This research was partially supported by the Office of Naval Research under
Grant N00014-67A-032l-006.
GAUSSIAN MEASURES ON L
P
SPACES, 1
~ p < ~
by
Balram S. Rajput
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 782
November, 1971
GAUSSIAN MEASURES ON L
SPACES
P
, 1 < p <
00.
<*)
by
llALRAM S. RA.TPUT
ABSTRACT
Let
(T,A(T»
a-finite measure on
be any measurable space and v be a nonnegative
(T,A(T».
Let
L
p
= Lp
(T,A(T),v>, 1 < p <
-
OIl,
be the
real separable Banach space of the equivalence classes of real Borel measurab1e functions on T whose p-th power is v-integrable with the norm
• <Jlx(t>IPv(dt»l/p. Let ~ be a Gaussian measure on L. One
T
P
of the main ideas in this pa,er is to establish a one to one correspondence
Ilxll
p
be~een
Gaussian measures on Lp
with ssmp1e paths in L.
P
and measurable Gaussian stochastic processes
This idea is used to prove a number of interestFor instance, the integrability
ina results about Gaussian messures on L •
p
of certrin functions of the norm of
covariance operator of
p
topological support of p
L
p
relative to
p
are given.
are proved; the
is defined; the characteristic function and the
are obtained in terms of its mean and the co-
variance operetor; five equivalent conditions for p
on L
~
These results
e~ntain,
as
to be non-degenerate
p~rticular
cases, the reou1to,
obtained by Vakhania [1965,66] and others; about GaussiB..."l meaom'ea on t
sp3ces, 1
~
p <
p
OIl.
Recently, ICa1U.anpur [1970] and Jain [1971] have proved a zero-one
law for Gaussian measures defined on certain function spaces.
Using the
result of these authors, a zero-one law for Gaussian measures on Freehet
spaces is proved, which is subsequently applied to obtain two other zero-one
laws.
In the first, it is shewn that the sample paths of a zero mean meas-
urable Gaussian stochastic process belong to
one; and in the second, it is shown that
~
Lp with probability zero or
certain random series converses
, uniformly, on any Borel subsct of tho roo I lino, with probability cero or ona.
AMS 1970 subject
classificat~on.
Primary 60B05, 60F20, 6OG15.
KeI words and phrases.
Gaussian measure, Gaussian process, zero-one
,
law, characteristic function, topological support, Lp space, Frechet
space.
(*)
This research was
part1~11y
supported by the Office of Naval
Research under Grant NOCOI4-67A-0321-006.
, ...
I
Foornons
#1
(p. 13).
We are grateful to Dr. L. A. Shepp, who suggested the
idea of truncating the stochastic process used in (3.11).
#2
(p. 19).
The proof of Sato's result is not available to us.
#3
(p. 42).
This application of Theorem 5.1 resulted from a discussion
with Dr. S. Cambanis.
/
il
INTRODUCTION.
Let T be any non-empty set, A(T) a a-algebra of subsets of T and
v a non-negative a-finite measure on the measurable space
Rand
8(R)
(T,A(T».
denote respectively the set of real numbers and the class of
Borel subsets of
R.
Let Lp (T,A(T),v), 0 < P
< ~,
be the space of equiv-
alence classes of real valued measurable (relative to the a-algebras
and A(T»
functions whose
p-th power is v-integrable.
<{lx(t)IPV(dt»l/P be denoted by
Then,for 1
11'11 p ,
Let
~
p < -, Lp(T,A(T),v)
and for
Ilxllp for every
x
8(R)
Let
~ Lp(T,A(T),v).
is a real Banach space with the norm
0 < p < I, Lp (T,A(T),v)
is a real complete metrizable
linear topological space with the metric generated by
the set N of positive integers, A{N)
II·II pP•
is the power set of
If T is
N and v is
the counting measure, then we write t p for Lp (N,A(N),v).
Unless stated otherwise the following notation and conventions are
fixed throughout this paper and therefore will not always be mentioned:
(A.!)
The measure space
(T,A (T) ,v)
hypotheses mentioned above.
Whenever
assumed that the a-algebra A(T)
with it.
L.
p
\:8
write the set T, it will be
and the measure v are always aoeociatcd
The measurability of real valued' function on T will always
mean relative to the a-algebras 8(R)
(A.2)
will be assumed to satisfy the
The space Lp (T,A (T) 'v), 0 < p
and A(T).
< •
Whenever we write L (T ,A (T) ,v), L
.
P
P
t
will usually be abbreviated by
or
I,
P
without specifying the
range of p, it will be understood that p satisfies the inequality
I
The Banach space Lp ' I ~ P < -, will always be assumed infinite
dimensional and separable. (For a sufficient condition and a necessary
~
P < ••
2
and sufficient condition on A(T)
of
Lp ' 1 .::. p < GO, see
space of L,
1 -< P <
p
be denoted by
[~
w,
1"1 Ip '
and
that guarantee the separability
v
pp. 168 and 170]).
will be denoted by
As usual, the conjugate
L;
and the norm in Lp will
q
When no confusion seems possible, we use the same
notation for a real measurable function on T and the corresponding
equivalence class.
(A.3)
Let
X be a real linear topological space; then the notation SeX)
will denote the a-algebra generated by the open sets of the initial topology
of
X.
The measurability of a real valued function on L, 0 <p < ., will
p
mean relative. to the a-algebras
(A.4)
The letters
S(R)
and
B(L )'
p
Nand R will denote respectively the set of positive
integers and the set of real numbers.
dictor function of F; and the letter
For any set F, IF will denote the in-
E will be used to denote the expected
value of random variables.
