1
This research was partially supported by the Office of the Air Force
under grant AFOSR 68-1415 and the Office of Naval Research under grant
N00014-67A-0321-0002 at the University of North Carolina at Chapel Hill.
Reproduction in whole or in part is permitted
for any purpose of the
United States Government
EQUIVALENT GAUSSIAN MEASURES WHOSE
R-N DERIVATIVE IS THE EXPONENTIAL OF
A DIAGONAL FORM I
Balram
s.
Rajput
University of North Carolina at Chapel Hill
University of Tennessee at Knoxville
Department of Statistics
University of North Carolina at Chapel HiU
Institute of Statistics Mimeo Series No. 865
March, 1973
EQUIVALENT GAUSSIAN MEASURES WHOSE
R-N DERIVATIVE IS THE EXPONENTIAL OF
A DIAGONAL FORMl
Balram S. Rajput
ABSTRACT
A simple necessary and sufficient condition, on a trace-class
kernel
K, is given in order for the existence of a measurable
(relative to the completed product a-algebra) Gaussian process with
covariance
K.
The integral of the exponential of a certain function
of a Gaussian process with respect to the corresponding probability
measure is calculated, explicitely.
Using these results, sufficient
conditions are given on the means and the covariances (relative to
two equivalent
(~)
Gaussian measures
P
and
so that the Radon-Nikodjm (R-N) derivative
of the diagonal form in
X.
PA)
dPA/dP
of a process
X
is the exponential
Analogues of the last two results in
the set up of Hilbert space are also proved.
Using one of these re-
suIts, a simple proof of the integrability of the exponential of a
certain multiple of the square of any continuous seminorm relative
to Gaussian measure on separable nuclear space is given.
AMS 1970 subject classification.
Key words and phrases:
6OGl5, 60G30.
Gaussian process, measurable process, RadonNikodjm derivative.
lThis research was partially supported by the Office of the Air Force under
grant AFOSR 68-1415 and the Office of Naval Research under grant
N00014-67A-032l-0002.
1.
Let
(T,T,v)
be an arbitrary a-finite measure space and
trace-class kernel on
condition on
INTRODUCTION
TxT.
We give
K a
a simple necessary and sufficient
K for the existence of a measurable (relative to the
completed product a-algebra) Gaussian process with covariance
K
(Theorem 1).
Assume that
K satisfies the condition of the above theorem, so
that there exists a measurable Gaussian process
space
S
(n,F, p) with covariance K.
of X belongs to
I ni Texp{1/2
Alf{t)I~{t,w)V{dt)}P{dw) where A is a certain number
Assume the hypotheses and notation of the
previous result and let, for each
covariance function
conditions on
SA
K on
A
and
a probability measure
(ii )
A , a function
T x T
PA on
P A '" P , and
SA
on
T and a
be given; then we give sufficient
K in order that
A
is of the diagonal form in
IT
Assume, further, that the mean
L2 {v); then we, explicitely, evaluate
(Theorem 2, Corollary 2).
Gaussian,
X on a probability
(i)
SA
and
KA determine
(n,F) with respect to which
(iii)
the
R-N
derivative
X
is
dP/dP
X; i.e., is expressible as
exp {1/2 Alf(t)lx2(t,w)}~(dt) (Theorem 3, Corollary 2).
The results of the previous paragraph are motivated by some of
the work of D. E. Varberg [10] and L. A. Shepp [9], and they are
generalizations of two results of the former author and are related to
similar results of the latter.
We may point out that these results are
central and are best possible in the sense that they are proved under
minimal hypotheses on the functions
1
S and
K (see Remark 2).
2
Analogues of Theorems 2 and 3, in the set up of separable Hilbert
spaces, are also proved (Theorems 4(i) and 4(ii».
Using one of these
Theorems, we obtain a result (Corollary 3), which provides a new simple
IE
proof of the integrability of
ble
nuclear space, p
and a Gaussian measure on
number.
~
and
exp{a p2(x)}~(dx) , where
E
is a separa-
are, respectively, a continuous seminorm
E , and
a
a suitably chosen positive real
This result is not new; it follows (using properties of
measurable transformations) from an important result of H. J. Landau
and L. A. Shepp [4] and also from a result of
X. Fernique [2].
All results are stated and discussed in Section 2 and their proofs
are given in Section 3.
2.
STATEMENT AND DISCUSSION OF RESULTS
We begin by stating a few definitions, notation, and conventions
that will be used throughout the paper.
(A.I)
(T,T,v)
denote an arbitrary a-finite measure space; whenever
we write
T, it is implicitely assumed that
with it.
