/v(a) for T = a
K 0 otherwise,

and let

Let

* b 2 * - .
L2<w)=[\r= [mm dw m<oo)
a
It is readily verified that

b *
PQ<s,t) = fa(v<s)x[alsl(s>) (V(tm[a,t]"‘”dw (T),

*
which shows that there is a congruence § from H(F‘Q) to L2(w ).

Hence any element g(°)€H(rb) has the representation

b
gm = v<t) [ax[a,t1‘T’§9‘T)dW*(T)

* 'k
with §g€L2(w ), and, conversely, any element ‘l’eL2(w )

determines an element g =§-1‘1’€H(I"Q) by the above relation.

We impose the following conditions
(B) 9',¢',u' and v' exist and are continuous on [a,b].

(c) e'¢—’e¢' > o and u'v- uv' > 0 on [a,b].

62

Theorem 2.2.1 (See [27], [28],) Suppose that

[X(t),t€T,P[ and )X(t),t€T,Q\ satisfy conditions (A), (B) and
(C). Then, P and Q are equivalent if and only if

(a) there is a function Y(T) such that

b
m(t) = V(t) _{‘ ijitfl(T)Y(T)dw*(T)
a I

with
b

I |Y(T)|2dw*('r) < 00 .
a
(B) 6’¢ - 6¢' = u'v - uv' on [a,b], and
(Y) u(a) and 9(a) are either both zero or both non—
zero.
Proof. Condition (a) is nothing but a restatement of
condition (1) of Theorem 1.2.3, using the representation Of

H(rb). We shall prove the equivalence of (B), (y) and (2)-

(4) of Theorem 1.2.3.

Assume (2)-(4) of Theorem 1.2.3. By (2),

r}fi°.t)€H(rb) for every t€[a,b]. Hence for all s,t€[a.b],

b *
(2.1) r;(3.t) = V(S)‘J;3([als](TY§F;('It)(T)dW (T).
Write pt(1') = §[;( ° .0 (T) .

*
If u(a) = O, [1.20, i.e., w = w. Hence, from (2.1)

e<a)¢<t) = Pp(a.t> = o.

63

Since ¢(T) > O for t€[a,b], we get 9(a) = 0. Similarly,
considering the representation of H(FE) and that of rb(s,t)
in it (using (2)), we Obtain that if 9(a) = 0, then u(a) = O.
This gives (Y).

One can actually show the explicit form of pt(T).

Define, for each te [a,b],

 

p:($) = { O for T = a and if u(a) = O
¢(t)9(a)/u(a) for T = a and if_u(a) # O
9' -9 '
HIE-ugh (T)¢(t)X[a,t-_](T)
¢' -¢ '
+ 11% (T)e(t)x(t,b](fl for a < 1 Sb.
* *
Clearly pteL2(w ), and
b
V(SJ 7C[a S]('1')pt(T)dw*(‘f)
a '
s
9'v-9v' e'v—ev'
= __ 9 ._
V(S)L ¢(t) u'v-‘uv' 7C[alt] + (t) u'v-uv' x(t,b] dw
e.
+ v(s)¢(t) u::}, 3::;

for 31S t

= V(S)¢(t)[9(T)/V(T)]: + v<s>¢(t)e<a)/v(a)

= 9(s)¢(t)

.64

for s >'t

v<s)¢(t) [9(T)/v(r)1: + V(S)9(t)[¢(t)/V(T)]:

+ V(8)¢(t)9(a)/V(a)

= 9(t)¢(S).
i.e., for all s,t€[a,b],
= r;(s,t).
Hence, pt(T) = p:(T) a.e. [w*], and, as elements Of
* *
L2(w )’ pt = pt.
Let
A(T) = O for T = a
{%é%;%%;(T) for a_< T g_b,

for T = a

e'v

B(T,t) = 0
-9V' 4) ¢
WM.) (T,[t)-T,(T)) +

¢ "’V (T)(— (t)-%(T)) for a.< T§.b,

 

 

'ETV:EVT
C(T,t)= O for T = a, if u(a) = 0
9(a) ..9 ) for T = a, if u(a) # O
u(a) v
6'v-6v' 9
u'v-uv' (T) V(t) for a < T g b