(A.S)
The underlined field for all linear spaces, considered in this paper,
is the field
R of real numbers; and all stochastic processes are assumed to
be real v..slu~d •.
Va1msnis, apparently inspired by the work of 110urier [18],
in £25
1,
[26
I and
defined on t
p
ha~
obtained
[27] various interesting rezults about Causaien measures
spaces, 1
~
Recently, de-Acosta L5 J has proved,
p < GO.
among many other results, two of the theorems of p6j using different methods
than those of Vakhania.
Almost concurrent to the paper of de-Acosta has
appeared the work of Kuelbs and Mandrekar [16] where analogues of Bochner's
theorem and Levy's continuity theorem are
of tha properties of lp
~8paces.
pro,~ed
0 < p < -.
i
x.,. x(n), n
~
N,
t p , 0 < p < •• Some
which have played a· crucial
role in the papers [25], [26], [27] and [16] are:
of the evaluation map:
for
(i)
from lp into
the measur&bility
a.
(ii) the fact that
a-
-
ip can be embedded in a
3
as a Borel subset. where
is the locally convex space of all real sequences endowed with the topology
.of pointwise convergence and (iii) particularly nice properties of the natural
(Schauder) basis.
The difficulty in extending the results of Vakhania [25],
[261. (27) and Kuelbs and Mandrekar [16) to Lp spaces arises mainly due
to the fact that the above three properties provide some useful structure
(from the point of view of studying probability measures) on l
necessarily available in the general case.
p
spaces not
Thus the methods, which are used
in [25]. [261. [27] and (16). do not seem applicable in the case of general
Lp spaces. . In this paper. we develop techniques suitable for studying
Gaussian measures on L
spaces. and obtain. £mOng other results, theorems
p
about Gaussian measures on Lp
l
'p
~
spaces anslogous to their counterparts for
spaces obtained by Vakhania in [25). [26] and (21).
We further hope
that similar techniques (see. for example. Theorem 3.1) will lead to the
generalizations of the results of [16] in the setup of
Lp spaces. With
the exception of part of Section 5. the whole paper is devoted to study
various properties of Gaussian measures on
L.
p
1
S
P < -,
Gauasian atcchastic processes vith sample paths in L,
p
not consider L.
p
0 < p
<
I,
mainly for
~~o
reasons:
and measurable
1 s p <
w.
We do
firstly our methods
of analysis heaVily depend on a number of Banach space properties which dou't
exist in L,
p
measure spaces
spaces,
for
0 < p < 1; secondly for a broad class of interesting
(T,A(T),v), the conjugate space of the corresponding L
0 < P < I,
p
is the trivial linear space (see the corollary and the
remark following it in [4, p. 820]) and therefore the st';'dy of Gaussian
measures on such spaces becomes
uninter~sting.
In the following. we summarize the main results of this paper:
Section 2 is preliminary.
In Section 3, we show that every measurable
Gaussian stochastic process with almost all sample paths in Lp induces a
4
Caussian measure on L
(Theorem 3.2), conversely for every measure
p
(not necessarily Gaussian) on L
p
there exists a measurable stochastic
(L .8(L ),~) which induces the measure
p
p
further, if the measure ~ is Gaussian then the stochastic
process on the probability space
Jl
on L;
P
process which induces the measure
is also Gaussian (Theorem 3.1).
The
contents of this section are basic to most of the results of Sections 4
through 6.
In Section 4, we prove that certain functions of the norm
II-lip
are integrable relative to Gaussian measures on Lp (Theorems 4.1, 4.2).
'Jhese results are related to the recent works of Skorokhod [24] and Sato
(Theore~
In SectioQ S, we prove three results.
1.14 of [17]).
In the
first result (Theorem 5.1), we obtain a Z9ro-one law for zero mean Gaussian
measures defined on separable Frechet spaces.
The proof of Theorem 5.1
depends on a recent result of Kallianpur [13] (indeed a slight extension of
it due to Jain [10]).
In the second result (Theorem 5.2), we obtain, using
Theorem 5.1, a zero-one law for measurable Gaussian stochastic processes.
A zero-one law for zero mean continuous co'.rariance
ces~e8
Shepp
Ga~sian
stochastic pro-
due to Varberg [28] and a zero-ane law for a Wiener process due to
~221
follow as corollari(>.8 to Theorem 5.2.
ktothe-c zero-one law for
zero mean continuous sample path Gaussian stochastic processes, rec2ntly
proved by Kallianpur (see Theorem 3 of [13]) is related to and partially
contained in Theorem 5.2.
In the third result (Theorem 5.3), we prove a
zero-one law for the uniform convergence in
a
of: a certain random series.
is the fact that the
One of the
Karhunen-Lo~ve expansion
t
on any Borel subset T· of
implications of this theorem
of a zero mean separable
measurable mean square continuous Gaussian stochastic process converges
uniformly in
t
ou a compact subset T of It with probability zero or one.
A particular case of this corollary is recently obtained by I<.al1ianpur
5
(Theorem 4 of [l3}).
In Section 6 we prove three results.
In the first
result (Theorem 6.1), we show that there exists a one to one correspondence
between the set of all Gaussian measures on L
where
x
€
p
and the set of pairs
(x.S),
L and S is an S -operator (see Definition 4.l) from L (~L*)
P
P
q
P
into Lp ' This theorem also gives the general form of the characteristic
function of a Gaussian measure on L p ' Theorem 3 of Vakhan1a [26] (same as
Theorem 7.1 of de-Acosta [5]) and Theorem 4 of Kourier [18. p.243] follow
from this result.