If
(r, A,y)
T
is a measure space, then
respectively, the completion of
A relative to
and
A
y
v
and
are associated
L2 (y)
denote,
and the Hilbert
space of real y-square integrable functions.
(A.2)
function
A real, nonnegative definite, symmetric and measurable
K on
TxT
IT K(t, t)v(dt) < m,
is called a kernel; if, in addition,
K is called a trace-class kernel.
be a trace-class kernel, and
{).
n
} and
{~n}
Let
K
be, respectively, the
positive eigenvalues (including multiplicities) and the corresponding
3
(normalized) eigenfunctions of the integral equation
(2.1)
then
K is called a Mercer kernel
(M-kerne~l)for short),
if
K
admits the representation
where
K1
~
and
are trace-class kernels such that
K2(t, t)
=a
a.e. [v] , and
K1(s,
where the series
(A.3)
If
t)
=
conve~ges
absolutely, for all
K denotes an
consistently, by
{A } and
n
M-kernel on
{~n}
s, t
E
T ,
s, t
E
T .
TxT, then we denote,
, respectively, the positive eigen-
values (including multiplicities) and the corresponding (normalized)
eigenfunctions of the equation (2.1), and by
related to
hence
{~n})
K as in (2.2).
K
l
and
~
We will assume that the set
the kernels
{An}
(and
is not finite, since it is the only case of interest here.
(A.4) We consider here only real linear spaces and real stochastic
processes.
Now, we are ready to state the first result of the paper.
4
THEOREM 1.
Let
K be a trace-class kernel on
TXT
then we
have the following:
(a)
Gaussian process
X on some probability space
is the covariance of
to
T
K is an M-kernel, then there exists a
If
X ; further, if
(n,F,
K , K , {~n}
l
2
p)
and
x
F -measurable
such that
{An}
K
are realted
K as described in (A.3), then X can be so chosen that
t e: T ,
Y and
where
K
l
and
~
Z are independent(2) Gaussian processes with covariances
, respectively, and
co
(2.4)
I:
n=l
where
in
Y 's
n
L (P)
2
n:-n
and also
Y( ., w)
N(O, 1)
=
I:
;r-n
where the series converges in
w
outside a
(b)
r.v. 's and the series converges
~
n
(.)
L2 (v)
t e: T
Finally,
Y ~w)
n
and also
a.e. [v] , for every
P-null set.
Conversely, if
K is the covariance function of a
measurable Gaussian process
then
n
a.s. [p] , for each fixed
n=l
-
n
are independent
co
.
~ (t) Y
K is an M-kernel.
X on a probability space
T
x
(n,F, p) ,
F
5
REMARK 1.
It should be noted that, for a given M-kernel, Theorem
4.1(a) guarantees the existence of a Gaussian process which has the
given kernel as its covariance and is measurable relative to the
completed product a-algebra.
The question, whether for every M-kernel
K there exists a Gaussian process which has covariance
.
K and is
measurable relative to the uncompleted product a-algebra, has a negative
answer.
This can be seen by constructing an example.
For the statements and the proofs of some of the following results,
we need a few more notation and conventions which we record in the
following:
(A.5)
K denotes an M-kernel on TxT
If
(A.3), {An} and
{~n}
are, respectively, the eigenvalues and
corresponding eigenfunctions of (2.1)) and
function on
e
>
a v-square integrable
onto the space orthogonal to the
{~n}
the linear space of
n
e
T , then we denote, consistently, bye, the orthogonal
projection of
1 - AA
(so that, in view of
0 , for all
,by
2
a real number such that
A
n , and by
L (v)-closure of
e
A
,and
KA the functions
defined as follows
.
-
= a(t)
(2.6)
00
+ A E A (1 -AA )-l<~ ,e>~ (t) ,
n=l n
n
n
n
t
e: T ,
00
E A (1 -AA )-1 ~ (s) ~ (t) + K2 (s, t),
n=l n
n
~n
n
where
<,>
as in (2.2).
is the innerproduct in
L2 (v)
and
K
2
is related to
Further, we consistently use the notation
respectively, for
D(A)
and
s, t e: T ,
K
W(A)
6
<Xl
(2.8)
(1 - AA )
II
n
n=l
and
"
(2.9)
•
II· 11
where
is the norm in
L2 (v).
The series in (2.6) and (2.7)
converge absolutely, respectively, for
t
E
follows from the boundedness of the sequence
T and
s, t
E
T.
