•
1
This research was partially supported by the Office of Naval Research
under Contract No. N00014-67-A-0321-0002.
2
And the Department of Mathematics, University of North Carolina at
Chapel Hill.
•
ON GAUSSIAN MEASURES IN
CERTAIN LOCALLY CONVEX SPACES l
by
BALRAM S. RAJPUT
2
Department of Statistios
of North Carolina at Chapel Hill
Unive~sity
Institute of Statistics Mimeo Series No. 797
January, 1972
•
-e
ON GAUSSIAN MEASURES IN
CERTAIN LOCALLY CONVEX SPACES
by
. BALRAM S. RAJPUT
*
~STRACT
. The purpose of this paper
~
i~.threefold: .
Firstly, the topological
s.upport of Gaussian measures on certain locally convex spaces are obtained.
Secondly, strongly convergent ser1esexpansions. of'elements in separable
Frechet spaces, related to Gaussian measures,
ar~obtained,
this'result
is applied to obtain Karhunen-Loeve type expansions for Gaussian processes.
Thirdly, it is shown that any zero mean Gaussian measure on a
separable Fr~chet space can be obtained as the a-extension of the canonical
Gaussian cylinder measure of a separable Hilbert space.
Other related
problems are also discussed.
AMS 1970 subject classification.
Key words and phrases.
Primary 60BOS, 6OG1S.
Secondary 60EOS.
Gaussian measure, Gaussian process, Karhunen-
Loeve expansion, topological support, characteristic function, locally
convex space, Frechet space.
·e
*Partially supported by the office of Naval Research under Grant
NOO014-67A-.03 21-0002 •.
1
§l
INTRODUCTION,
The main purpose of
thi~
paper is threefold:
Firstly, .it is shown that the topological support of an "inner regular"
Gaussian measure
space
(LCS)
E
~
on a complete Hausdorff separable locally convex
is the
algebrai~
sum of the mean of
is also proved) and a closed subspace of
(whose, existence
U
E; for certain
LC
spaces, this
closed subspace is described in terms of the reproducing kernel Hilbert
space (RKHS) of the covariance function of
4.1 (a), 4.2 (a) and 4.3),
u
(Theorems 3.1, 3.2, 3.3,
These results unify and extend some of the
work of Garcia, Posner and ROdemich [12], Ito [14J.and Kallianpur [18].
Secondly, str.ongly convergent expansions of elements of separable
Frechet spaces, related to Gaussian measures, are obtained (Theorems 4.1 (c),
4.2 (c».
Applying one of these results, it is shown that every Zero mean,
continuous sample paths, measurable Gaussian
-e
in any a-compact metric space
stochas~ic
process with parameter
T admits Karhunen-Loeve type expansions,
which
converge almost surely uniformly on compact subset.s of the metric
space
T
(Theorem, 4,4),
These results contain, as particular cases, two
results obtained by Jain and Kallianpur [16, 17] and also by Kuelbs (20].
Thirdly, it is shown that. any zero mean Gaussian measure on a
separable Frechet space
F
is the c-extension of the canonical Gaussian
cylinder measure of a separable Hilbert space
F
is a closed subspace of the Frichet space
functions on a a-compact metric space
be taken as the
RKHS
H
(Theorem 4.2 (b»).
C(T)
of real continuous
T, then it is proved that
of the covariance function of
If
~
H can
(Theorem 4.1 (b).
These results extend two results by Jain and Kallianpur [17, 18], one of
which is also due to Kuelbs [20] and Sato [24].
Theorem 4.2 (b) is related
to and is implied from a theorem of Dudley, Feldman and Le Cam [8].
·e
2
Two other results, which deal respectively with the characteristic
.
function of a Gaussian measure on
C(T)
(see previous
p~ragraph)
and with
the continuity of sample paths of a Gaussian stoehastic process are discussed in §5 (Theorem 5.1 and Corollary 5.1).
For the sake of brevity, we shall assume that the reader is familiar
with the notions of Gaussian measures and Gaussian cylinder measures on
linear topological spaces [2, 8, 24] and also with the notion of
§2
PRELIMINARIES.
RKHS [22].
In the following, we collect the necessary
notation and the conventions that will remain fixed throughout
t~e
paper
unless stated otherwise; we also list a few known facts that are used throughout the paper often without explicit mention:
(A.I)
If
Y
is a topological space, then
a-algebra generated by the open sets of
(A.2)
logical space, then
x*
Y.
The underlying field for all linear topological spaces, con-
sidered here, is the field of
and
A (X)
~eal
numbers.
If
X denotes a linear topo-
will denote the algebra of
c
will denote the topological dual of
The letter
E
complete Hausdorff separable
sets of
x
X.
(with or without subscripts) will denote a
LCS; if
E happens to be a Frechet space
(respectively a Banach space), then the letter
will be used for
(A.3)
(note that
E.
The letter
T
will denote the
It is known that
(respectively the letter
= Bc (F)
C(T)
[1, p. 100].
T will denote a a-compact metric space [11, p. 18]
is separable).
LCS
R(F)
F
If
T
is a a-compact metric, then
of all real continuous
fu~ctionS
topology of uniform convergence on compact subsets of
that
cyli~e3r
Bc (X) will denote the a-algebra generated by Ac (X); and, as usual,
(A.2)
B)
will denote the
B(Y)
C(T)
on
T endowed with the
To
It is well
kno~
is a separable Frechet space [11, po 205], and that if ~ ~ C(T) *,
3
then there exists a unique regular Borel measure
on
~(x)
T such that
= !x(t)A(dt),
for all
x
A with compact support
€
[19, pp. 126, 127]
C(T)
T
or [11, p. 203].
(A.4)
If
X is a linear space of real functions on some set
then, for a fixed
(A.5)
If
S
€
S, 0
will denote the evaluation map at
s
s.
H is a Hilbert space then the norm and the inner product
II· IIH
in H will respectively be denoted by
(A.5)
S,
The letters
Nand
and
<'>H.
R will respectively denote the set of
E will be
positive integers and the set of real numbers; and the letter
used to denote the expected value of random variables.
Now we list a few definitions.
DEFINITION 2.1.
negative measure on
-e
Let
(X,B(X».
is inner regular if t
is a compact subset of
DEFINITION 2.2.
where
X and
for every
D
~(U)
~(D) =
B(X),
€
sup
K
~(K),
~
Let
be an inner regular measure on
G of
X such that
measure on
'('"Hned hy
Let
U of
G.
LeS; and let
X be a Hausdorff
(X,B (X».
The characteristic function
X(~) = !ei~(x)~(dx), for every ~
(X,B(X»,
~(G)
€
=
~(X)
The set
G
~.
will be called the topological support (or simply support) of
l'.1hility
K
Then it is easy to show
> 0, for every non-empty open subset
DEFINITION 2.3.
where
D.
that there exists a unique closed subset
and that
be a non-
We say, following Bourbaki [2, p. 45], that
are as in Definition 2.1.
B(X)
~
X be a topological space and
~
be a pro-
X of
~
is
X*.
X
"",
"""'·'~~":~.E2ir~~~~~',~J.
there exists an element
·e
~
€
X*, then
e
e
€
B(X)
and
X such that
~
be as in Definition 2.3.
~(e)
= f~(x)~(d~),
If
for every
X
is called th6 mean of
Hahn-Banach theorem, if the mean of
~
~.
Note that. as follows from the
exists then it is unique.
4
DEFINITION 2.5.
{~
Let
stochastic process; then the
k
{~
of
s
: s
€
S}
s
~
1J
sl, S any set, be a second order real
function
6
and the covariance function
6 (s) - E(~ )
s
of
€
and
S.
be a Gaussian measure on
Let 1J
is a closed subspace
function of
€
are respectively defined by
k(s,t) • E(~s~t) - 6(s)6(t), s,t
REMARK 2.1.
: s
(F,8(F», where
F
C(T); then by the mean function and the covariance
respectively we mean the mean function and the covariance
function of the stochastic process
{6 : t
t
T}.
€
Using characteristic
functions,it is easy to show that the mean function and the covariance
function of
1J
are continuous respectively on
T and
TxT
(see, for
example, Lemma 5.1 of [5]).
§3
THE SUPPORT OF GAUSSIAN MEASURES ON
E.
In this section, we prove the following three results.
THEOREM 3.1.
mean of
1J
1J
be a Gaussian measure on
Let
1J
be a mean
(E.8(E».
subspace of
Then the support of
1J
Let
LCS
measure on
·e
6 + G, where
is
RKHS
E
= RS
of all real functions on
G is a closed
(E.8(E».
Then the support of
of the function
is the E-c1osure of
k(s,t) ..
hI (S)
..::. ~)~,
.!h!.
be the complete Hausdorff
S endowed with the topology of
H be a mean
Let
pointwise .. convergense.
H(k)
inner regular Gaussian measure
S be any non-empty set with
cardinality of the continuum; and let
separable
6
E.
THEOREM 3.3.
is the
(E,8(E»; then the
exists.
THEOREM 3.2.
on
Let
6
H is
inner regular Gaussian
6 + H(k) , where
J 6 (x)6 t (x)1J(dx) - 6(s)6(t)
E s
H(k)
and
H(k).
First we state and prove two preliminary results; namely, Lemmas 3.1
and 3.2.