Then

pt(T)/V(t) = A(T')7C[a t](T) + B(T,t)x[at](1’)

+ C(T.t)X%mb b]( T)

65
. * * 'k -1 *
Define the Operator S on L2(w ) by S =§S§ . Note that S
maps xl‘a t] (T) into pt(T)/v(t), since S maps PQ(-,t) into

* ‘k *
r'p(-,t). Define operators S , S , and S by

l 2 3
Slx[a t]( = A(T)x[alt] (T) I
$2x[a.,t] (T) = E(T't)x[a,t] (T), and

$3x[a,t] (T) = C(T,t)x[a'b] (T).

' *
Since x[’a,t] (T), t€[a,b] span L2(w ), these relations

* 'k

'k *
determine 31' 82 and S3 completely, i.e., for every h€L2(w ),

S:h(T) = A(T)h(T).

 

sghm = o for T = a
6 HV-GV )'¢ v- ¢v'
h d
u' v-uv' ———'-T( )Lb -'l'(8)x( ,(b] )u' v-uv' ——(S ) VHS)
_ b
¢'v-¢v' e'v-ev'
u'v—uv' (T) {h(s)x‘(1_:] (S)u'v-uv'
a
(s)dw(s)
for a < T S b.
S;h(T) = f O for T = a, if u(a) =0
b
{MNI h( )m(s)dw(s)
u(a) u‘v-uv'
a
+ h(a)_6_____,(<'=1) . Ma) . u(a)

u(a) ‘u(a) V(a)

 

for T = a, if u(a) + O

66

b
[ .ELX:EXL(1) S h(s)3;3:31; (s)dw(s)

 

 

u'v-uv' u'v—uv'
a
L for a (T .g b, fi u(a): O
b
u_u ¢u_¢n
.ETX_EXT(T) I h(s)_TX__XT (S)dW(S)
u v-uv u v-uv
a
+ e'v—Gv'(T)h(a)¢(a) u(a)
u'v-uv' u(a) V(a)

for a<t§b, if u(a) + O.

'k *
Hence $2 and 83 are Hilbert-Schmidt Operators with the following

Hilbert-Schmidt kernels chr,s) and K30r,s):

G'v—ev' ¢'v-ev'

K2(TIS) = m(T)X(T,b](S)W(S)

(T>7¢(T,b]<s>%<s>

¢'u—¢u'
u v-uv

 

for a <T , s.§ b

 

L 0 otherwise

9"V-9V' ¢'u—¢v'

K3(T,S) = 'ETVIEVT(T)ETV:EVT(S) for a (T , s'g b

O for T = a or s = a, if u(a)=0

'V4 V' 6 a
ér‘71%r(‘f)7é3;- for a <1- g b, s e a

6(a) T'v—qw'
u(a)'u'v-uv’(S) for T = a,a<s§b
9(a) 9(a)
u(a) 'u(a)

 

for T=s=a.

 

if u(a)f0

67

'k 'k 'k *
It is clear that S = Sl + S2 + S3 . By (3) and (4) of

*
Theorem 1.2.3, I-S is Hilbert-Schmidt. Hence

'1' * * *
I-S = (I-S ) + S + S is also Hilbert-Schmidt.

l 2 3
Consider the case u(a) = 0. Let
* 9'¢-9¢'
A (T) _'ETV:EVT(T) for a £,T g_b.
* * .
A (T) = A(T) a.e. [w ], since uEEO in this case. Hence

* *
Slh(T) =7A (T2h(T)

* *
for all h€L2(w ). Note that the measure w = w is nonatomic.

The argument used in the proof of Theorem 2.1.1 shows that

'k

*
I-Sl is zero Operator and A (T) = 1.