This
{(I - AA )-1}(reca11 that
n
<Xl
E An < <Xl) , Cauchy inequality for sequences and (2.3). Since
<Xl
n=l
E A < <Xl , we have that
1 - AA > 0 , A > 0 for all n , and
n
n
n
n=l
o < D(A) < 1 . From this and the boundedness of the sequence
{(1 - AA )-l}
, it follows that
n
W(A)
is a well defined positive
real number.
In Theorem 2 and 3 and Corollaries 1 and 2, it will be assumed
that the space
~(v)
is separable.
We are now ready to state the following two results.
.-
THEOREM
2.
there exists a
space
(n,F, p)
Let
T
x
such that
F;
e
and
K
t;
e
E
L2 (v) ; then
on a probability
are , respectively, the mean
further, if
e and K as in (A.5), then
(2.10)
and
TxT
F -measurable Gaussian process
and the covariance of
to
be an M-kerne1 on
K
A and
W(A)
are related
7
THEOREM 3.
and let
Let
K,6,
A ,6 , K and
A A
~
W(A)
(n,F, p) be as in Theorem 2,
and
6 and
be related to
K as in (A.5).
KA is a covariance function, and there exists a probability
Then
measure
mean
SA
dPAldP
P A on
(n,F)
~
such that
and covariance
K
A ' P
tV
P
A
is Gaussian on
(n,F, PA) with
' and the
derivative
R-N
is given by
~
(2.11)
2
(t,w)v(dt)}
a.s [p].
.. ~ '.
REMARK 2.
It is clear, from (2.10) and (2.11), that in order'to
obtain results similar to Theorems 2 and 3 the functions
•
6
and
K
appearing in these results must guarantee the existence of the process
which is measurable and whose almost all paths are
integrable.
v-square
Since, in view of Proposition 3.4 of [7] and Theorem 1,
these conditions on
are equivalent to the facts that
~
M-kernel and that 6
is
K is an
v-square integrable, it follows that Theorems
2 and 3 are best possible, i.e., they are proved under the weakest
possible hypotheses on
6
and
K.
In order to point out the relation between the above two theorems
and the corresponding results of Varberg (Theorems 1 and 2 of [10])
and Shepp [9, p. 352], we now state two corollaries.
These corollaries
are, essentially, the restatements of Theorems 1 and 2; nevertheless,
their inclusion is necessary in order to compare our results with the
corresponding results of Shepp and Varberg.
8
COROLLARY 1.
and
p
f
Let r
be a kernel on
be measurable functions on
(see (A.2)), and
TxT
=
T such that (i) K(s, t)
l 2
l
res, t)lf(s)l / 2 If(t)l / , s, t E T , is an M-kernel, and
l 2
(ii) e(t) = P(t)lf(t)l / , t E T, is v-square integrable (both of
these conditions are satisfied, for instance, when
P E L2 (V) , and
there exists a
r
e
and
~
x F -measurable Gaussian process
such that
and the covariance of
to
is an M-kernel,
is bounded, this follows from Theorem 1).
f
(n, F, p)
space
r
and
p
on a probability
r , are, respectively the mean
Further, if
~
Then
A and
W(A)
are related
K as in (A.5), then
(2.12)
COROLLARY 2.
Let
r, p, f, K and
assume, in addition, that
f(t), 0 , t
be as obtained in Corollary 1 and let
to
e
PA
,-
and
on
K
(n,F)
as in (A.5).
such that
If(t)I-1 / 2 eA(t), t
PA ~ P , and the
E
R-N
e
~
E
be as in Corollary land
T.
Let
A,e , KA and
A
~
and
W(A)
(n,F, p)
be related
Then there exists a probability measure
is Gaussian on
with mean
If(s)l-l !2If(t)l-l !2 KA(s,t) , s,t
T , and covariance
derivative
(n,F, PA)
dPAldP
is given by
a.s. [pl.
REMARK 3.
v
= the
If
T
= [a,
b] ,T
Lebesgue measure, and if
= the
r
class of Borel subsets of
is a continuous kernel on
T,
TxT,
€
T,
9
then, by Mercer's theorem,
f
r
is an M-kernel on
is any bounded measurable function on
Corollary 1,
res, t) If(s)
TxT.
Now
if
T, then, as indicated in
Il/2~f(t) 11 / 2 , s, t
e: T , is an M-kernel.
From this it is now clear that Theorem 1 and Theorem 2 of Varberg [10]
are special cases, respectively, of Corollary 1 and Corollary 2.
These
corollaries are also related to two results of Shepp that are given
on pp. 350 and 352 of [9].
to
prove a
0-1
Finally, we mention that it seems possible
law for Gaussian process in our general setting
similar to those obtained by Shepp [9] and Varberg [10]; however, since
such a
general
0-1
0-1
law in the present setting follows from a slightly more
law recently obtained in [7], it will not be attempted here.