In the second result (Theorem 6.2), we show that the
topological support of a Gaussian iI1:!a8ure
lJ
of the sets
maL~
in L
P
{a}
and D where
is the
e
on L p
of
of the range of the covariance operator of
(see also Garoia et
is the algebraic sum
lJ
and D is the closure
lJ.
A theorem of Ito [9]
at [7]), where the topological support of a Gaussian
measure on a Hilbert space is
obtain~d,
follows from Theorem 6.2.
In the
last result (Theorem 6.3), we obtain, using Theorem 6.2, five equivalent
conditions for a Gaussian measure
conditions in the case 'of l
...... -
p
lJ
to be non-degenerate on L •. 8i1l11&r
p
were obtained by Vakhania in [27].
6
§2 SOME DEFINITIONS.
Let X be a real separable Fr;chet
convex) space and let
x*
(complete metr1zable locally
be its conjugate space.
A cylinder subAet of
X 1s a set of the form
{x ~ X: (C1(x) ••••• Cn(x»
where n
~
N, tl ••••• Cn t X*
Euclidean space R(n).
(n)
B
is a Borel subset of the n-
The a-algebra generated by all the cylinder sets
X will be denoted by Bc(X). Note that, as is shown by Ahamad in
of
[1. p. 100). Sc(X) • SeX)
~
and
, B(n)}.
(see (A.3) for tIle definition of
(X,B(X».
be a probability measure on
Sex»~.
Let
If there exists an element 8
of X such that t(e)· ft(x)~(dx). for every C € X*. then e is called
X
the
~ !.lem~nt
(or
~
value)
of
1.1 •
Note that 1£ the mean value of
exists. then it is unique. The chnracteri~tic function X of ~ is defined
by x'(e)· Je iC (x)~ (dx). for every C ~ X*. The probllbility measure lJ
~
X
is called Gaussia~ if every element of X*
considered
(X,B(X).lJ)
variable on the probability space
8S
a random
is Gaussian (possibly
deger..erp.~(?).
Although. the above definitions make cenae in any linaar
topolc3i~al
space. however. since the most general spaces considered in this paper
are tbe Frecbet
spaces. we bave restricted these definitions only to
such spaces.
Let {C : t (T} be a stochastic process defined on a probability
t
space (0,F .P);
it is said to be !!~m:abJ'!. if the map t fr01ll
T )( n into R defined by
a-algebras
8(R)
(t,lAl) .....
and· A(T) )( F.
stochastic process such that
C(t.w)
is measurable relative to the
Let' {( t: t
t(·.lll)
t!
T}
be a mensurable
c Lp ,a.s. [Pl.
Then the map defiuod
7
by
t; (. ,w) E: Lp}
T(w) •
if
is measurable relative to
F.
and
t (. ,ld) • L
(2.1)
p
Define the probability measure
(2.2)
for every B € 8(Lp ).
stochastic process
Let
function
on L.
P
and the autocorrelation function k of
covariance function K of
e(s)e(t).
and
k(s,t)· E(tstt)
{tt:
A stochastic process
€
N and for all
.
.
a,
{tt:
tl ••••• t n
.
are jointly Gaussian <tt'S
....
t
(T)
{E;t:
€
t (T)
are
respectively; and the
is defined by K(s,t)·
It is not bard to prove that i£
measurable, tben the functions
n
called the measure induced by the
be a second order stochastic process; then the !!!a
defined by a(t)· E«(t)
k(sft) -
is
liE;
{tt: t ( T}
t (T}
{E;t:
e
Then
£tt: t (T)
is
k and K are also measurable.
t (T)
is called Gaussian if. for every
T, the random variables
tt ••••• tt
n
1
are permitted to have degenerate distribution) •
.
J
8
13 A CORRESPONDENCE BETWEEN GAUSSIAN MEASURES ON L
P
AND MEAStmABLE
GAUSSIAN STOCHASTIC PROCESSES WITn SAMPLE PATUS IN Lp '
As mentioned in the" introduction, the contents of this section are
basic for" the rest of the paper.
Theorems 3.1 and 3.2.
The main results are contained in
Analogues of Theorems 3.1 and 3.2 under a few
restrictive hypotheses in the case ofL spaces, were obtained in [3J.
2
The proof of Theorem 3.1 is obtained by means of Proposition 3.1,
which constitutes an important part of the theorem; however, we have
aeparated the proposition from the theorem for convenience and clarity.
In Proposition 3.1, we make use of the fact that there exist
bases in L
p
spaces.
Schauder
For a justification of this fact see [23. pp. 16
and 17].
PROPOSITION 3.1.
Let J!.
{+j: j
.£!.! probabilitI
!!.!!. Schauder
€ N}
meaaure
x
€
Lp (T); ---for e~ch n € N,
-....-..
8 (t,x).
{+j: j
respect
n
11 €
€
.!J.,!£E..!.!.S!!.
'n
r
j-1
N}.
1!:!!!l, ill each n
!2..E!! a-algebras !li!l and
N}
rroof.
clear.
= Lp (T,A(T) ,\I). 1!S
j
€
.!1
l!!..!. repre-
N.
~
t
€
T
~j(t)tj(x),
(3.1)
t j (x) .!! the j-th coefficient E.!. .!. .!!!..!!! expansion .!!l terms
where
"{a :
p
define
--
n
.!!
L (1)
basis" i2!. Lp (T), and ~
sentati~le E!!!!!. equivalence clst9s
and
-
~
.h!!..!.