L~a 3.2 will
be used for the prod! of
theofem
3.1; and
5
Lemma 3.1 will be used for the ,proof of Lemma 3.2 and
a1"80
for the proofs
of some of the later results.
LEMMA 3.1.
(a)
Let
F b-e a closed subspace of
C(T).
Let
U(F)
be the a-algebra generated by the sets of the form
«\ (x), ... ,<\ (x»
{x E: F:
R(n) ,and
-
space
Let
(b)
(depending on
.n -+
;n(~)
--
-Then
U(F) = B (F).
c__
be a Gaussian measure on
;) 'in
(F iB (F) ). where
F
is the
; E: F* 'there exists a sequence '{~ : n E: 'N}
:rhen for every
n
sp {at: t E: T}, the linear span of
conver.ges to;(x), for all
{at:t E: T}, such
x E: F. and also in
L2(~)
as
00.
Proof of (a).
A be the regular Borel
Let
corresponds to a continuous linear extension ,of
be the (compact) support of
A.
be the open balls of radius
l/n
j-l
and
= [li n,j \ i=l
u li
]
n,i
A
n,j
rn
u A
j=l
= T
n,j
is the metric on
Choose any
T.
For each
;
to
T .
I
Set
j = 2, ...• r .
n
s,s' E: A j' then
n,
~easure
on
T that
C(T); and let
n E: N, let
covering
n TI , for
Further, if
l'
; E: F*,then ;
It is sufficient to prove that if
B(R)/U(F) measurable.
i~
D}, _,w_h_e_r_e__n-..,;;,E:_N
...........;D;;.........;a-...,;B;;.;o;.;r;.;e;.;l;;....,;s;.;;e;.;t;;.....;;:;o;,;;;f~t;;:h:=e....:;;n:..,-;;Eu;;c,;:;,l,;:;,i,;:;,d::;.;e:;,;a::.=n
tl, ••• ,t n E: T.
~
same as in (a).
~
E:
n
1
li n, j ' j
An,l
T
I
= 1, ••• , r n ,
= lin,I
n
TI
Clearly
d(s.s')
<
2/n, where
Without loss of generality, we assume that
d
An, j
:/-0.
t
j E: A j ' Let x be any arbitrary but fixed element of
n,
n,
define the functions' {x : n E: N} on T by
n
I
r
n
L x(t
x (t) =
u
j=l
·e
where
I
x
T
I
on
A
n,j
n,
j)IA(t) ,
n,j
is the indicator function of
and an
E: - a
An, j '
(3.1)
Uniform continuity of
argument show that
x (t) -+ x(t)
n
<3.2)
F',
6
~
as
~ ~,
n
for all
t
€
T1 "
From (3.1), we have
Ixn(t)\ ~ [sup Ix(s)IIIAI(T1 ) <~, (here and he~eafter
S€T
IAI
denotes the
1
total variation of
A).
Using this, (3.2) and the dominated convergence
theorem [10, p. 151), we have
~(x) •
r
J x(t)A(dt)
- lim En 6 (x)A(A j)'
n+co j-1 tn,j
n,
T
1
x was arbitrary, (3.4) holds for all
Since
B(R)/U(F)
measurable.
Proof of (b).
A
n,j
(4),
's
~
n
and
(x)
T
1
€
Let
for all
F* be
~ €
x
Set
F, as
€
arbitr~ry
~
rn
(x) • E 0 (x)A(A j)'
n
j
tn,j
n,.
n
and let
111· llO~-l.
€
F.
~
Let
Gaussian measure on
is
B
2
C(T );
1
measurable and 111 is a
~ is B(B) /B(F)
= C(Tl)'
Then, by
(3.5)
-0,
be the natural projection of
Then
A,
Thus
+~.
rn
x
~
and fixed; and let
1itlll~(x)-E 0t(x)A(A j)1
n+coj=l
n,j
n,
for .a11
F; and hence
This completes the proof of (a).
be as in (a).
+ ~(x),
x
(3.4)
F
into
Using the change of variable formula for
a measurable transformation [13, p. 163], we have
III
F T
l
Let
k(s,t).
x(t)A(dt) -
J 0s(x)Ot(X)ll(dx);
rn
E
j=l
then
2
°tn,j
(x)A(A j)1 ll(dx)
n,
k
is continuous on the compact set
F
T x T
l
1
(Remark 2.1).
Therefore
J J J ly(s)y(t)1
.~
BX T x T
1 1
<
IAI(ds) x \Al(dt) x
J f lina,s) Ik(t,t) IAI (de)
T1xT1
x
Ill 1 1(dy)
IAI (dt)
< "'.
7
III....
III
x~lp11 • Il x l x P1 1.....
integrable [10, p. 192]; and hence, by Lemma 18 of [10, p. 113], it is
Thus the map
>
(y,s,t)~
T~,-~'"
l x l. x P -integrab1e.
1
'._'.
y(s)y(t)
is
,':
x
"
From this, Theorem 13 of [10, p. 193] and from the
".
change of variable formula, it follows that
J[ J J ~(B)x(t)l (ds)
x
F T1xT1
=f
-
[J J y(s)y(c)~(ds)
x l(dtglJl(dY)
:8 Tlx Tl
• Jf
f!y(s)Y(C)P 1 (dY)]l(dS) x l(dt)
LB
T1xT1
ff
=
~ (dt~ P (dx)
k(s,t)l(da) x l(dt).
(3.7)
TlxT1
Using (3.7) and arguments similar to thoQe used to obtain (3.7), it
follows that the left hand member of (3.6) is hounded by the real constant
2M[l l
I2 (T1 )
+ Il l(T 1 )], wheX'e
M ='BUp
Ik(s,t)l.
Using this, (3.5),
s,teT
l
(3.6) and the dominated convergence theorem, we obtain
.lim fl;(x) - ; (x)12p (dx)
0; i.e.,; +; in L (P) as
2
~ F
n
n
....:c·
1;1
completes the proof of (b).
.
.
~,
' •.. ~,
-~
~.to
'.
n +~.
This
: :,"
.
The above pX'oof is given in some detail, because the aX'guments
used in it are typical and will be used later at several places.
LEMMA 3.2.
of
Let
H
be a Gaussian measure on
(B.8(:8»; then the mean
P exists.
Proof.
proof of this result follows from Corollary 1 of [1,
(" '..
p. 112] and a recent X'esult of Skorokhod [26, p. 508]. Here we present
~
a proof of the lemma which does not use either of the results mentioned
"':.
... -~
in t1:u~ pr.~~o•• sentenc:_. Let P1 be a Gaussi~n mea~.~li'e .~n .C[O,l].
.',
·e
tJ~f:itle
8(t)· .
f 6~CX)~~Cdl:);
C[O,l]
then e
(C[o.:t:1
(ttetiaad: fa)
arid,
8
clearly, 0t(6)
~
if
f
0t(~)~l(dx).
C[O,l]
..
=
r
C[O,l] *, then
€
It follows,from Lemma 3.1 (b),that
~ ~(x),
11 a
D'
jOt(x)
j-l n,
n,j
for all
x
€ C[O~l],
and
f
~(6)·
From this we immediately have
r
and also
n
€
N.
~(x)~l(dx).
C[O,l]
By the Banach-Mazur theorem there exists. a norm-preserving isomorphism
~
of
then
B onto a closed subspace Qf
~o
~o exists.
the mean of
~.
projective
li~it
'Il
relative t.o the maps {gae: a,e
E into
.
Let
; then
~a· ~o'ITa
B , for
a
6· (6a) ad.
a,e
€
Then
gaeBe and
6
6
53])~
is the
E•
~~
gaeaa'
I; a ~ e}.
€
€
G~ussian
I.
{'IT a(xn):n
"
We
~~IBa"
is the mean of
M.
on
is
N}
'IT a
be the
B(Ba~/B(E)
(Ba ,B(B
. a
Indeed if
€
Let.
{x : it
n'
n.
€
Sine.
N}
is
is dense itl ~ . B.
a
Den9te t;his by
will show that
Noting that
6a , a
6
€
I,
Thus,
and
indeed bel()ngs
~l
~a· ~Bogea'
'
for
I, e ~ a, and using th. change of variable formula [13, p. 163], we
have t f or every
~
€
B* and
a
e
~
~(6 a )
a,
•
f~(x)~ (d~)
B
a
·e
a
exist,.
~a
€
is
~a
e~ch
E, the.n
from Lenuna 3.2,the mean of
l!~
of [25, p.
B; then, clearly, 'ITa
a
"':1,
a countable dense set in
to
(~eorem S~4
LC,S
denotes the projective limit of a family of Banach spaces
E is separable, so is
let
by the previous paragraph,
Since every cQmplete Hausdorff
of Banaep spaces
natural projection of
measurable.
-1
~ ~o~;
o
This completes the proof of thf! lemma.
where !!-m g(leBe
a
Hence~
~
60~ the~, clearly, ~-1(60) is
Denote this by
Proof of Theorem 3.1.
{B : a €
C[O,l]~
is a Gausstan measure on
the mean of
[6, p. 93]; let
C[O,ll
•f
(1,,;,
,
~ogea(y)~~(dy)
Be
• tOS ea(8e)·
9
~
= ~cgBa(~6)'
~(6a)
Thus
for every
I
points of
~,let
~
and
a
= gea(6 e ).
€ E* ; then,
B, we have 6a
is the mean of
*
~ € B
follows that there exists an a € I
Thus
e ~ a.