We shall now state two more results (Theorems 4(i) and 4(ii)).
Theorem 4(i) is important in that it is needed for the proofs of
Theorem 2 and Corollary 3.
Theorem 4(ii) is included here to show
that the analogue of Theorem 3 can be forumated for Gaussian measures
defined on abstract separable Hilbert spaces.
We assume that the reader is familiar with the elementary
properties of Gaussian measures in separable Hilbert spaces.
In the following theorem,
Hand
B(H)
denote, respectively,
a separable Hilbert space and the a-algebra generated by open sets
of
H; and
<,>
and
and the norm in
H.
THEOREM 4.
Let
l.I
II' I I
denote, respectively, the inner product
be a Gaussian mea.sure on
mean m and covariance operator
S.
(H, B(H))
Denote by {o}
n
and
with
{$},
n
the positive eigenvalues (including multiplicities) and the corresponding
10
normalized eigenvectors of
00
n
8, by
0
, a real number such that
< 1 , for all n , and, by m , the orthogonal projection of
m
onto the space orthogonal to the closed linear space generated by
{1jJ }
n
Define
=
00
E
n=l
0
n
(1 _
00
n
)-1 <1jJ , x> 1jJ
n
n
,
X E
H ,
and
u( 0)
Then we have
00
= [n~l (1
(2.14)
00
-oon )]-1/2exP [1/2 o{
_ U(o)-l <
.
(ii )
~s
00
the G
'
(3) on
auss~an measure
covariance operator 8 , then
0
d~o/d~
11~112+ n~l (l-oon )-l<1jJn ,m>2}]
~o ~ ~
(H, B(H)) with mean mo and
, and the
R-N
derivative
is given by
a.s.
[~]
,
11
as in
U( 0 )
where
(i )
The final result is a corollary to Theorem 4(i), which we now
state.
COROLLARY 3.
B(E)
Let
E be a separable nuclear space
the a-algebra generated by the open sets of
Guassian measure on
(E, B(E)).
E , then there exists an
a > 0
If
p
[8, p. 100],
E, and
~
a
is a continuous seminorm on
such that
(2.16)
3.
PROOFS
Proof of Theorem l(a).
three parts.
For clarity, we devide our proof into
In parts (i) and (ii), two auxilary processes
yl
and
Zl
are defined; and ,in part (iii), these are used to construct the required
process X .
(i)
covariance
yl
T
There exists a
Kl
x
F -measurable Gaussian process
l
defined on a probability space
can be so chosen that, for every fixed
00
yl = l:
t
n=l
.,r;::- ep
n
n
where the series converges in
are independent
N(O,l)
t
€
(nl,F , PI) •
l
T ,
yl
Further,
(t) yl
n
L2 (Pl )
r.v.'s on
and also a.s. [PI] , and
(n l , Fl , PI) . Further
with
12
where the series converges in
L2 (V)
a.e. [v] , for every
and
w
outside a PI-null set.
Proof
2!-
(i).
{yl}
n
Let
be a sequence of independent
r.v.'s defined on a probability space
two processes
(Ol,Fl , PI) •
then define the required process
yl
We first define the process
We now define
An 's ''I'n
~ 's
and yl 's , and
'
n
in terms of ~ and ~
in terms of
and
N(O, 1)
For each
~.
n , let
(t ,w) e: T x 01 •
co
Since
r
A
j
j=l
(0
n,m
<
=;r;rn
m
and
co
is the Kronecker
on,m
0 ) , it follows that
m
=
" j=n
r
n
as
n, m
~
Thus,
co.
{r
j=l
there exists a subsequence
converges in
$j}
~(v x PI) ; and, so,
~1k
{ r
$j}
which converges pointwise off
j=l
a
A.
v x PI-null set
~(t ,w)
Define
A
on
A
=
°
then, clearly, ~
there,exists a
off
is
T x Fl-measurable.
v-null set
Tl
Further, by Fubini's Theorem,
such that, for every fixed
t ~ Tl '
13
At -
the set
w ~
(t,w)
{w:
€
A}
has
PI-measure zero, and, for every
At '
~(t,w)
= lim
k
Now we define the process
Since for every fixed
~
t
€
T
co
r
<
co
(see (2.3))
n=l
and
are independent mean
co
variance
r
r.v.'s,
1
n=l
also pointwise off a
For each
t
€
;r4> (t)
n n
rn
and
Bt ' for each
PI-null set
t
€
T [6, p. 147].