Since
€
.!!. product measurable!!!£!!.
N, sn
S(L (T»; ~!!l!. sequence
p
subsequence which converges.!.:....!.. {\I l( }.I 1.
t n 's
€
Lq (T)
= L*(T),
p
Thus we need only prove that
whicb converses a.e. [\I
)C
}.I].
ACT)
x
themeasurab11ity of
S~'8
...
{a: n «N} has. subsequence
n
1a
9
Since v
is
a~flnite,
we can find a sequence
{T: m f N}
m
of
eo
A(T)
disjoint elements of
v(T)
such that
11
<
and
eo
u T •
mal m
T.
Denote
by v
the restriction of v to T, by A(T) the a-algebra
m
m
m
(B n T : B ~ ACT)} and by sCm) the restriction of s
to T x L (T).
n
m
Let
x.
m
p
be a representative of the equivalence class.
Then we denote by x(m)
to T.
m
y
and let y
x E: L (T)
p
n
the equivalence class of the restriction of
We divide the rest of the proof into three parts for the sake
of clarity.
<a l 'l1t :
Thare exiets -a subsequence
--
(a)
{s1~~:
. Proof of
F
n
converges
k (N}
J!l.
Let
• {(t,x): (t,x)
for each n
€
N.
Since
and
> 0
£
~.
R.
E:
[VI
N
k
E:
x
~~
N}
_of .......
{s:
n ( N}
n
_
be arbitrary but fixed.
n+t
E:
T x L (T)tl 1: .j(tH (x) I > d,
j
I
P
j~
s(l)
is measurable relative to
n
Set
(3.2)
S(R)
and
A(T ) x. B(LP (T», it follows that Fn E: A(T ) x B(LP (T». Let n f N
I
I
and x ( L (T); and let IF be the indicator function of F and
P
F (x)
n
n
€
n
be the set {t
f
U
F}.
n
By Tonelli's theorem, for each
P : (t,x)
n
€
• If
IF (t,x)VI(dt) x ~(dx)
n
N, we have
TlxLp (T)
• I
L (T)
P
Let x
o
a (·,x )
n
0
E:
v1(F (x»~(dx).
n
(3.3)
be fixed; then, by the definition of
L (T)
p
s 'a
n '
x (.) in L (T) as n ~ -. Since
o·
p
;
fls (t,x) - x(t)IPv (dt) ~ fls (t,x) - x(t)IPv(dt),
(3.4)
T .n
m
T n
m
.
for each x ( L (T) and 11 € N, it follows that s(l)(.,x) ~ x(l)(.)
~
P
in L (T ) :: L (Tl,A (T ) ,vl)
p l
p
l
n
as
n'" -.
Thus
0
0
8~1) C, ,xo ) ... x~l) (.) in
10
vl-measure as
n
(s(l)(.,x ): n
n
0
which implies that
+ -,
in vl-measure; i.e. vl(Pn(xo» + 0 as
we have that,for each x ~ L (T),
n
N}
1s Cauchy
o was arbitrary,
Since
+ -.
£
X
P
f (x)
n
as
= v 1 (Fn (x»
0
+
(3.5)
n + -; and, from (3.3), it follows that, for all
€
N, f 's
are
n
measurable functions from
Lp(T)
the fact that
are finite measures allow us to use the bounded
v
and
1
~
into R.
n
Equations (3.3), (3.5) and
convergence theorem to get
lim v l x p(Fn >· f
lim vl(F (x»u(dx) • O.
t (T) nn
n+"D
Since 1
and
p
were arbitrary, we conclude, froia (3.6), tb4t the
O
E:
eequence . (e~l): n
in
£ If}
"1)( li measure.
{8~~~:
~
k
is cauchy in
of A(T ) )( 8(L (T», where
1
a.e.
'~t
-
Pro2! E! ill.
n
l,~
.)
,
E:
N}
E:
N}
eon'"erg~1i
•
The sequence
h!!..! subsequence
a.~.
--
{Sl
,~
of
\/1
x(2)(.)
+
{slz k
Denote by
A
2
'l\:
k
€
N}
pl.
-=---
["2 x
_.
: k (N}
is a subsequence of
in L (T ) :: L (T ,A(T ), "2)
p 2
p 2
2
by' "2' we conclude that
,~
2
P
ve repaat the proof of Part (a) replacing
aDd
{e
This fact along with (3.4) imply that, for all x c L (T),
{a : n (N).
s(2) (x
It
pl. Denote by ~ the element
1,~
(b) . ~ seguence . {sly~: k
{o~2):
["1)(
(s(l): k (N} does not converge.
p
8t.1Ch t'!1at
-.-..- - - -
measure, hence it converges
"1)( p
Which implies the existence of a subsequence
that converges
H}
(3.6)
~
N} such that
the~e
{8~2)
'iit
{s; n
n
E:
N}
by
{sl
e1ti,.te a subsequence
,~
Now if
: It
{S2,~;
€
converge.
.
a.e.
N}
k ~ N}
f
; It € U}
the el_ent of A(T 2) )( 8 (Lp (T», where
fails to converge.
k + -.
&8
["2)( u].
{8 (2) : It c. N}
2 -I\;
11
There exists -a subse,guence -of _,;,;n
{s: n
(c)
Proof
~J£l.
'lit
:
k
€
s
,1\
:
kEN}
• s , for all
o,~
k
{s : n
E
n
N}
of {s: n
N}
.
converges on Tj )( L p (T)
j
j ~
If we continue the process as in (b). at step
we will have a subsequence {Sj
Moreover, {8
which converges
N)
_
E
off a
such that
€ H}
n
1
null set Aj •
"'j)( II
is a subsequence of
kEN.