6 € E.
B* separates
Since
In order to see that
6
using Theorem 4.3 of [25, p. 137], it
with
~
the change of variable formula once more,
~
= .~/' a o~a ,where
~
*
€ B.
a
a
Using
have
= ~ a (6 a )
•
~
a
o~
a
(9).
= ~ (6) •
This completes the proof.
REMARK 3.1.
The above proof actually shows that if
(E,B(E»
probability measure on
.~
each
~
a
=
~o~
-1
a
then the mean of
the mean of
(~
exists
E = lim gasBs
with
(9)
a a€
E
into
I' where
6
a
is
~a.
NOTATION 3.1.
~
Let
(E,B(E».
be a zero mean Gaussian measure on
A = {~ € E*: v(~) - I~2(x)~(dx) -
Define
e·
exists and is equal to
~
and if the mean of
the natural projection of
a
i8 any
~
a}
and let, for
~ € A,
E
A~ • the null space of
~
~.
Then the notation E(~)
(or simply
B(~)
n
~€A
A~.
'"
If
and B)
E
.
=F
(respea~:ively
will be used for
,}-.
I"'tJ
- B),· 'then
E(~) , and
'"
E
F(~)
and
'"
F
LEMMA 3.3.
Let
~
(respectively
respectively.
The following two lemmas will be needed for the proof of
-e
E, when
is known from the context) will be used, throughout this paper, for the
set
on
'"
Theor~
3.3.
be an inner regular zero mean Gaussian measure
(E,B(E»; then H(E). 1.
Proof.
lemma.
Let
We use here the
A
o
notat~~n
given before the statement of the
be the class of all finite subsets
n
of
A directed by
10
n
A = n A~. Now Theorem 2 of
n
i-I ~i
[2, p. 46] and Corollary (b) of [2, p. 16] imply ~(E). ~(n A) =
n€A o n
inclusion.
n· {tl, ••• ,t n }
If
=1
n
• inf ~(A )
n
(note that
Ao ' let
€
A 's
n
and
.....
E are closed and thus belong to
B(E» •
LEMMA 3.4. 'Let
H be a zero mean Gaussian measure on (B,B(B»;
then B (see Notation 3.1) is the support of . ~.
Proof.
Every probability measure on a complete separable metric
space is inner regular (Theorem 3.2 of [21, p. 29]), therefore, by
= 1.
Lemma 3.3, ~(B)
C[O,l]
is a closed subspace of
Remark 2.1).
--
~
But the support of a zero mean Gaussian measure on
o
€
B.
B\G.
t
vanishes on
contradicts the fact that
is easy to verify that
then
~.
B, the null space of
Let
~e.(D). ~ (6
1, vet) • 0; there.....
contains
B.
This
.....
B. G.
+ D), for every D
€
B(E).
is zero mean inner regular Gaussian measure
~e
E.
Let
~e
if and only if
G • E(~e) • E
e+s
(see Notation 3.1);
E and, by Lemma 3.2, ~e(E)
0 is any open subset of
E
As in the proof of Theorem 3.2, E •
liIn
U ~
t
t(x ) ; 0; and therefore
o
is a closed subspace of
show that if
~(G).
B* such that
~ €
Thus it is sufficient to prove that the support of
is a closed subspace of
E
So, since
Moreover, S is the support of
is the support of
~e
G.
.....
Proof of Theorem 3.2.
(E,B(E» •
is not equal to
~
By the Hahn-Banach theorem there exists a
fore, by the definition of
on
G of
~(B). 1, G is properly contained in B•
B is closed with
t(xo ) ; 0 and
It
If the support
.....
B, then,since
x
(see Theorem 3.1 of [12] and
So, by the Banach-Mazur theorem [6, p. 93], the support of
is a closed subspace of
Let
C[O,l]
•
then ~e(U) > O.
1.
We will
This will
finish the proof.
·e
natural projection of
E into
B, G
a
a
gaBBB.
the closure of
Let
1T
a
1T
a
(E)
be the
in B
a
and
11
e;
to
the restriction of
G
""
E
then, since
is a closed set of
we have. by Corollary (ii) of [3, p. 49], E.!!!! 8 G •
ae e
(not.e 'that
8 (E) • {E n D: D
mean Gaussian measure on
£
(E.8(E».
limit topology, for the given
open set
in G
a
U
a
o
is zero mean Gaussian on
lla
'II' -1 (u ) c U.
a
o a0 -
o
(G ,B(G ».
a
a
= ""lle(U)
lle (U)
""-1
= II a
-> lle('II' a (Ua »
0
0
0
o
Let
and a non-empty
I
£
lla
-1
• lle0'll'a ; then
o
o
Since
(U ), the proof of
a
lle (U)
0
>
will be
0
o.
lla (Ua ) >
implied by the proof of
o
0
0
0
is zero
From the definition of the projective
U there exists an a
such that
Let ;:;e· lle/E
Then ""
lle
[13, p. 45]).
8 (E) }
E,
We give the proof of
lla (Ua ) > 0
o
0
0
in the following:
0_
Since
that
G*
a
B
a
E is separable, it follows, as in the proof of Theorem 3.2,
G
a
and hence
0
Ga
0
sep~rable.
2 ...,
; (X)lla (dx)
- f
with v(;)
is
;
be an arbitrary element of
• o.
Thus, by the change of variable
0
0
So, by the definition of
formula, L[;o'll'a (y)]2;:;e(dy) • O.
E
Let
0
0
...,
vanishes on
on
Ga
.
E.
Therefore, since 'II'
Thus, since
~
~ai
a (U
. a ) > O.
o
·e
is dense in
arbitrary and since
is the
G
a
Ga ' ;
o
suppo~t
is separable, it
G
a
vanishes
o
of
and therefore
o
The proof is now complete.
0
REMARK 3.2.
if
(E)
o
o
follpws, from Lemma 3.4, that
II
a
II
The above proof also shovs that if
is zero mean inner regular Gaussian meas(are on
E(ll) • lim g~aBB(llB)' where
E(ll)
E -
lim
gaeBB
and
(E,8(E»; then
(respectively ~B(llB»
is the support
12
of
~
to
BB(~B)'
~B)' ~B
(respectively of
B ~ a, and
REMARK 3.3.
*~
-1·
....
8aB is the restriatibn of gaB
• ~owB •
proje~tion
is the
of
E
into
BB'
Using Theorem 3.1 of [12] aad the Banach Mazur theorem,
we have shown in the proof of Lemma 3.4 that the support of a zero mean
Gaussian measure on a separable Banach space
B is a closed subspace
B.
A proof of this result can be given by using Theorem 4.2 (c) without
appealing to the above mentioned result of [12] (note that, as will be clear
from the following section, the proof of Theorem 4.2 (c) does not depend
Thus the proof of Lemma 3.4 and hence that of
on any result of [12]).
Theorem 3.2 can be made
ind~pendent
Proof of Theorem 3.3:
Let
of the above result of [12].
be defined in the same way as in the
~e
beginning of the proof of Theorem 3.2.
Then, in view of the proof of
H(k). E(~e) • E.
Theorem 3.2, it is sufficient to show that
H(k) ~ E; which, by the definition of
we show that
is equivalent to showing that if
E
(see Notation 3.1),
is an arbitrary element of
~
First
E* with
v(~) - f~2(x)~e(x) - 0; then ( vanishes on H(k). Let ~ be fixed with
E
v(~)
• 0; there exist
~(x)
..
n
N, a j
E
E
R, t j
n
E aj&t(x) [19~ p.120]. The fact
j-l
j
n
2
f[ j-l
E a & (x)] ~e(dx) - 0; therefore
j tj
E
n
t
0=
0
implies
(3.8)
be an arbitrary but fixed element of
s; then it follows, from
(3.8), that
n
j~laj&t(x)&t~x)" 0
n
E a & (k(t,'»
j-l j t j
-e
v(~)
a.s.
E aj&t(x)
j=l
j
Let
S, j • 1, ... ,n, such that
E
we have that
facts that
•
~
~(k(t,'»
.. O.
vanishes on
sp {k(s,'): s
E
0=
0
a.s. [~e]'
Since
t
sp {k(s,'): s
S}
is dense in
Therefore
was an arbitrary element of
E
S}.
From this and from the
H(k)
[22, p. 301] and that
S,
13
the convergence in
imp1i-es ,the convergence in
-
~
foHows that:
H.(k)
.~-
[22, p. 304]. it
E
.
"
This,complet~ ~he proof of the fact
vanishes on H(k}.
.,'"
that
Now we show that
H(k)..=. .E.
wise, by the Hahn-Banach
H(k)
and
R, t j
ES,
vanishes on
n
N, a j
E
E
[19, p. 120].
~'
theo~em~
~(x)
£or same
x
j • 1, ••• ,ft, such that
Thus ,v(I;)"
JI;
E
2
equal to
i8t~eed
there exists a
P ,0,
o
H(k)
(X)lle (dx)
n
1:
=
i-I
0
~ E
E
E\H(k).
z;(x)
ail
*
E
n
1:
j=l
....
E.
For other-
such that
~
There exist
n
= j:laj<\j (x),
ajo
tj
X E
E,
(k(t ,·)]
i
n
1: ail;(k(t ,·» = 0,
= i=l
i
....