T , define
n
i:
j=l
1
~'j4>j(t) Yj(w) ,
o
where
B~ denotes the complement of
Clearly, if
t
€
T~ , then
~
B
t
PI (A~
, ~(t, w) = ~(t, w) .
.
n B~) = 1
{j~l ~j(t, w)}
; further, if
is a subsequence of
Thus, for every
t
€
Finally, define
if
o
(t,w)
€
TC x n
1
1
TC
1
14
We now show that
yl
is a required process.
Since, from (3.2), the set
in
v x PI-null set
it follows that
series
Tlx Ql
yl
{(t, 00): yl(t,oo)
~
' and since
converges to
L
n=l
Z';t
Tx
is shown
T x Fl-measurable.
is
+~(t,oo)}
Fl~easurable,
Since, as shown above, the
~(Pl)
in
a.s. [PI]
and also
t e: T ,and since, from (3.1) and (3.2), y~
for each fixed
is contained
= Z';t
a.s. [PI] ,
00
for each
L (P )
2 1
and also
a.s. [PI] ,for each
yl
t
of the series to
yl 's
n
n~l Y\:'4>n (t) y~ converges to ylt in
t e: T , we have that
t e: T
yl
t e: T , we have that
is Gaussian (recall that
yl
K2 (s,t) = n~lAn4>n(s)4>n(t)
is
00
,
n~l An 4>n(S)<P n (t)
where
converges absolutely for
converges to
L (V)
2
00
t e: T,
LIn:n=
n
<P
n
(t) y~
convergence of
n~l
L
I\: <Pn (. )Y~(oo)
to
n=l
n:-4> (.) yl (00 )
n n
n
L (V)
2
Now we show the
00
Y1 (. ,00)
a.e. [v]
.
Since
00
A (yl)2}P (doo) = L A
l
n n
n=l n
<
00
00
we have that
L A (yl(oo))2
n=l
n
Since,
00
00
fQ{n~l
oo.
y~ a.s. [PI] , we
is shown to converge to
yl(. ,00) a.e. [v] , for almost all
00
.
n~lYT <Pn(')Y~(oo)
a.e. [v] , for almost all
and
have, by an application of Fubini's theorem, that
to
s, t e: T
00
To complete the proof of (i), it remains to prove that
for
convergence
00
are Gaussian) and that the covariance of
s, t e: T
~(Pl)
Also, from
n
<
00
a. s. [PI] .
Therefore,
converges
15
co
a.s. [PI]
in
n, m -+
as
co
r
; consiquently,
converges
n=l
Now using the fact that
L (V)
2
L2 (v) convergence
implies the existence of a subsequence that converges to the same
co
function a.e. [v]
r
and the fact that
n=l
n:-n
</>
yl (w ) converges to
n
co
n
w.
covariance
Proof
K
2
2t
a Gaussian process
vanishes off
T2'
2
By Kolmogorov's existence theorem, there exists
with covariance
n
(n2 , F2 , P2 ) .
space
(n2 ,F2 , P ) •
defined on some probability space
(ii).
The proof of
Let
T2
K
2
defined on some probability
be the v-null set of
T
such that
Define
Zl(t,w) = n(t w) X
' T2
n (t,w)
x
2
X
is the indicator of T x n . Then, clearly,
2
2
T x n
2
2
is Gaussian with covariance ~ ; further, since Zl(t,w) = 0
where
a.e. [v
x
(iii)
P ]
2
,
Zl
Let
Il
(n l , Fl' PI) , yl , y~ , s , and (n2 , F2 ' P2 )
Yn
= y~
0
III ' Y
= yl
0
(n,F, p) = (n
Let
be the projection of
j
Zl
T x F2 -measurable.
is
(i) and (ii), respectively.
and
K2
III ' Z
T
x
= Zl
n
0
onto
Il 2
and
x
1
T
x
,
n2 ' Fl
nj
Zl
x
be as in
F2' PI
, j = 1,2 •
x
Let
P ),
2
16
x,
then the processes
satisfy the required
r. v. 's the Yn 's
and
Y, Z
properties of Theorem l(a).
Proof of (iii).
(n, T
x
F)
onto
yl
(i) and (ii )
respectively,
Y
It is clear that
(T x n , T x
j
and Zl are
Fj ), j
and
T
Z are
T
proof follows from (1) and (ii )
p{(y
tl
, •.. ,Yt)EA,(z
n
1
sl
x
TI j
= 1, 2.
F
x
is measurable from
and
1
Therefore, since by
T
x
F-measurable.