Then the subsequence
..
converges on T)( L (T)
p
'" )( \J(A) • j:l"'j )( \Jj(Aj ) • 0, since
{sk,k= k E H}
off the set A.
~j'S
•u
A~.
j-1
are disjoint.
of
But
oJ
The proof is
complete.
THEOREM 3.1.
~
l!. J!!..! probability pteJl!lure!m
.! measurable stochastic process
{tt: t
E
(Lp,B(L
»'
p
T} defined
.!h!.!! there
ex:f.s~~
~~~robabi11ty
(o.l,PH::(Lp ,8(L ),\J», almost.!l!.'sample 'paths '.2!. which belong'~ ~.
p
The m~asure.,~ 'induced.!?l {tt: t £ I} (eee (2.2».2!! ~ coincides
sEBce
~
~.
Furth~r,.!!.
Pro~f.
Let
.l:!.
.!!!. ~.1an,.
{sk,k: kEN}
Part (c) of Proposition 3.1.
for all
(t,x) • A.
\1)(
T} .!! Q!.ussian.
{Sk,k(t,x): k
Define
sk,k(t,x)
€
N}
converges,
cA
if
(t,x)
if
(t,x) c A •
(3.7)
(t,x) •
0
Since
t E:
and A be the s&me as in the proof of
We have that
{~
l;
H: t :
th~.B.
1J(A) • 0, it follows that the map
~
from T)( L
p
into R is
measurable relative to B (It.)
and A (T) )( B (Lp )' Set At·' {x: (t,x) f: A}
An application of Fubini's theorem and the fact
and A • {t: (t,x) € AI.
x
that "')().I (A) • 0 guarantee the existence of a "'-null set To and a
null set
no such that if t
¢ To.
then sk.t(t.x) ~ ,Ct.x)
8S
k
+ •
).1-
•
12
for all x off the
~-null
as k
t
for all
+ .,
set
~
At; and if x
sk,k~t,x) +
no' then
t(t,x)
off the v-nul1 set A •
x
Now define
(3.8)
t(t,x) • t(t,x)I(T xo)c(t,x),
o
where (To .x n)c denotes the complement of T0 x O. Then the map t
froll T)( Lp into R is measurable; and therefore by Fubinis's theoreua,
for each
t
T}
€
is measurable from Lp into R. Thus
is a measurable stochastic process defined on the probability
T, the map
tt(')
{tt: t
£
space
(Lp ' B(Lp) ,U). Moroover, it followa,from (3.7), (3.8) and the defini-
¢ 00 ,
tions
of To, 00' At and Ax, that if· x
.-.. ..,.
';,
th~n, ~ff
the v-null set
AUT,
x
0
and if
~(t,x)
~
no'
Since {Sk,kC"x): k
£
'
is a subsequence of {Sn("x): n
N}
(see Proposition 3.1 for the definition of
converges to x(')
(3.10)
• lim sk k(t,x).
k....
Let x
(3.9)
t(t,x) • lim sk k(t,x);
k- '
t . To. then, off the \.I-null set At'
in L·
p ,
we have th"t
s)
and since·
n
{sk ,k (. ,x): k
{s
E:
n
(',x): n
£
£ N}
N}
converges
N}
to xC, )
in L p • Which implies that {ok. ,1-.. (. ,x): k E: N} haa a 6ubsequence which converges to xC, ) a.e. [v] • From this and (3.9), we have
that tC' ,x) • x(')
belong to L.
p
.easur~ble,.
a.e.
[v].
Thus almost all sample paths of . {tt: t
In view of this fact and the fact that
the map
T froUl 0
into L
p
{tt: t
£
T}
(see 12) io well defined.
£ T}
is
In
U induced by . {tt: t ~ T} coincides
t
with lA. we only need to verify that the two measures agree on the algebra
order to prove that the
lIl~asure
of the cylinder sets of Lp •
we omit the details.
Since this verification is straightforward,
13
The only thing that remains to be shown is that
if
is Gaussian.
~
In order to prove that
is sufficient (note that
snd tl, ••• ,tn
tt •••• ft t
1
(3.1»
k
T0c
E:
tt
=0
{tt: t
E:
{tt: t
is Gaussian,
T} is Gaussian. it
on To) to prove that. for any n
E:
N
(the complement of T).
the random variables
0
Since the family
are jointly Gaussian.
{t: n
n
0
€
N}
(see
of random variables 1s jointly Gaussian, we have that, for esch
N, the random variables Sk,k(tl,·), ••• ,sk,k(t n .·)
E:
T}
E:
Using this. (3.10) and the fact
tl~t
are jointly Gaussiao.
a.s. limit of Gaussian random variables
is Gaussian (see,for example,Lemma 5.1 of [3]), we have the tt •••• ,tt
n
1
are jointly Gaussian.
The proof of the theoran is now complete.
THEOREM 3. 2.
1.!E. {tt:
•
~ ..!2!!:.
t
.!!!..!. !..e4surable
T}
€
.!! t (. ,CI)
E!0bahi1ity space .!q,F .p).
measure
~
induced
~
{tt: t
E:
9aueslan stochastic process defined
~ ~ ~
p ~.
!!. Gaussian.
E
L
ltl.
th,!'].!h!.
,
The proof of the theorem will
foll~~
from the following proposition:
PROPOSITION 3.2.
ca,F .P). 1£l
.2!!..!. rrobabi11tI .!E.aCe
ill,
and.!!!!h!. sample path
integra~
l; (. ,w) £ Ll~T2
It
:.: L1 (To ,A (T) tV) .!:.!.•
(t,ro)v(dt) i! denoted ~ ~l
T
!.:.!.
ill.