E imply that
of
I;(X )
o
-e
+O.
since
I;(x) =0
I;
vanishes on
for all
Therefore H(k): E.
x
....
E
H(k).
This and the definition
E; this oontradicts the fact
14
§4
THE SUPPORT OF GAUSSIAN MEASURES ON
OF ELEMENTS IN
F, AND RELATED EXPANSIONS
F.
In the following we state the first two results of this section.
the last two results, whose proofs partly depend on the following theorems,
will be given in the later part of this section.
THEOREM 4.1.
where
F
is a closed subspace of
covariance function
(a)
H(k2
~/Ac(H(k»
of
H(k)
k
of
C(T).
Let
-1
of the
F; and the F-c1osure of
~.
i
, where
of
H(k)
into
F is continuous and
is the canonical normal distribution
~(k)
n
r.v~'s
N}
E:
{e : n E: N}
n
of independent
(. Gaussian zero mean variance one random variables) on
ClO
r
with the property that the series
in the topology of
~
n-1 n
(x)e
Let
~
converges to
x
be a zero mean Gaussian measure on
F is any separable Frechet space.
Hilbert space
n
of
N(O,l)
(F,B(F),~)
a.s.
[~]
F.
THEOREM 4.2.
where
~CONS)
For every complete orthonormal set
H(k), there corresponds a seguence . {~ : n
(F,B(F»,
Then there exists a separable
H with the following properties:
(a)
H is contained in
(b)
The injection map
F and
th~
F-c1osure of
H is the support
~.
~/Ac(H)
•
(c)
·e
RKHS
Then we have the following:
~.
The injection map
• ~(k)oi
be the
H(k).
(c)
of
H(k)
is separable and contained in
is the support of
(b)
H be a zero mean Gaussian measure on (F,B(F»,
Let
sequence
~oi
-1
, where
For every
{~ : n E: N}
n
~
i
of
H into
F is continuous and
is the canonical normal distribution of
CONS· {e : n
n
E:
of independent
N}
of
N(O,l)
H.
H, there corresponds a
r.v. 's on
with
15
the property that the series
the topology of
e
~
E t (x)e
n-l n
n
converges to
x
a.s.
[~]
in
F.
REMARK 4.1.
is
00
If the mean of the measure
~
in Theorem 4.1 (a)
0, then, by the argument used1.n the beginning of the proof of
Theorem 3.2, it follows that the support of
denotes the F-closure of
REMARK 4.2.
H (k) •
~
is
e+
H(k), where
H(k)
:S·imUar remark applies to Theorem 4. 2 (a).
Theorem 4.1 (a) ,f.or
T compact met-ric, and Theorem
4.2 (a), for separable Banach spaces, are recently obtained by Kallianpur
[18].
Theorems 4.2 (b) and 4.2 (c), for separable Banach spaces, are
obtained by Jain and Kallianpur [17] and also independently by Kuelbs [20].
The former authors have also proved in [16] the analogue of Theorem 4.1 (c),
for
T compact metric.
Theorem 4.2 (b), for separable Banach spaces,
is also proved by Sato [24].
REMARK 4.3.
Theorems 4.1 and 4.2 contain and unify all the
results mentioned in Remark 4.2.
It is worth pointing out that the
techniques used in [16], [17], [18], [20] and [24] for the proofs of the
above mentioned results heavily depend on various properties of Banach
spaces; since these properties are not necessarily available in Frechet
spaces, some special treatment is required for the proofs of Theorems 4.1
and 4.2 in the set up of Frechet spaces.
We must also mention that
Theorem 4.2 (b) has a point of contact with and follows from Theorem 4
of Dudley, Feldman and LeCam [8]; our proof of Theorem 4.2 (b), however,
uses somewhat different set of ideas than those used in the proof of
Theorem 4 of [8].
Finally, we refer to Theorem 4.6 of Dudley [9] which
has some relation to our Theorem 4.2 (c).
The proofs of Theorems 4.1 and 4.2 are obtained by means of
Propositions 4.1 and 4.2 which will be proved first.
following notation:
We will need the
16
NOTATION 4.1.
where
F
Let
~
be a zero mean Gaussian measure on
is a closed subspace of
H~
this section,
classes modulo
will denote the
~-null
C(T); then, throughout the rest of
L2(~)-closure
*
F.
functions of
the same notation for a
B(R)/B(F)
equivalence class modulo
~-null
PROPOSITION 4.1.
Let
(a)
~
and
H(k)
functions of
be the
into
For convenience, we will use
(F,8(F».
RKHS
H(k)
of
Let
k.
closure of
k
F and the
C(T)
and
~
be the covariance
Then we have the following:
is separable and
C(T)
on
~.
~
H(k)
C(T); further,
is continuous.
H(k) ~ F(~) (Notation 3~1), where
(b)
~
F be a closed subspace of
The Hilbert space
the injection of
"e
H(k)
of the set of equivalence
measurable function
be a zero mean Gaussian measure on
function of
(F,B(F»,
H(k)
denotes the C(T)-
H(k).
(c)
There exists an inner' product preserving isomorphism
_H..:,(_k...
) -.;o~n_t~o_
H~
of
wi t~ the following properties:
~(x)· <~(x)'~>H
(i)
~
' for every
x
€
H(k)
and
~ €
F·* •
~
(ii)
and (b),
>€
If
~/H(k) €
F*
and if
~/H(k). <e'·>H(k)
(note that, by (a)
~(e)· ~.
H(k», then
,
Proof of (a).
Separability of
H(k)
follows from Theorem 2C of
[22, p. 271], Theorem 5D of [22, p. 302] and the facts that
(see (A.3»
and
k
is continuous on
TXT
of [22, p. 303] and from the continuity of
H(k)
~
C(T).
is continuous.
with ·1
-e
IXn
the RKHS
(Remark 2.1).
k
on
Now we show that the injection map of
Let
x
€
- xl IH(k) ~ 0 as
[22, p. 100],
and . {x: n €~}
H(k)
w~
n
n ~~.
have
T is separable
From Theorem 5E
TxT, we also have that
H(k)
into
C(T)
be a sequence in H(k)
Then using a well known property of
17
Ixn(t) - x(t.) I
- I<xn
- x, k(t,," »'H(k)I
n - xii H(-k)
< Ilx
Let
T
l
be a compact subset of
on the compact set
(k(.t,t).
T; ,.then ·using (4.1), continuity of
k
IIXn
as
Tl x Tl , and the fact that
sup Ix (t) - 'x(t) I + 0
tE'T
n
n + m, we conclude that
(4.1)
as
- xl IU(k) + 0
n + m; i.e.,
1
x +x
n
as
n+
m
in
C(T). 'This completes the proof of (a).
Proof of (b):
It follow&, from the .definition of
H(k) ~.F(lJ)
the p.roof of the fact
,tic
. •E; E: F
.such that
.,is equivalent ·to showing that if
v(E;)· 0, then .E;
withv(-E;)- .0; and let
T.
tE:
o
F(lJ), -that
vanishes onH(k).
Let
Let
Abe the regular Borel measure on
T that corresponds .toa .continuous linear extension .of
veE;) -0, Jmplies
J[Ix(s)A(dB)]~lJ(dx) -
~F
be f,j;xed
E;'
E;
.to
C(T).
NOll1
Ther-efore
O.
T
.x(t ) IICt(sHI.'(ds) -0
a. s. [lJ]
.aT
(4.2)
Using .(4.2) .and arguments similar to ,those used :to obtain (3.7), ,we have
(4.3)
I(Ix(t )x,(s)A(ds)lJ(dx) - Ik(t ,s)A(ds).
F TOT
0
But the right hand member of (4.3) is zero by (4.2).
E;(k(t
o
,·n -
0, and thus
sp {k(t,·):t
continuity of the injec.tion of
H(k)
Proposition) and the fact that
sp{k(t,'): t
301], it follows that
Define
Q
j
E:
from M
l
~
R, t
j
E:
into
C(T)
Let ~ • sp{k{t,'): t
E:
T}
n
onto
M2 by
T, j - l, ••• ,n.
serving isomorphism of
T}.
Using ,the
(Part (a) of this
T}
E:
E:
[22,
1's dense in H(k)
vanishes on H(k).
E;
Proof of (c):
-e
vanishes on
E;
This shows that
M
l
~(j;lQjk(tj"» -
Then, clearly,
onto
and
H •
2
~
F*
M2 - sp{6 t
:
t
E:
T}.
t Qj 6 ' where
.t j
.
J-l
is an inner product pre-
Since HI
since, by Lemma 3.1 (b), M is dense in
2
n
~nd
is dense in H(k), and
hence in H, it follows
lJ
18
~
that the unique extension of ,
isomorphism of
to
H(k)
onto H.
1.1
Now we prove (1). Let
Let
of
is an inner product preserving
H(k)
and
t€ T
~e
arbitrary
A be the measure on T corresponding to a continuous linear
~
to
C(T).
;fixed.
bu~
!
exten~10n
By the arguments stmilar to those used to obtain (3.7),
we have
!k(t,s)A(ds) • ![!x(t)x(s)1.1(dx)]A(ds)
T
T F
• ![x(t)!x(s)A(ds)]1.1(dx)
F
T
• JcSt(x)~(x)1J(dx)
F
• <'(k(t'·»'~>H·
1.1
Thus
t(k(t,·».