F2-measurable,
The rest of the
and the observation that for any
'H.' zs
E:
B}
m
1
1
=P { ( \ , ... ,Y ) EA},
l
t
1
n
1
z )
P {(Z , ... ,
2
sl
E B} , where
is the class of Borel subsets
sm
of the k-Euclidian space Rk
We omit the details.
Proof of Theorem l(b):
This follows from Theorem 1 of [1] due
to S. Cambanis.
Proof of Theorem 2.
Let
X be the Guassian process on
(n, F, p)
as constructed in Theorem lea) subject to the additional condition
that
E(X )
t
=0
, t
E:
T.
Note that, as follows from the proof of
Theorem 1, this additional condition is satisfied by
the process
Zl
(3.4)
e
in the proof of Theorem lea) to have zero mean.
= Xt
F;t
then, clearly,
F;
IT
+
is a
and covariance
Since
X if we choose
e(t)
T
x
E:
T
F-measurable Gaussian process with mean
K
K(t, t) v(dt)
a.s. [pJ , and since
, t
Let
L2 (v)
<
00
and
e
E:
L2 (v) , F;(. ,w) E L2 (V)
is assumed separable,
~
induces a Gaussian
17
measure
~
w I~ 0
if
on
L2 (v)
via the map
~(',w) ~ L (v)
2
w I~ ~(·,w)
[7, Theorem 3.2].
if
~(',w) £ ~(v) ,
For each
f
£
L (V) ,
2
define (pointwise)
=J
S(f)(s)
K(s, t)f(t)v(dt) .
T
Then it follows from Lemma 3.2 and Proposition 3.5 of [7]; that
and the operator
operator of
~
S
8
are, respectively, the mean and covariance
Further, it is clear from the definition of
S
that its eigenvalues and corresponding eigenvectors, are respectively
{An}
and
{~n} (see
(A.3)).
The proof of (2.10) now follows from Theorem 4(i),
the above observations, and the following equation
which is a direct consequence
of the change of variable formula
[3, p. 163].
Proof of Theorem
1.
Define, for every
then it is clear, from (2.10), that
on
(r2,F) , and, from (3.5), that
dPA/dP
B
E
F,
P A is a probability measure
PA '\,P
with the
equal to the right side of (2.11) a.s. [p] .
R-N
derivative
Thus, the proof
18
of Theorem 3 will be complete, if we can show that
with mean
e A and coV&risnce
K
A
characteristic function, where
PA ,and
R and
sl' ... , sn
and
is Gaussian
We prove this in the following
n
EA[exp{i j~l Sj ~t}]
j
by showing that
~
is the right n-dimensional
EA is the expectation relative to
t l , ... , t n
are arbitrary elements
of,
T, respectively.
~t = Yt + Zt + e(t)
Recall that
co
and that
Ll;r- ep (.) Y (w)
n=
n n
n
,
t
£
T ,(see (3.3) and (3.4)),
y(. ,w) in
converges to
L2 (V) a.s. [p]
(see (2.5)).
Using these, the independence of the families
{Yt : t
{Zt: t
£
T}
E(Zt) = 0 , t
£
T}
£
(see footnote #2), and the facts
E(Y ) =
t
E(Z~) = 0 a.e. [v] , we have
T ,and
co
co
= n_~1\nYn2(w) + 2 n=l
L;r-n <ep n ,e>yn (w)
+
I
lei
1
2
a.s. [p] .
Using (3.5) and (3.6),we have
m
E\[exp {i j~l Sj~t }]
j
m
..
= W(A)E[exp{i
j~l Sj~t
+ 1/2
\JT~2(t,w)V(dt)}]
j
m
= W(\) E[exp{i j~lSj~t }
j
co
Noting again that
converges to
Yt
~t = Yt + Zt + e(t), t
£
T , and that
L ;r-ep(t) Y
n=l
n
n
a.s. [p] (see(2.4)), the right side of (3.7) is
19
m
k
= w(A)E [iim [exph L Sj(Jl
j=l
k
k
k
2
L
x exp{1/2 A~~l An y n + 2 n=l
.;r-n
<j>
;r--n
<<j>
n
(t ) Y + Zt
n
j
n
+ 8( t
j
,8> Y + "8"
n
j
0}
2) }]J
which, by the dominated convergence theorem, is
(3.8)
= W(;»
where
B =
n
I:\:
n
j~l Sj¢n(t j ) - iA
Now using the independence of the
r.v.'s
;r-n
<<j>
n
,8>
Yn's and the independence
of the two families (see footnote #2) {Yn : n = 1,2, ..• }
and recalling that
E[exp{i Y B + 1/2 AA
n n
n
it follows that the expression in (3.8) is
, {Zt: t
T},
20
00
= W(A) D(A)-1/2 exp {-1/2
(3.10 )
E (1 - AA )-l B2}
n= 1
n
n
m
x
m
exp{i j~l
SUbsituting the value of
00
I Ie/ 12 = n~l <~n,e>2
+
m
Sje(t j ) - 1/2 j~l k~l SjSk K2 (t j , t k )}
Bn
from (3.9) in (3.10) and observing that
I lei 12 (see (A.5)) , we see that the expression
in (3.10) is
=
m
W( A) D(A)-1/2 exp [i .E Sj{8(t ) + E AA (1 - AA )-1. (tj)<~ ,8>})
j
n
n
n
n
n=l
J= 1
x
m m
exp [-1/2 j~l k~l Sj Sk{K (t ,t ) + n~l A (1 - AA )-1~ (tj)~ (t )}]
2 j k
n
n
n k
n
00
•
(3.11)
00
00
x
exp [1/2 A{
II~ 11
2
+
n~l (1 - AA n )-1<epn,8>2}] ,
which, in view of (2.6) - (2.8), is
..