~
Proof
.l .!! ~ GIlUSS!!!l random variable.
.2!. Proposit!2n. 1:1:.
First we prove the proposition under the
additional assumption that veT) < -.
Let k boe the autocorrelation
function of ,: {tt:
€
t €
T}.
t (n) (t,CI)
Then.for each
1'l f:
N,'
process and . t (n) (. ,CI)
For each n
,
• t (t,CI)I{ s:k(s,fJ)<n) (t).
{t~l'l):
C
Nt define(l)
t
L (T)
l
€
T}
a. s.
. (3.11)
is a m:aaurable Gaussian 8tochastic
[P].
Denote by k
n
the autocorrelation
14
(I;~n): t
function of
we have k (t,t) < n.
n
Let
n
N be arbitrary but fixed.
E:
Using this and the fact that
-
.
T}.
E:
E(!lt(n)'t,w)IV(dt»2
T
<
-
E:
E(!lt(n)(t,w) 12 v(dt»
manifold generated by the random variables
(l~,p.
K -
[Pl.
(3.13)
< -.
T
2
defined an integral (see
a.8.
L (T)
be the closure in L (O)
(T)
we denote by
(3.12)
=L2 (T,A(T),v)
2
P is a finite measure imply
t(n)(.,w)
Thi8 fact and the fact that
t
e
T n
From (3.12), we have that
H(t~n):
.
veT) < ., we get
E(/lr.(n) (t,w) /2 V (dt» • !k (t,t)v(dt) < -.
T
Let
By (3.11),
=L2 (O,F,P)
(t~n):
of the linear
t E T}.
Karhunen has
63]) in the quadratic mean sense which
!t(n) (t,w)v(dt)
to differentiate it from the sample path
T
.integral
It(n) (t,lAl)v(dt). We use the notation
ten) (w)
for
!t(n) (t,w)vCdt)
T
T
and
n(n)(W)
for
!t(n)(t,W)V(dt).
K -
The integral
K
- !,Cn) (t,w)v(dt)
T
T
H(t~n): t
exists as an element of
i£ and only 1£ (see Lemma 6 of
E T)
(14, p. 33])
0
2
n
=TXT
If kn (s,t)V(ds)
Further, if the integral
Since n
(n)
fact that
( Cn).)
H
E:
0
't .
2 < -
n
t E
n(n)
!t(n)(t,w)V(dt)
.
T , i.e £oJ.lovo tha.t
exists, then
n(n)
Now we will shaw that
is Gaussian, it will follow that
0
2•
n
.
..~ ~
...
) •
Thus
,(n)(w). n(n)(IAl)
,en)
2
on.
The
nCo)
.
a.s.
is GauQsian.
2
on' we have
By applying a property of Karhunen's integral (see (14,
p. 30]), Fubini's theorem and (3.13), we obtain
"
(n) 2
is GausRian.
F,rom Fubini's theorem, (3.13) and the definition of
E(t(n»2 •
fen
follows, from (3.12) and Tonelli'. theorem.
exists and is Gaussian.
Since
K -
(3.14)
x V(dt) < -.
!
[Pl.
•
15
E(C;(n) '" (n»
•
fEr t (n) CfA»
1; (n) (t .w) ]v(dt)
T
• !!E(~(n)(s,w)t(n)(t,w»)v(da)v(dt)
TT
2
(3.15)
• an •
From (3.15) and the fact that
f(lt(n) - ",(n) 1)2 • 0; i.e.
r,;(n)
is Gaussian.
E(",(n»2 - ECr;(n»2.
t(n)(w). n{n)(w)
8.a.
e,2,
n
we conclude that
[Pl. Thia ahows that
Since n was arbitrary, we conclude that
r,;Cn)(w) • !r,;(n) (t,w)v(dt)
ia Gaussian, for all n c N.
From the definition
T
of
{C~n): t
E
T}, we have that, for eaeh fixed w
Ir;(n)(t,w)I ~ 1r;(t,w)I, for all
.s n"'· and
the fact that !I~(t,w)lv(dt) < . a.s.
[P]
E
a, r;(n)(t,w) ... t(t,w)
t c T.
These facts and
allow us to use the Lebesgue
T
•
dominatGd convergence theorem to conclude
a.8.
(P]
!t(n) (t, wV(dt) ... !r;(t,lIJ)v(dt)
T
T
asn'" -. Since each !r,;(n) (t,w)v(dt) is Gaussian, andsince
T
•••• limit of Gaussian random variables is Gaussian (see Lemma 5.1 of [3]),
it follows that !C(t,w)v(dt)
is Gaussian.
T
Wa complete the proof of the proposition by dropping the hypothesis
that vCT)
is necessarily finite.
-
(see (A.l».
of
Recall that v is assumed to be a-finite
Choose a non-decreasing sequence {T:
III
ACT)
11 E
N} of elements
such that vCT) <- and u T • T. From what we have proved
m
m-1 m
above, wa. have that, for each Il c N, the random variable
J t(t,w)v(dt)
• !c;(t,w)lr (t)v(dt)
T
Dl
T
is Gaussian.
This, Leama 5.1 of [3]
m
and an application of the dominated convergence theorem show that,
t(w) • !r,;(t,w)v(dt) is Gauesian. nlis completes the proof of the proposition.
T
Proof
q In order to prove the theorem,we
have to show that the random variable !C(t,W)f(t)V(dt) ia Gaussian.