In case u(a) = 0,115? O, i.e., there is a point mass
at the point a. Applying the same argument, we can conclude
that ACT) = 1 for a‘< T‘g_b, 1 is the only nonzero eigenvalue

*
of I-S

1' and its corresponding invariant subspace is

l-dimensional (i.e., if Y(T) is its eigenvector, then
Y(T) = O forvr + a). Condition.(5) then follows immediately.
It remains to prove (2)+(4) Of Theorem 1.2.3 assuming

*
(B) and (y). Define the functions ptCT) (tela,b]) and the

'k

* * * * *
Operators S , 51’ 82 and S 32 and 83 are Hilbert-Schmidt,

3.
'k
and, by condition (8), I-Sl is also Hilbert-Schmidt (zero

* * * *
operator if u(a) = 0). Hence I-S = (I-Sl)—S2 - S3 is

68

. . . -l * .
Hilbert-Schmidt, and so 15 I-S = I—§ 3 § Therefore, (3)

and (4) hold. By virtue of (y),'we have, for all s,teIa,b],

bw * *
V(S) fa Xla'slhmtwmw (T) =Pp (s,t).

which shows that r;(-,t)€H(rb). Since S corresponds tolfip,

there exists a finite positive constant c2 such that

r1>< < czra. By interchanging the roles of rap and r5, we

can show the existence of a finite positive constant cl with

clrb‘< < FE. Hence (2) also holds. This concludes the proof.

Remarks. (i) Assume that m(t) is differentiable and its
derivative m'(t) is continuous on [a,b]. Then condition (a)
may be replaced ([28]) by

(a') If u(a) = 0, then m(a) = 0.

Proof. By (a), if u(a) = O,

t t
m(t) = V(t) f §m(T)dw*(T) = V(t) f §m(T)dW(T).
a . a

Hence m(a) 0. Conversely, assume (G'). Define V(T) by

Y(T) = O for T = a if u(a) = O
m(a) /v(a) for T = a if u(a) # 0
m'vlmv' for a (T g b.

(T)

u'v-uv‘

69
*
Y(T)€L2(w ) and

t
m(t) = V(t) ‘jr Y(T)dw*(T),

a
which shows m(-)€H(rb).

(ii) Theorem 2.2.1 is an improvement of D. E.
Varberg's result [27], [28], and our proof is entirely dif-
ferent from his. He assumes the existence of second
derivatives 6", ¢", u", v" and their continuity, and his

proof is based on Baxter's strong limit theorem [4].

(iii) Observe that ([4], [22])

9'¢-9¢'(t) = lim B(Slt)-r;(tlt) _ lim Pp(slt)-f‘p(tlt)
s-§t- s-t s—)t+ s-t

 

 

and, similarly,’

u'v-uv'(t) = lim [10(S.t)-r6(t,t) _ lim [h(s’t)-[§(t’t)
Sat“ 8-12 s-‘It+ s-t

 

 

(iv) The following examples are taken from [27].
Example 1 Let {X(t), 0g t < b g 1,P } be the process
with mean 0 and covariance kernel
r}(s,t) = cs(1-t) for s‘g t
[ ct(1-s) for Sig t,
where c is a positive constant, and 1et»[x(t), o.g rig b < 1,Q}

be the process with mean 0 and covariance kernel

70

[10(s,t) = min(s,t) = s for s.g t
i t for s.) t.
P and Q are equivalent if and only if c = l.
E e 2. (Ornstein—Uhlenbeck processes) Let
{X(t), o g t 3 LP) and (X(t), o g tg 1,0) be Ornstein-

Uhlenbeck processes with means 0 and covariance kernels

2
PP(S.t) = 0P exP(—BPIs-t| )
and
2
[20(s,t) — OQ exp(-Ble—t|).
. . . 2 , 2
Then P and Q are equivalent if and only if OPB P: O 0 80.
Any element g€H(rb) has the following representation
1 (.8 t
28 T 2 -8 t
_ 2 Q-~ QT dT + 0 e Q ?(0L
9(t) — 2003 Q'{O e ;C[O,t](T)Y (T)e Q

* * 2 250T
with YeL2(w ), where w = w + u, dw = ZGQBQe dT and

2

u(O) = 00'

2.3 Stationarngaussian processes
Let [X(t), - a) < t < OO,P[ and \X(t),-oo<t (an ,QSbe
stationary Gaussian processes with mean functions zero and

covariance kernels PP(s,t) and F'Q(s,t) , and let FP(7\) and

FQ(A) denote their spectral distribution functions. It is

71
well known thatr'lQ (similarly forij) can be represented in

the form

oo-i(s—t)«7\
["Q(s,t) = [e dFQ(M'