m
= W(A) D(A)-1/2 exp{i
x
exp [1/2 A{llel1
by the definition of
2
m
j~l
SjeA(t j ) - 1/2
00
+
m
j~l k~l
n~l (1 - AA n )-1<epn,e>2}]
W(A) (see (2.9)).
Thus,
SjSk KA(t j , t k )}
21
m
m m
exp{i j~l sjeA(t j ) - 1/2 j~l k~lSjSkKA(tj,tk)}'
.
as desired .
Proof
2t Corollary 1.
Since
K is an M-kernel and
there exists, by Theorem 2, a
T
a probability space
with mean
~t
= If (t ) 1-1/2
I
~t'
(n,F, p)
t
Gaussian with mean P
€
x
e
L2 (v)
€
~
F-measurable Gaussian process
e
T ; then, clearly,
and covariance
~
K.
on
Let
~~measurable and
is
and covariance r ; further, the proof of (2.12)
follows immediately from (2.10).
Proof of Corollary~. Define PA as in (3.5) replacing ~t
1 2
by ~(t j 1 / Z;t. Then by (2.12), P A is a probability measure, and,
by the definition of
PA ' PA ~ P with the
equal to the right side of (2.13) a.s. [p]
by
R-N
derivative
dPA/dP
.
F;
Since the process
l 2
l;t end since i t
~t = If(t)l /
of Theorem 3 is related to
r,;
is shown to be Gaussian on
(n,F, PI.)
e A and covariance
with mean
KA ' it follows that r,;
is Gaussian on (n, F, PA) with mean
1 2
If(t)l-l / 2 6 (t),t € T, nnd covariancel f (s)j-l/2If(t) 1- / KA(s,t)
A
s, t
T .
€
Proof of Theorem
of
H so that
iLLL:
{~n}U{~kJ
Choose an orthonormal set
is a Hilbert basis of
finite or
+00
r. v. 's on
(H, B(H), ll) , that
that
are Gaussian with mean
~
n
's
It follows that
{~n} LJ{~k}
{~k:
H ,where
R,
k
<~
n
,m>
is
is a family of independent
are degenerate at
~' 's
= 1,2, •.. ,t}
k
and variance
and
0
n
Using
these facts, Parseval's relation and the monotone convergence theorem,
we have
22
f H e1 / 2A1 Ixl 12~(dx) = f R exp{1/20(ji1<~j'
= 1im{!H
n
x>2 +
j~l<~J' x>2)}~(dx)
n
exp{1/2 0j~l<~j' x>2}~(dx)}
R,
x {!H exp{1/2 0
= lim
n
j~l<~j, x>2}~(dx)}
n
{j~l !H exp (1/2 o<~j' x>2)~(dx)}
.
R,
x
exp{1/2 0 j~l <~j, m>2}
~
t.
= [j~l(l
- OOj)]-1/2 exp [1/2 o{1
= U(o)-l
<
~
Proof of Theorem 4(ii).
Define, for every
(H, B(H)), and, from (3.12), that
j~1<~j,m>2(1
B
£
B(H) ,
P A is a probability measure on
~ ~
Po
equal to right side of (2.15) a.s.
element of
~
•
then it is clear, from (2.14), that
dPo/d~
I~I
2
1 +
with the
[~].