<'(k(t'·»'~>H
; therefore the same relation holds for
1.1
x € H(k); there eXists a sequence
.~
every element of M • Now let
l
. {x : n € N} in Ml such that x + x in H(k) as n'+ - [22, p. 301];
n
n
therefore, by (a), we also have that x + x in F as n + -. From
n
what we have prO'\fed above, we have <' (x ), t>H • ~ (x ), for all n € N.
n
1.1
n
Since x + x in H(k) as n +. and , is continuous from H(k) onto
n
~"
~
it folloWs that
F as
ltm <'(x )'~>H •
n
1.1
~
n + -, it .follows that
<'(x)'~>H
1.1
<'(x ),t>U •
n
1.1
~(x
n
and since x + x
n
) +
~(x)
as
n + -.
• ~(x). This completes the proof of (i).
1.1
That (ii) holds for any element of the form
n
Therefore
<'(x)'~>H
I: la 13 '
j-l.j t j
where n € Nt
-e
Q
j
€ R, t j € T, j • l, •• o,n, follows from the fact that
<k(t,·),x>a(k) • x(t) • (\(x), for all
t € T and x € R(k).
in
19
Now let
{~
n
: n e N}
= If -1 (l;).
e
~n/H(k)
~
n
• <e, ·>H(k).
n
implies
t
:
t e T}
n
as
= 1;.
n
Then 'JI(e)
- <en' ·>H(k)·
00
in H
lJ
-+- ~
n
~/H(k)
-+-
sp{c5
of elements of
x e F, and
all
'*' ,
t e F ; then,by Lemma 3.1 (b),there exists a sequence
-+-
sua.h that
00.
Let- e
n
1;
n
(xl-
-+- ~
= 1f-1 (1; n )
(x) , for
and
From the previous paragraph, for all
Clearly If(e)
-~.
n e N,
Thus we need only verify that
TOH see this we observe that
en
lim <en'Z>U(k) - <e,x>H(k)" for all
-+-
e
in
H(k)
as
x e H(k); and
n+-
~ ~n (x) • t (x), for all
f-or all
x
E:
~n (x) • <en,x>H(k)'.
x e H(k), and the' fact. that
H(k) .. imply that.
<e'X>H(k)' fO.r all
I; (x) •
x
E:
H(k).
This
completes the proof.
PRQPOSI'l1IOK
(F,B(F», where
the
RKHS
4~
F
2.
Let
l.l
be a ze·ro mean Gaussian measure on
is a closed subspace of
of the covariance function
k
of
C(T)i and let
H(k)
be
~.
E: N}
is a· CONS in' H(k) and ~ • If(e ), where
n
n
n
is as in Proposition 4.1. then ~ n 's are independent N(O,l) r.v.'s
(a)
on
If . {e : n
(F,B(F),lJ).
(b)
If
e n 's
and
~
n 's
n
are as in (a) and if, for every
1: ~j (x)~ (e )
j
j;-1
.::i~n...lH~-_m;;.;;e;,;;;a;;.;;;s_u_r_e...,;a;;.;;s
n__
-+-_oo_, then
the topology of
F.
REMARK 4.4.
4.1 (c) and 4.2 (c).
00
1: ~j (x) ej
j-1
Proposition 4.2
-+- ~ (x)
converges to
(4.4)
x
a.s. [HJ
is crucial for the proofs of Theorems
This proposition resembles certain results of Ito
and Nisio [15J where various equivalent criteria for
a.s. convergence for
series of symmetric independent B-va1ued random variables are given.
·e
in
These
results of Ito and Nisio are used by Jain and Ka11ianpur [17] and Keu1bs [20]
to obtain Theorem 4.2 (c) for separable Banach spaces.
20
The proof
adaptati~n of
and a careful
Nisio [15].
~£ Propo~ition
4.2 depends on a result of Tortrat [27]
a few results
an~
the techniques of Ito and:
The following two elementary lemmas will also be needed.
measures on
{~ : n €
Let
LEMMA 4.1.
(F,B(F».
p. 40J, then
~n
If
be a sequence of zero mean Gaussian
~
converges weakly to
-
n ~ ~
as
[21,
is also zero mean and Gaussian.
~
Proof.
N}
n
Proof easily follows by using characteristic functions (see,
for example, Lemma 5.1 of [5]).
LEMMA 4.2.
If for every
Let
be a zero mean Gaussian measure on
~
F*, there ex!sts an
~ €
.;;t,;,;;h,;;;e_c;;.;h;;;;a:;;:;:r;,;:a;;,;;c;,;:t_e_r,;;;i,;;;s_U...c~f;;.;u;;;:n;;,;;c;,;:t_i;.;;o,;;;n . .xu..-_o;;.;f
degenerate at the null element of
Proof.
'e
The
n
o
£
(F,B(F».
N such that, for all
~__,;;;i_._u,;;;n_i_t
..y'_ooo;a~t
F.
1
~,
-
then
~
n > n ,
-
0
is
condition implies that v(t)· IE2(x)~(dx) • 0 for
F
every ;
€
*
F.
The rest of the proof follows from Lemma 3.3.
Proof of Propos:f,.tion 4.2:
of Proposition 4.1
prove (b).
the fact that
~nd
~j(X). ~j(x)ej'
Let
~j'8
since
are independent
~
used in [15, p. 41], that
on
(F,B(F),~).
of
S
Let
S
n
[21, p. 52].
n
The proof of (a) follows from c (ii)
's
(x)·
Since
€
H , fer each n E N.
Now we
~
[1, p. 100] and
r.v.'s, it follows, from an argument
are independent F-valued r.v."S defined
n
E
j-l
~
n
Since Be(F). B(F)
N(O,l)
j
~
tj(x)
and let
~n
dt~tribution
be the
is a tight measure on F
[21, pp. 28, 29]
and since Bc(F). B(F), it follows, using independence of
tj's
and
arguments similar to those used in the proof of (e) . . (d) of Theorem 4.1
of [15], that
{~
n
Let' {n : n
n
-e
: n
€
€ N}
N}
is uniformly tight .[2l,p. 47].
be an independent copy of
{~:
n
n·
€
N}.
Set
n
Un • j:lVj ' where Vj • ~j - nji then Vj's
be the distributilb of
Un.
are independent.
Let
"n
Now i ' fol1GWI, by repe.ting par. of the proef
21
of Theorem 3.2 of [15] and using uniform tightness of
~
{~
[21, p. 40].
n
: n € N}, that
{v :
n
n € N}
converges weakly to some
Therefore
{v :
n
n € N}
is conditionally compact and hence uniformly tight [21; p. 47].
be the distribution of u - U , m < n, m,n € N. Since Vj's
m,n
n
m
are independent and {v: n €N} is uniformly tight, it follows, by
n
Let
v
repeating part of the proof of (c) . . (b) of Theorem 3.1 [15], that
{v
m,n
:
m < n, m,n € N}
is uniformly tight and therefore has a subsequence
v.
that converges weakly to some measure
Now we show that
{U: n € N}
converges in
n
then there exists a neighborhood W of
~-measure.
0 in F and
€
>
Suppose not,
0 such that
(see equation (3) of [15, p. 38])
v(W)
"-
(4.5)
€.
Again it follows from [15, p. 38], that
(4.6)
~
for every
€ F* , where
functions of
Gaussian on
and
seque~tial
continuity of
null element of
{Un : n € N}
It is clear that
X~,
~ach
that, for every
= 1.
it follows, from Lemma 4.2, that
F.
vm,n
is zero mean
is also zero mean
~.measur..
{U: n e:' N}
*
€ F ,
there exists an
So, since
v
v
is zero
is degenerate at the
there exists a sequence
in
pa~agraph
{c : n
n
F.
U.ing thil _nd Theorem 3 of [27,
converges
n
the argument used in the last
[~]
~
But this contradicts (4.5); and thus we have that
converge. in
a.s.
Are respectively the characteristic
~
n > no' then 'xvL(-)
n
p. 231], we have that
converges
~v
XA(0) .. 1, it follows, from (4.6) and from the weak*
Since
Gau~sian,
and
and therefore, by Lemma 4.1, v
Gaussian.
mean
X~
v.
(F,B(F»
n0 € N such that i f
-e
1 -
<
€
Thus
{S
n
..
[~].
This along with
of Theorem 3.2 of [15] show that
in
N}
a. s.
F such that
{S
n
- c : n
n
8n - cn - [-Sn - cn ]
2
€
n e: N}
N}
22
in. y~
converges" &.s.. [p]
Therefo1."e.~ sinet.e" g::'(x)·,-
a..
fQ11oW&~' i»OII' (4.4) aDd tha'Bahn-!aueh" theer.,. that:..
.
to· x ;, a. 8. .hl]
n
I:. t,,(z)e , it
j
j~l''''
j
!it-;tCa)e
j
....
cOllVerges
in. the topology of. It•.
Now we, are"-ready to. prove:· Theertilta 4 .J.. and 4.2£
In,v1ewof.Parts (a) and (b) of Pre-
Proof of Theorem 4.1 (a):
position 4.1, we need only show that
F(P).
il(k)
is DOt a"proper subset of
But this fQ~laws .by an, araument similar. to that used in the end
of the proof of·TRearem 3.3.
Proof of Theorem 4.1 Cb):
Let
tion
tl"" ,tn
€
.F
*
and let
Let
'I'(hj~
and
If-
R
U
- t j • where.- h j
4.1 (c) ,tj/B.(t)I!I!I<hj'·>RM~ aQd' <t.! ,tj>R
be as
€
in,propos1t~u 4.1.