~
Theoreml:.!:. Let
f c L •
T
16
Define
l;(t.w)· t(t,w)f(t)'; then
stochastic process with
we have that
t(·,w)
~his
a.s.
T}
is a measurable Gaussian
[Pl.
point it 1s worth pointing out a result apparently due to
which partly states that if {tt:
R, ia a measurable stochastic process with
€
From Proposition 3.1,
T
Doob (Theorem 2.8 of [6, p. 64)
a,b
L
l
€
!t(t,w)v(dt). !~(t,w)£(t)v(dt) is Gaussian.
T
At
€
{tt: t
t(·,w)
€
t €
[a,b)},
tl([a,b], Leb.)
.b
a.s.
[Pl, then the sample path integral
/t(t,w)dt is the limit in
a
probability of certain finite linear combir~tion8 of the random variables
b
{tt: t
€
[a,b]}.
From this and LemmA 5.1 of [3] , i t follows that
!r;(t,w)dt
a
is Gaussian, if the stochastic process
{tt: t
€
(a,b]}
is Gaussian.
Usins
this and an argument similar to thaC used in the last paragraph of the proof
of Proposition 3.2, we obtain the analogue of Proposition 3.2 for the case
yhen T 1s a Borel subset of
and
v
R, A(T)
However, Theorem 2.8 of [6, p. 64) does not
the Lebesgue measure.
apply in our general setup.
A careful study of Doob's proof of Theorem 2.8
of [6, p. 641 shows that it devends
n~bcr6; a~d
the class of Borel subsets of T
hC~111y on~arioU8
properties of real
we don't know if a result GUlilar to Theorem 2.8 of [6, p. 64)
can be proved for the measure space
(T,A(T),v)
satisfying the
h)~otbea$3
of (A.l) without appealing to Xarhunen's integral even in the case of
Gaussian measurable processes.
Note that, as follows from the proof of
Proposition 3.2, the sample path integral !t(t,w)v(dt) of Proposition 3.2
.
T
1s the a.s. limit, and hence limit in probability of finite linear combi~Ations
of the random variables
3.2) contsinG part of
{tt: t
Theor~
€
T}.
In this sense our result (Propooition
I
2.8 of [6. p. 64] in the casa when the
stochastic process i8 me&8urnble and
G~uos1au.
•
17
In the following, we state three results which will be used later.
The last two.results were used by Vakhan!a in [25]; the proof. of the••
results are elementary, and therefore omitted.
The first result 1s
well known and its proof can be found, for example, in [8, p. 163].
LEMMA 3.1.
!:!! ! 1?!..! measurable
transformation
~.! measurable space
(0l'Fl'P l )
!!2!. .!. fIleaBur~
BRaee
-1
(02,F 2); ~ ' 2 · ' l ef • !1
.!!..!!!!! ~alued ~asurable function.5m
.a.
~,!!!!!!
JS(OJ 2)P2(dc.J 2) • f S(f(wl»P(dw l )
°2
°1
!!!! ~ .!!!. egual.
L'D-D'aA 3. 2.
!!:i
{~n:
n
!!:!!. .increasing
p{t
:>
a }
n- n
LIMKA 3.3.·
:> €,
-
€
N}
E.!..!.
sequenee
for all
--
~
n
E
seguen~1!
posit1.ve
Pi FandC11l
!!!! number.. .!E! E
N, then P{t ......
-
variables, {an: n.E
aa
:>
.!!
O.
n + .}
:>
O.
-...;;;,---------n
, ..
/
.n
18
54
TIlE
INTEGRABILITY OF CERTAIN FUNCTIONS OF THE NORM
II· II p
WITH
RESPECT TO GAUSSIAN MEASURES ON L.
P
The main purpose of this section is to prove Theorem 4.2 (b), which
states that if u
&> 0
is a Gaussian measure on L , then there exists an
f e£llxll~ ~(dx)
such that
1
< -.
L1
achieved by showing that. for all k
The proof of Theorem 4.2 (b) is
.
N, the integrals
€
k
Ilxlll~(dx)
f
Ll
finite and bounded above by suitable constants.
measure on L,
p
I
L
Ilxllkp~(dx)
1 < p < -.
for
< .,
p
Let now
are
be a Gaussian
~
It turns out that the proof of the fact that
1 < P
<., k
£
N, is not much different than
the
P
corresponding result in L , which is needed to prove Theorem 4.2 (b); and
l
therefore we also prove that all non-negative integral powers of
(and hence all non-negative real powers of
to the measure
in Theorem 4.1.
~.
II' lip)
are integrable relative
This and the corresponding result for L
are contained
1
Further, applying Theorem 4.1 and the techniques similar
to those used in the proof of Theorem 4.2 (h) t we easily obtain Theorem 4.1 (a) ,
which states that if
exists a
u is Gaussian meesur.e on Lp J
6 > 0 such that
f
L
1 < p
~
2,
the~
there
e61lxllp u(dx) < -.
p
Theorems 4.1 and 4.2 (a) are not new; they follow from a recent result
of Skorokhod [24], who proved the analogue of Theorem 4.2 (a) for any real
separable Banach space.
The only reason to include Theorems 4.1 and 4.2 (8)
here is that their proofs are obtained,
8.S
indicated above, without much add i-
tional effort than the one reqUired to prove Theorem 4.2 (b).
Very recently •
• result due to Sato is mentioned (\1ithout proof) by Marcus and Shepp in
[171; specifically, this result states that the analogue of Theorem 4.2 (b)
19
holds for every real Banach space X whose conjugate space
separable.