Let
R-N
x
derivative
be a fixed
H, then, using arguments similar to the ones used in the
23
proof of Theorem 3, it can be shown that
,This shows that
Po
operator
Therefore, since in a separable Hilbert space the mean
So
is Gaussian on
H with mean mo
and covariance
and the covariance operator determine the Gaussian measure uniquely
(see, for example, [7, p. 399]), it follows that
is now
2! Corollary 1. Since E is nuclear and p is continuous
E, there exists a continuous hilbertian seminorm
a constant
Denote by
k > 0
such that
p(x) < kq(x, x)
q(x, y) , where
=x
x
Eqlq-l(O)
+ q-l(O)
and
Hilbert space obtained by completing
is separable, since
Denote,by
y
=y
+ q-l(O) •
N in
Eq
onto
$
Therefore,
~l
=Q 0
i
=~
$-1
0
The map
N
is a continuous seminorm on
continuous,
E
£
[8, p. 102].
N , the
Let
<~,~>
=
H be the
The space
<,>
E
hence
E onto
Eq
i
H is separable.
Eq , and,by
Therefore, since quotient map
is continuous linear map from
is a Gaussian measure on
I 1$(x)1 12
Q, the
is continuous, since
is the a-algebra generated by the open sets of
E , q(x, x) =
E and
E is; therefore, since the quotient space of
i, the identity map of
quotient map of
X £
x
on
with the inner product
a separable normed space is separable, N and
B(H)
for all
q
Eq , the semi-inner product space(E, q) , and by
quotient inner product space
t.
The proof
complete.
Proof
on
= ~o
Po
, where
I,. II
E into
Q is
H.
(H, B(H)) , where
H.
is the norm in
from the change of variable formula [3, p. 163] that
q
Since, for
H , it follows
24
The proof of the corollary now follows from (2.14), (3.13) and the
'I
fact
p2(x) ~kq2(x,x) , x
E
E
We conclude by noting that if
(not necessarily nuclear) and
p
E is any linear topological space
a continuous Hilbertian
~~
in Corollary 3,
then using Theorem 4 and a proof similar to that of the corollary, one
can easily show that the integral in (2.16) is finite.
is not Hilbertian and
However, if
E is not nuclear, our method does not seem to
give a proof of the finiteness of the integral in (2.16).
p
REFERENCES
[1]
Cambanis, S. (1973).
Representation of stochastic processes of
second order and linear operations,J. Math. Anal. Appl., to appear •
.
[2]
Int~grabilit~ des vecteurs Gaussiens.
Fernique, X. (1970).
C. R.
"
Acad. Sci. 270, 1698-1699.
[3]
Halmos, P. R. (1950).
Measure Theory.
Van Nostrand, Princeton,
N. J.
[4]
Landau, H. J., and Shepp, L.A. (1970).
Gaussian process.
[5]
Sankhya. Sere A 32, 369-378.
Mourier, E. (1953), E'l~ments al~atoires dans un espace de Banach.
Ann. Inst. H. Poineare.
[6]
Neveu, J. (1965).
Probability.
[7]
On the supremum of a
13, 161-244.
Mathematical Foundations of the Calculus of
Holden-Day, San Francisco.
RaJput, B.S. (1972).
Gaussian measures on
Lp
spaces, 1
~p < m •
J. Multivariate Analysis ~, 382-403.
[8]
Schaefer, H. H. (1970).
Topological Vector Spaces.
Springer,
New York.
[9]
Shepp, L.A. (1966).
Radon-NikodYm derivatives of Gaussian measUres.
Ann. Math. Statist. 1[, 321-354.
1.
[10]
Varberg, D. E. (1967).
Equivalent Gaussian measures with a
particularly simple Radon-NikodYm derivative.
38, 1027-1030.
25
Ann. Math. Statist.
FOOTNOTES
1.
This terminology is motivated by the classical theorem of Mercer,
•
which asserts, in the present terminology, that every continuous (hence
...
trace-class, relative to Lebesgue measure) kernel K on
[0, 1]
x
[0, 1]
I
admits expansion of the type given in (2.3)
2.
That is,for any
t l , •.. , t m ; sl' ... , sn
in
T and any A
B(Rn ) , we have P{(Yt ' ... , Yt )
€
€ A; (Z
, ... , Z ) € B}
m
sl
sn
P{(Y ' ... , Y ) € A} x P{(Z , ... , Z ) € B}, where
B(Rk )
t1
tm
sl
sm
is the class of Borel sets of the k-Euclidean space Rk • As
B
€
1
B(~),
=
follows from the proof of this theorem, the families of r.v.'s
{Yn }
and
{Zt' t
€
T}
are also independent.
These two facts will
be important for us in the proof of Theorem 3.
3.
Note that, since
So
is a bounded, linear, nonnegative, self-adjoint
and trace-class operator on
Hand mo
....
26
€
H ,the measure
~o
exists