R(k). Then, by Proposi-
- <hi ,hj>B(k)' for all
p..
~
e
i,j - 1, •• .,n.
<hI' '>B(k)"" ,<hn'·>H(k)
are. the ssma.
Proof.: of· Theorem 4.1 (c)t
where-' 'I'
*
F.
is: as·
tl~I" .tn
Thus. tba:.jo1nt dc.trilnitlons of
1nP.ropoe~t1on
Let. t
4..1.
and
This completes the proof.
j
-. 'I'(e
t
);,.
~an'
Lat. t,
for every
j€ N.
arbitrary e1.ent of
Then"
II t(j u1:-1tj. (~)ej' . -
x)
F
-
h·
Ilj~tjt(.,.1?
-
I2p (dx)
.
Z
tllRy.
- " fl' e'",<tj"t>u1J -.·E;lli,1J
I
,,- '• .&.
by 'liOpo.:tt:1on. 4..·1 (c.).
.a.t.. ' {t j :. .j
€
right handuteillb* of (4.'h convergea to
t(
.e
~.'. t j. (X~.j) + l(x).
j"l ..
~s :it+
GO
Nl
is· -.. CONS
aHO
ihtliean. .quir~,
hom th$-, and Propositios' (4.2) ,we haYe,· that
x-
a.8. [llI
in the topology of
i1:h R , t;hus t:he
ll
a& n+·1I.; 1.'e:"
P.
aiJd h~~", ia.p-measure.
t.· t (~e4
j.-1 j
-l
converse. to
23
Proof of Theorem 4.2:
F onto a closed subspace
Then
RKHS
~
Let
C
o
8
he the topological isomorphism of
C _ C(R)
of
is Gaussian on
o
of the covariance function
define the inner product
<X'Y>H
k
(C ,B(C
o
0
of
in
(see '[19, p. 218]); and let
~.
o
».
e
and
that this
H(k)
HOc)
H to be
be the
H ... e- 1 [H(k)]
If we let
and
<ex,eY>H(k)' then H
becomes a Hilbert space with the inner product
of
Let
<'>H'
Using definitions
and Theorem 4.1, it follows, in a straightforward way,
H satisfies ,(a), (b), and (c) of Theorem 4.2.
We omit the
details.
For the statements arid the proofs of the'lasttwo'resu1tsof this section,
'we will need the following notation and assumptions:
Let
v
be a non-negative a-finite measure on
assigns non..,zero "mass for every non-empty open set
(T,B(T» "which
of T. 'Let L2 (v) be
the 'Hilbert spaee"oLal1 equivalence classes of real ;B(R)/B(T)
urab1e 'functions on
T is separable,
Let
(~t:
T whose squares 'are v'ltintegrab1e. 'Note that, since
~2(v)
(G,F,p).
t €iT)
Let
291~:
is, also separ,ab1e!,,[6, p.
be'a zero
mea~m~~surab1e-Gaussianstochastic
process with a1most.;a11 sample paths
space
meas-
k
cont~nuous defiD~on
be the covariance function of
a probability
{~t: t € T}, and
assume that
!k(t,t)v(dt) <~.
(4.8)
T
(4.8) implies that
~(.
,Ill)
Proposition 4.1 of [23]).
E:
L2 (v) .ta.'S. IP]
(6ee,; for ,example,
Let .,fAn: n € N}and Jti,,: n € N}
be the
non-zero,j!!genvalues and corresponding normalized'eigenfunetions, of the
integral equation
-e
A f(a) • !k(s,t)f(t)V(dt),
T
where
f(8)
is'defined potntliise.
Using
(4~8)
(4.9)
and the continuity.of
k
24
(see Remark 2.1) it follows easily that
on TxT
on T.
e:
Let
Or+ OCT)
41 n 's
0(~). ~(·,w),
be defined by
are continuous
~(.,w)
if
is
Subject to the above notation and assumptions we have:
THEOREM 4.3 (a).
B(C(T»/F
The
~ps
Sand
0
B(L (V»/F measurabl~; and
2..
and
are respectively
1
~ _ Po0~1 and
.
are ;z:ero mean Gaussian measures respectj,velyon
(C(T),B(C(T». and
(L (v),B(L (V»)).
2
2
.
;
(b)
~
The supports of
'e
sp{4l : n
n
L (V) ...closure of
2
THEOREM 4.4.
CONS . {en: n ~ N}
of independent
<;>f
's
are re.spectively. the C(T)-:-c1osure
N}.
RKHS
of
k.
Then for every
{~ : n ~ N}
H(k2 .. there c::.0rrespon.ds, a sequence
r.vj'j~S
N(O,l)
I; ~ (w)e (t)
converges
n-l n
n
of T. Moreover. if e n
.;;;.=,_::.:..--:.;;;;;.::..=.;:;",;..:.=.ir.-:::.::..
n
~
\.11
be,"~he
H(k)
Let
co
~
~an~
(O,F.F)
on
to~(t,w)
= ~41
n n,
'n
such that the series
a.s. [P]· uniformly on c:ompact subsets
for each .n ~ N. then-the random variables
are given by
~ (w)
n
=
1
.
~ ft(t,w)~ (t)v(dt),
Y).n T
(4.10)
n
almost surely (note that. as will be shown in the proof of Theorem 4.3,
{~41
nn
: n
€
N}
is
a
CONS
in
Proof of Theorem 4.3 (a):
H(k».
The measurability of
0
L
follows from
Fubini's theorem, measurability of the process and the fact that the
L (V) coincides
2
[I, p. 100]; and the measur~bility of 0 follows from
a-algebra generated by the class of cylinder sets of
with
B(L 2 (V»
Lemma 3.1 (a).
-e
For the proof of the fact that
~l
measure see Theorem 3.2 and Proposition 4.2 of [23]
simpler proof is also possible).
is a zero mean Gaussian
(an alternative
The proof of the fact
tha~
~
is a zero
25
mean Gaussian measure follows fa:-om' (3.. 4) and the. f.act,that
a.s·.
limit
of Gaussian random variables is Gaussian.
Proof of. Theorem 4.3 (b):
defined by (4.9).
L (V)
2
on
Let
be the bounded, linear
I<
It is well known that
op~rator
{4> n : n E: N}
is a
CONS in K(L (v»,the L2 (v)-clasure of the range of 1<. Choose {4>' € N}
2
m
.L
N} is a CONS in L (V)
from K (L (V», so that {4> : n. € N} u {4>'
'm : m E:
2
2
n·
(note that
L (V)
2
separab~e).
is
(4.10); and for each m
For each
n E: N, let
~
n
be defined by
N, 'let
€
~'(w) .. !~ (t,w)4>,' (t)v(dt)
m
T
m
almost surely.
~'
m
4>~.L K(L (V» ,. it" follows that
Since
2
- 0 a.s. [P], for all mE: N.
it follows that
is the
~n
(4.11)
E: H
(~t:
t E: T), for all
sP{~t:
L (P)-closure of
2
From the proof of Proposition 3.2 of [23],
t E: T}
H(~t:
n E: N, where
t E: T)
(a direct proof of this fact,
without using the proof of Pr.oposition· 3.2 of [23], is also possible).
{~ : n
Now we prove: (i)
{;r-4> : n E: N} is
(11)
CONS
n n
~
Let
E:
H(~t:
t E: T.
Since
and
~
Ig(s) - g(t)1
~ .L ~
~ IE(~s
is continuous, it follows that
g
H(k)
RKHS
n
,
a a.s. [Pl.
-
~
n
's
of
for all
Set
11.
E: N.
-e
all
n
m E: N,
{4> : n
n
E:
€
N.
80
N} u
But
~
n
is continuous on
T.
and
In order to
for all
k
Further, since
g E: L (V).
2
Using Fubini's
we get
n
I>::T
n
.L ~,
for all
by (4.11) and (4.12)
E: T}
T, and since
€
E(~~ ) • -!- !g(t)4> (t)v(dt),
for all
: t
= E(~t~)'
get)
s,t
'
t
k.
~t)2 ~,
-
!g2(t)V(dt) ~ E(~2)!k(t,t)v(dt) < co, we have
T
T
theorem and the definition of
H(~
is complete in
in the
t E: T)
prove (i), we must show
E: N}
n
(4.12)
n
n E: N, and
g
~'.
m
0
a.s. [Pl, for
is orthogonal to
{4>-:
m E: N}; arid therefore g. ri
m
a.e.
[v].
~ut sinct
g
26
v
is contiQuous on T and
subset of
non~zero
mass.to avery non-empty open
T, it follows thatg(t) .. 0, for· all
E(E;tE;) .. 0, for all
and hence to
(i).
assigns
t E T; Leo., E;
H(E;t: t E T).
t E T.
is. orthogonal to
".~.
Therefore
Therefore
SP{E;t: t E T}
0 a. s. [P].
Thi-s proves
(This proof is an adaptation of an argument used in the proof of
Theorem 4 of [4].)
Now.·we prove ,(ii).
Since
E[fIE; E;t~ (t)lv(dt)] ~ Ik(s,s) Jik(t,t)l~ (t)lv(dt) < ClO,.for all
T s
n
n
T
sET,
we get by rubini's theorem,
f(E; E; ) • E[....L
for all
sET.