-OO
This relation gives immediately a representation of H(Fb),
°. ., H ~L F .
1e (PO): 2(0)

In this case we have the following complete
characterization for the equivalence of P and Q in terms of
their spectral distribution functions. The proof is similar
to that Of Theorem 2.2.1 and is, therefore, omitted.

c d c d
Let FP(A), FP(A), FQ(A) and FQ(A) denote the
continuous parts and the discontinuous parts of FP(A) and

FQ(A), respectively. Let

W
ll

d
FP(Ai) — F (Ki - O)

b.
1

DD: "UQJ

d
i.e., point mass at the discontinuity point Ki.

Theorem 2.3.1 ([8]). P and Q are equivalent if and only if

 

c __ c
(a) FP(7\)::FQ(?\), and

(5) their discontinuity points are the same, and

.2 J((ai/bi)-l\2 < 00-
l

72

If, in_particular, both processes have continuous

spectral distribution functions, i.e., FP_=_ FP and
c . .
FQ-EFQ, conditions (0L) and (5) are reduced to FF = F0. It

might be Of some interest to note that there is a very simple
proof to this special case.

If FP and FQ are continuous, then, by Maruyama's

theorem, P and Q are ergodic. Hence they are either identical
or singular. (This is a trivial consequence of Birkhoff’s
ergodic theorem.)

Suppose now that T = [-a,a],i.e., a finite interval,

A
5% is the Osfield generated by a subclass of random variables

[X(t),t€T\, and P and Q are the restrictions of P and Q on

3; (1,jv, No general criterion for the equivalence of P and
Q is known at the present time. However, many partial results
have been obtained (e.g., [1], [8], [19] and [29]). To
indicate the applicability of Our method, we consider a very
special case in the remainder of this section.

Assume that {X(t),P}has rational spectral density and

[X(t),Q] is an Ornstein—Uhlenbeck process with covariance

kernel PQ(s,t) = Ozexp(—B]s-t|). Since PQ(s,t) is of the

triangular form, the result stated in.§2.2 applies to this

73

* *
case, and we have H(PQ) g.L2(w ), where w = w + m,

G- - .
B'dT and u(-a)=02e 25a (see Example 2 in § 2.2).

dw = ZOZBe2
To obtain a necessary condition for the equivalence
of P and Q, we assume (2)-(4) of Theorem 1.2.3. The argument
is similar to that used in the proof of Theorem 2.2.1. By
(2), we have PP(.’t)€H(Fb) for all t€[-a,a]. Hence, letting

*
§denote the congruence from H(r‘Q) to L2(w ), for all

S,t€ [" aIa] I

a
rust) = 2023] ...-55x

-a

S] (T)I>PP( . .t) (wezmdr

{-al
+ OZe-ZBa e- 3S§[-;( ° , t) (-a) .
Noting that FP(s,t) is infinitely times differentiable for

s + t, since [X(t),P) has rational spectral density, define

 

 

 

 

O O
— (Tit) =Y — r1 (Tit) T < t
81- PP ] 51' P
w P (s.t)-I"(t.t)
lim P s—tP T'= t
[5-,]:-
9—7F‘P< .t) =' 2— F‘P(T.t) T> t
OT OT
4
‘ PP(SIt)-Fp(tlt) T: t
lim s-t

s +5t+

74

 

 

 

a2 a2
W_FP(TIU) = pr (Tl 11) u > T
PP(T.u)_ O PP(T,T) u=T
1m 61- a1“-
u-%T+
u - T
62
PP(T.u) = ["P(T.u) u < T
au’a“ his:
3 (Wm) B P(Txr)
iim_‘Ta-~P ‘WP u=T
u-iT

 

u—T

Define p(t,T), for each t€[-a,a], by

 

P(T,t) = O -eBar£)(-a,t) T 4= -a
-31;
e a O }-
2529 [ E(T'tH—TT guns) 7C {-a,t1""
-(3t

+——2-é‘[9|;(s,t)+-§—+ ["P (T when ah)

—a<T ga.