=f
s.
n
(t)v(dt)]
It .is well known that there existsan inner product
preserving isomorphiSm a
a(n)
It; ·E;t~
~~T
s n
if and only if
H(~tJt
between
f(t)
= E(E;t Tl )..,
E T)
and
H(k)
such that
fo-r.a1l ·t E T, [22, p. 302].
Thus, from (4 ..13), aCE; ) . vr-~ , for each n E N. Sinc-e, by (1), {E; : n E N}
n
n n n
is a
CONS
CONS
in H(k).
in
H(E; : t
t
€
T), it follows that
'.
(b)~
Now we complete the proof of
~
support of
is the C(T)-c108ure of
1<.
is a
n n
From (ii) and 'Theorem 4.1, the
sp{~
n
: n
€
N}.
~rom
Theorem 6.2
:ts the L2 (v)"'c10sure of the
From this and from the well known fact that . {~ : n € N}
of [23] , i t follows that the -support of
range of
{vr-~: n ~ N}
~1
n
1<,
is an orthogonal complete set in the L (v)-c10sure of the ranse of
2
it follows that the support
Proof of Theorem 4.4:
~f
is the L (v)-c10sure Qf
2
~1
Let
the change of variable formula, k
e
and
~
sp{~
n
: n
be as in Theorem 4.3.
is also the covariance function of
€
N}.
By
~;
27
and 1:herefore, by Theorem 4.1,. to eyery
C6HS
{e: n E:N} ,of
It
corresponds a sequellCe . {~n :. n E: It}, of iDdepell'4eat
. . . . ~(G.l)
t..
= C(T}.B(C).p)
(C
t n· - 1; n
Let
(n,F ,P).
Let
dense. subset of
.e.
T
to
t (. ,IAI)},
on
T
to
x(·)}.
...
D -
coover.as to x
T.
La
be a compajct subset ofT
l
J}l -
Let
and
{i
Thi!n i t
n:.
E:
and
foll~fh
S
be a countable
n!len ( •.l~ (CIa) . converges uniformly
k; C: .n-l.
! e 0...'(')l; n (x)
D· {x
colWerges uniformly
firom the density of
S
in
T
l
and the
e 's, that
n
...
CD
nun
n {x:.
r-l j-l i-I S€S
j+i l
In=ln'
r , (s)t (x)
n
- 6 (x)
8
I
and
I~
l/r}
I
I
It
...
CD
j+i~
1: ~ (elt (w) .. t (w) I < l/r}.
n-l: n
0..
a.
-1·
P-(ltt) - poe (P) - u (D) .. 1. This completes the proof
D - nun
n {CAl:
1
r-l j-l i-I s€5
Thus
on
('a aJ'e 1nd•.pe~ N(O,l). r.v. 's
then clearlr'·
Tl •
T
l
continuity of
,
r •v. ' s
I
on
l
~le~t~(x}
uniformly on compact su~e.'s of
a. s. [u]
on
•
such that the se~i.s
there
B(k)
D E: F and
l
I
I
of the first part of the Theorem.:
Now we prove the last
par~
as in Proposition 4.1; and let
of the theorem.
~(~t:
t
~
T)
Let
~
and.
B
II
be
be the Hilbert space
I
obtained by completing
H - H(6 : t
we have
ll
t
€
sp{6 : t
t
I~ T}
T).
i n E:
Let
~(ll). Then from Lemma
in
3.1 (b)
N be arbitrary aDd fixed; define
I
a.sJI Ill].
Since
i
~
f
f[f~
I\n TxT C s
(x). (s)6tGx), (t)lJ(dx)]v(d.) x v(dt)
nl
n
!
- ,~ I fk(S,t)'n(s)~nKt)v(d.)
n TxT
I
n
n
E: H(~
t
: t
€
x
v(dt) - 1 < CD,
T), by Proposition,3.2 of [23].
I
we need only show that
Thus to complete the proof
~(~'n)r·nn' In view of Theorem 5D of [22] and
28
e
Proposition 4.1 c (i), it is sufficient to showcthat
;r-~ (t) • !6 (x)n (x)~(dx), for all
.c t
n n
definition of
n
n
t
€
T.
Butth~s ~0110ws from the
and an application of Fubini's theorem.
n
.
Note that Theorem'4.3 genrea1izes the main conclusions of Theorems 2.1
and 3.1 of [12]; and that Theorem 4.4·contains Theorem 1 of [16] and
Theorem 4.1 of [20].
REMARK 4.?
The result of Ito [14] also follows from Theorem 4.3.
It must be
point~d
out that condition (4.8) on the
covariance of the ,process is not·.;8 ·.r<estrictionfor the existence of the
Karhunen-Loeve type expansions .discussed in Theorem 4.4.
!k(t, t)v(dt) •
ClO,
For, if
one ·can define a fin·ite no~negative measure
v'
on
T
(T,B(T»
v'
such that
!k(t,t)v'(dt) <
v
is absolutely continuous with respect to
and
(see, .for example., Theorem 1 of .[4] or the proof of
ClO
T
Lemma 6.1 of [23]).
The above remark and the last part of Theorem 4 imply, in particular,
that if
{;'.:t
t ..
E:
I}, I
an interval of
R, is any zero 'mean measurable
Gau&sianstocha&tic .process with almost all sample paths oontinuous, then
;'t
;'t
admits Karhunen-Loeve type expansions which converge to
uniformly on compact
;~
v'
and
~ubsets
of
I.
Indeed if
kt
is a non-negative finite measare on
is the covariance of
(I,B(I»
tinuous with respect to the Lebesgue measure with
a. s. [P]
absolutely con-
!k(t,t)v'(dt)
< ... ,
I
then
{~':
n
'f If'
~' (t);' ,(w)
lil
n
n
n=l
n
E:
N}
.is one such expansion, where
E:
N}
Af(s)· !k'(s,t)£(t)v'(dt)
I
;'(w) • ~f~'(t,w)~'(t)V'(dt)
{A'I
n
n
and
are the non-zero eigenvalues .and the corresponding normalized
eigenfunctions of the integral equation
n
p,': n
n
almost surely.
and
29
-;t
§5
THE CHARACTERISTIC FUNCTION OF A GAUSSIAN MEASURE ON
The main result of this section is
Th~orem·
5.1.
C(T).
The proof of this
theorem essentially follows from Theorem 4.t. Proposition 4.1 and two
results ofi [8].
A particular case of Theorem 5_1 (a)
is recently proved
by de-Acosta [7].
We will need the following definition and notation:
DEFINITION 5. L
space
H; then-
11·11
Let
II' II
is
be a given seminorm on a Hilbert
•
exists a finite dimensional $ubspace
where
o
such that i f
H
is a
G
o
H: inf
€
of
G
there
> 0
£
Hand GiG, then
finite dimensional subspace of
~{x
ii for every
~al1ed n~easurable
y€G
.L11x- yll
~ d ~ 1 -
£,
nH is the canonical normal distribution of:: H.
This is a
particular case of a mor. general def1nition due to Dudley, Feldman and
'"
t.,
LeCam [8].
NOTATION 5..L
of
Let
{T:
m
m
€
N}
be a sequence of compact subsets
T sueh t'ha.-t - XlIl-i + T;. 'then, for each •
the seminorm em:
- C(T)
i-'
defined by
of
:l.C 6
Pm
CCT).
m
~.
In the following th.eorem we will n,ot distinguish
betwe~
C* a.d the me,sure
Riesz repres,entation.
THEOREM 5.1. (a)
and let
N, we wit! denote by
Pm(X) • sup Ix(t) I,
.- t€T
;
~
€
Let
e(t) - !Ot(x)~(dx)
C
-
aS8~ciated
X
~
and
to
~
in
~he
be a Gaussian measure on
(C
r(s,t) - fOs(x)Ot(x)~(d~).
_I
~.
for every
X
seminorm Pm
€
C*,where
C(T),B(C»;
Then the
C
characteristic function X of ~ is given b!
•
X(X) • exp{i!e(t)X(dt) f J~(~,t)X(ds) x X(dt)},
T
an element
t
(5.1)
TxT
k(s,t) - r(s,t) - e(s)e(t).
is continuous and n-measurable on the
Moreover, every
RKHS
H(k)
of
k.
30
(b)
a
€
Conversely, a function
C and
k
function on
TxT
such that every seminorm
H(k)
RHKS
Proof of (a):
k, is the characteristic function
(C,B(C».
A.
A be an arbitrary but fixed element of
*
C.
2
= J[Jx(t)A(dt)]2~(dx)
Since
MOreover, a(t) • J5t(x)~(dx)
C
- p2(A).
Let
CT
of
is continuous and.n-
X(A). exp{ip(A) - ~ a (A)}, where p(A)=J[Jx(t)A(dt)]~(dx)
CT
It is clear that
2
a (A)
C s
Let
Pm
(x)5t(x)~(dx).
f5
k(s,t) + a(s)a(t) •
of
H on
of a unique Gaussian measure
and
C* of the form (5.1), where
is a Symmetric, non-negative definite real continuous
measurable on the
and
X on
and
peA) = Ja(t)A(dt)
k
be the compact support
T and
l
respectively, we can use Fubini's theorem to conclude that
T x T
l
l
a
T
l
and
are continuous (see Remark 2.1) on
2
a (A) =
T
J Jk(s,t)A(ds) x A(dt).
TxT
So
X is of the
form (5.1).
pm's
are clearly continuous on
they are n-measurable is proved now.
D
€
Let
and
B(C).
Then
~a
~a(D) = ~(a
Let
by
H(k)
~a'
C
is dense in
~a
and
= ~a/C
and from the fact that every
is continuous on
is n-measurable on
Proof of (b):
»,
H(k), it follows that
Applying Corollary 2.1 of [8] to the n-measurab1e
normal distribution of
(C ,B(c
Pm
From this, Corollary 2.2 of [8]
H(k)o
and continuous seminorms
to
C
is the a-extension of the
H(k).
~o
E by
k.
Then it follows, from Proposition 4.1 (b) and Theorem 4.1)
canonical normal distribution of
Pm
+ D), for every
C be defined the same way as in Notation 3.1 replacing
~
That
is Gaussian with mean zero and covariance function
~
that
I'
H(k), by Proposition 4.1 (a).
pm 's
H(k)
where
C
0 0 0
on
H(k) , it follows that the canonical
extends to a zero mean Gaussian measure
is the closure of
H(k)
in the topology
31
{p : m €N}of seminorms.
induced by the family
covariance
k.
m
~o
Extend
~(D) • ~ (D - 6), D € 8(C).
o
6(t). f&t(x)~(dx) and
.to
!,
.C1ear1y
~
0
has the
(C,8(C»' ia thenatar'al'way and define
Then
~
is Gauss!aaon
:k(s,:t) +6(s)6{t) •
C
(C~8(C»
J& S (x)l5t(x)~(dx).
with
Now
given by (5.1) is the charactecistic function of
~,
by Part (a). If ~1
is some other measure with the same characteristic function, then .
and
~1
B(C)
=
X
C
coincides on cylinder ~ts' of
C
and hence onB(C)
~
(note that
B (C» , h e ••. ~,,' is ..unique •.
c
Using Lenma 3.1 (a), Proposition 4.1, Th. .rem 4.Land,Corollaries
2.1 and 2.2 of [81, oneiDDDediate1y obtains the followitig Corollary.
COROLLARY 5.1.
Let
6€,C(T} Jandk
definite real continuous function on
TxT.
be a symmetric non-negative
In order for the existence
of analmo..!t all s@le pa~hs' coutinUOl1$ GaUssian stochastic process
with mean
64nd
cevariance function
,
k, it is necessary
and sufficient that eve.ry seminorm"'P
is' continuous. alid n-measurab1e
m
on the
r
RKHS
R(t}" of
k.
32
UPERENCES
1.
S. Ahmad, Elements aleatoires dans les espaces vectoriels topologiques, Ann. Inst. H. Poincare, Sect. B2 (1965), 95-135.
2.
N. Bourbaki, Elements de Mathematiques, Livre VI, Integration,
Chapitre IX, Hermann, Paris, 1969.
3.
N. Bourhaki,Elem¢nts of mathematics, general tOJlO1op,y, Part I,
"'rlrliFlon-Hesley~ lvfaRRf1.chusetts, t 0 66.
~.
:~.
5.
S.~a~~anis
6.
M. M. Day, Normed linear spaces, 2nd ed.,
New York, 1962.
7.
A. D. de-Acosta, Existence and convergence of probability measures
in Banach spaces. Trans. Amer. Math. Soc. 152 (1970), 273-298.
8.
R. M.Dud1ey, J. Feldman, and L. Letam. On seminorms and probabilities, and abstract Wiener spaces, Ann. of Math. 93 (1971), 390-408.
9.
R. M. Dudley, The size of compact subsets of Hilbert space and continuity'of Gaussian processes; J.FuJ;1ct .. ·ADal.' 1 (l967),.290-330.
10.
N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience Publishers, ' LDC., New York, 1957.
11.
R. E. Edwards, Functional analysis, Holt, Rinehart and Winston,
New York, 1 9 6 5 . ' , · ' :",
12.
A. M. Garsia, E. C. Posner, E. R. Rodemich, Some properties of
measures on function spaces induced'by Gaussian processes, J. Math.
Anal. Appl. 21 (1968), 150-160.
13.
P. R. Halmos, Measure theory, Van Nostrand, Princeton, New Jersey,
1950.
14.
K. Ito, The topological support of Gauss measure on Hilbert space,
Nagoya Math. J. 38 (1970), l8l~183.
15.
K. Ito and M. Nisio, on the convergertce of sums of independent
Banach space valued random variables, Osaka J. Math. 5 (1968).
35-48.
16.
N. C. Jain and G. Kallianpur, A note on uniform convergence of
stochastic processes, Ann. Math.:Statist. 41 (1970), 1360-1362.
"
I
,
r.f1.mhnnis, and F.. Masry, On the representation of weakly continuous stochastic processes. Information Sci. 3 (1971), 277-290.
and B. S. Rajput, Gaussian stochastic processes and
Gaussian measures, 'available in the preprint form.
l "
Academi~
Press, Inc.,
33
11.
~.
C.Jain and G. Ka11ianpur, Norm conver~eDt eXDansi0R6 for
processes in Banach spaces, Proc. Amar. Math. Soc. 25
(1970), 890-895.
(~ussian
18.
G.
~allianpur, ABstract Wiener procefilsses &Il6 their repr04iucia:g
kernel IU1hert spaces, Z. Wahrscheinlicftkeitst.heorie und Verv.
Gebiete, 17 (1971), 113-123.
19.
J. L. Kelley, 1. Namioka and eo-authors, Linear
Van Nostrand, Princeton, New Jersey, 1963.
20.
J. Kue1bs, Expansions of vectors in a Banach space related to
Gaussian measures, Proe. Amer. Math. Soc. 27 (1971), 364-370.
21.
K. R. Parthasarathy,Probability measures on metric spaces, Academic,
New York, 1967.
22.
E. Parzen, Time series analysis papers, Holden-Day, San Francisco,
California, 1967.
23.
B. S. Raj put , Gaussian measures on
In the preprint form.
24.
H. Sato, Gaussian measure on a Banach space and abstract Wiener measure,
Nagoya Math. J. 36 (1969), 65-81.
25.
H. H. Schaefer, Topological vector spaces, 3rd printing, Springer,
New York, 1970.
26.
A. V. Skorokhod, A note on Gaussian measures on Banach spaces,
Theor. Prob. App1. 15 (1970), 508.
27.
A. Tortrat, Lois de probabi1ite sur un espace topologique comp1etement
regu1ier et produits infinisa termes independants dans un groupe
topo10gique, Ann. lust. H. Poincare, 91 (196~), 217-237 •
L
P
to'Po1~ical
spaces, 1
~
spaces,
p < 00, available
,
•
UNCLASSIFIED
Security Classihc~[ion
C:>CUMENT CON:-ROL DATA· R&D
(Security dessif,c"t" n of title, bod,- of
1. ORIGIN" T1NG AC T! VI Tv (C
nstrsct lind ;ndtHdn .. Itnnotation mu.'it be entered when the ow'rall f#!'por' I,,; claaslfied)
>t'por.te lJuthor)
2a. REPOAT SECURITY
Department of Statistics
University of North Carolina
Chapel Hill, North Carolina 27514
2b.
CLASSIFICATION
Unclassified
GROUP
3. REPORT TITLE
On Gaussian measures in certain locally convex spaces
4. DESCRIPTIVE NOTES (Type of report and incllls;vf" dllte ... )
Research Report
s' AUTHORIS! (First neme, middle Inltie/, lest ne."e)
Ba1ram S. Raj put
II, REPORT DATE
7e. TOTAL NO. OF' PAGES
January, 1972
ea.
CONTRACT OR GRANT N.).
NOOO14-67-A-0321-0002
b.
PROJECT NO,
I'b.
NO. OF' "EF'S
-35-
-27-
98. 0 RIGIN A TOR'S REPOR T NUMBE RtS»
Institute of Statistics Mimeo Series
Number 797
NR042-214/1-6-69 (436)
c.
91l. OTHER REPORT NOISl (Any orher numbers thet mey be essl"..sd
thl. report)
d.
10. DISTRIBUTION STATEME.T
The distribution of this report is unlimited.
II
I. SUPPLEMENTARY NOTE:
12· SPONSORING MILITARY ACTIVITV
Statistics and Probability Program
Office of Naval Research
Washington, D.C. 20360
13, ABSTRACT
The purpose of this paper is threefold:
Firstly, the topological support
of Gaussian measures on certain locally convex spaces are obtained.
Secondly,
strongly convergent series expansions of elements in separable Frechet spaces,
related to Gau.ssie.n measures, are obtained, this result is applied to obtain
Karhunen-Loeve type expansions for Gau.ssian processes.
Thirdly, i t is shown
that any zero mean Gaussian measure on a separable Frechet spac.e can be obtained as the a-extension of the canonical Gaussian cylinder measure of a
separable Hilbert space.
•
Other related problems are also discussed.
UNCLASSIFIED
Security
CI""~lftciitlOn
UNCLASSIFIED
Security Classification
14.
KEY WORDS
.
•
LINK
LINK A
ROLE
WT
ROLE
e
WT
Gaussian measure
Gau.ssian process
Karhunen-Loeve expansion
topological support
characteristic function
locally convex space
Frechet space
UNCLASSIFIED
Security Classification
LINK
ROLE
C
WT