Then

2025 f e-Bsx[_a S](T)P(T.t)e2 ZBT dT+ 02 e HZEae fisp(-a.t)
—a ’
for s1§ t

= e'aSJPgm, t)+a,r — _Pp (T, t)}e m on + e-B(S+a)Pp(—a,t)

75

e‘BS (Pp(s,t)efis— FP(-a,t)e"3a} + 65‘5“” [13(-a,t)

Pp(5.t)

for s 2 t

S
-BSL{Er‘P(T,t)+—g-IE—; PP(T,t)} eBTdT

[;(T,t)} eBTdt + e

 

t
+e'fi8[ Pryce. 3.-
-a

Fp(S.t).

Hence, p(T,t) =§[;(-,t)(T) a.e. [w*]. Rewrite p(T,t), for

-a (T .g a, as follows.

f3t

 

 

 

p(T,t) = 1 {O - [‘p(t.t)- 3.; Pp(t,t) } e— 'x[—a,t] (T)
20 5 T 5T
-51 'k
e 5 ..O. }
+70 5 { Q‘T't’ + a? Ppmu 7C[_a,a]('r),
where
*
Argon.) 4.5.: FP(T.t)- efi‘T't’{__:3___ PP (t t)- -—;p[' (t t)5
5T aT OT
{ ‘1' § t
_=F Pp (T t) t.
OT T 2

-5(s+a)
PP(-a

.t)

76

Noting the stationarity of {X(t), P} , write

D=—5—F'p.(t t)-——.F["p.(t t)
5T 57

4‘12“ (0-) - Pf,“ (o +)

Define
Pl(T,t) = O T = -a and t < T g a
@025le e"at —a < T g t
P2(T,t) =10 '2 eBa P(-a,t) T = -a
P
l 6431'
{BPU,%..U+ :pr'('r,t)}~a<'r<a

 

 

20 25

* _
Then p(T,t) = pl(T,t) + p2(T,t). Define S = §S§ l.

. _ Bt .
Since Sr'Q(-,t) = r'P(-,t), S*_7C[ -a t]( T) - e p(T,t). Define
* *
operators S1 and S2 by
s*”x ( ) - (zozafl nzx ( )
1‘ [-a,t] T ’ (-a,t] T
* 5t
SZDC[_a't] (T) = e p2(T,t) .
'k 'k *
Clearly S = S1 + $2. Define

77

Ze-BTe-Bu { 2

K2(‘r,u) = ”(2025)" B gnu)

5 a
+ 5 ‘SG: rpm-mm “a.“ (u) + 8—1;: rp('r,u) X(T.a] m}

+B{:T+ M(TumMTJmH-éfr‘p'u.uncHa 1m}
t I

2

+P(T .uDC u)

["aT](

 

+ Di 5 _
Bu 5T+

 

 

< au+aT
for -a < T, u.§ a ,
26a -B(T+a)
e2 'ET——{BP (TI-a) +§—;P (TU-a);
O 20 5 p 5T p
for-a<T,-a<t§_a,u=-a,

eZBa . e-3(u+a)

.7. __2_ wry-am) +i¢Fp(-a.u)}

 

o 20 3 bu
for T=-a, -a<u§a,
0-4 e2661 P (-a,-a)
P
g
for T = u = -a .

Elementary but somewhat tedious calculation shows that K (t,u)

2

*
is the Hilbert-Schmidt kernel of 82. Since I-S is Hilbert-

*
Schmidt, so is I-S

1' Again applying the argument used in the

proof of Theorem 2.1.1, we can conclude that

77

D = 2026,

Pél)(o-) 4‘19) (0 +) =Pél)(o -) —Fél)(o +),

This condition is equivalent to the following
requirements (see [5]).

m n
If ( Z Akkk)/ ( ZBkkk)(2W) is the spectral density of
O O

2
‘X(t),P\, then n - m = 2 and ‘Am / Bn‘ = 20 5.

This is a special case of Slepian-Feldman's result

(see [8]).

[l]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

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TVLB

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3 1293 03146

3080