X*
is
Clearly Sato's theorem is stronger than that of Skorokhod;
however, since the separability of
X*
implies that of
X and since
there exist separable Banach spaces whose conjugate space is
not separable
(for instance X· L ), it follows that the hypotheses of the Sata's theorem
l
are more restrictive than those of Skorokhod. Note that Theorem 4.2 (b)
is not implied by any of the two results mentioned in the preceeding sentence.
In passing we note that Skorokhod's short proof(2) of his theorem mentioned
above depends on some ideas from the theory of Markov
processes, whereas
our proofs of Theorems 4.1 and 4.2 rely
on the tech-
niques'and two rather elementary results (Lemmas 3.2,3.3) used by Vskhania in [25],
where he (Vakhania) proved the analogue of Theorem 4.1 for
t
p
spaces.
This section also contains two more results; namely Propositions 4.1
and 4.2.
In the first proposition, we essentially obtain necessary and
sufficient conditions (in terms of the mean, autocorrelation and covariance
{~t:
functions of a measurable Gaussian stochastic process
order that almost all sample paths of
second proposition,
V~
{~t: t £ T}
t
€
in
T})
belong to Lp '
In the
prove the existence an.'! the uniqueness of the
covariance operator (see Definition 4.2) of a Gaussian measure on L.
,
P
Recall that the conventions and notation given in (A.I) through
(A.S) are in effect throu.ghout this and the following sections.
The proof of Theorem 4.1 iaobtained by means a series of three
lemmas.
LOO'aA 4.1.
·Let. {t : t e:
---
t
T}
be a measurable Gau8s1an stochastic .froc:ess
----
- ·defined -.<l.!t .!'probab11ity space
~
!'
.m..L!2. :!!!h
E.!.!2!.!!!.!!.'funct1on.2£, {(;t:
't ~ T)~
t C· ,(/)
~
L
Eh!!!. e
~
L •
p
2
a.s.,:.
ill·
20
Suppose on the contrary 'that o •.L Lp'. then we can find an
Proof.
f (Lq (; L*)
p
such that
(4.1)
!O(t)f(t)v(dt) • +-.
T
Denote by k the autocorrelation function of {C: t
t
a non-decreasing sequence {T: n
•
veT ) <. and
u T • T.
nlllli n
n
t
(n)
of elements of A(T)
T}
€
For each
T}j and choose
n
€
Let en
such that
N. define
(t,w)· t(t,tIl)IT(t)
let).
n I{ s.'kCs,s.)<
~n
(4.2)
{t~n):.t (T} 1s measurable, Gaussian and t(n)("",)
Then
of
n
€
€
a.8. [Pl.
L
p
and kn
be respectively
the mean and the...
autocorrelation
·
,
.
f~nctions
{t~n): t (T). Using Tonelli's theorem, the inequality
n> Ik (t,t) > feltt(n)l)
-
n
and the fact that v(T) < -, we have
-
n
E(~lt(n)(t'W)f(t)lv(dt»)
• f({ It(n)(t,(j)f(t)lv(dt»)
n
~ nJ If(t)/v(dt) < -.
(4.3)
Tn
From Theorem 3.2, we have that !t(n) (t,w)f(t)v(dt) 1s Gaussianj and
T
from (4.3) and Fubild's theorem, it follows thst this random variable
IeT n (t)f(t)v(de).
has the finite meen value equal to
p{w € Q:
It'n) (t,w)£(t)v(dt) ~
T
. Since
fe
Tn
Therefore
(t)f(t)v(dt)} • 1/2
(4.4)
> O.
en (t)f(t) ... e (t)f(t) as n"'. and since
..
en
(t)f(t)
• e (t)f(t)L
,
~
.
n
(t)I{ s."k(s,s )<}
n (t),
-
it follows that
[0 (t)f(t)]+ t [O(t)f(t»)+
n
(4.5)
.I
and
[9 (t)f(t)]n
as
,..'
D"'.,where s+.max(s,O)
+ [6(t)f(t)]-
(4.6)
and s-·max(-s,O), for any seR.
.
21
Using (4.5), the monotone convergence theorem and the fact that
(see (4.1», we have
![9(t)f(t)]+v(dt) • +T
1[0 (t)f{t)]+v(dt) t f£a(t)f(t)]+v(dt)
TnT
n ~ -; and using (4.16),
as
the dominated convergence theorem and the fact that !£e(t)f(t)]-v(dt) < _,
T
we have l[e (t)f(t)]-v(dt) t f£O{t)f(t)]-v(dt)
TnT
Ie
clear that the integrals
Ie
Tn
(t)f(t)v(dt)
as
n
Nov it is
+ -.
are eventually positive and
.
(4.7)
(t)f(t)v(dt) t lo(t)f(t)v(dt) • +-
TnT
88
n
From (4.4), (4.7) and" Lemma 3.2, we have
+ -.
p{w: lim
n+et
It (n) (t.w)f (t)v(dt)
• +-) >0.
(4.8)
T
E-quation (4.2) and the dominated convergence theorem imply that
11m
0-:-
f~(n)(t.w)f(t)v(dt) • f~{t,w)f(t)v(dt)
T
But (4.9) and the fact that t(',w)f(')
which
a.s. [PI
(4.9)
T
c~pletes
L
1
€
contradict (4.8).
[p]
A.8.
the proof of the lemma.
LEMMA 4.2.
Let
{tt:
t
E:
T}
2.!!. !. prababi1itI !2.t.lce
be.!.
(O,f..a.!2. ~£h that
8Ce). l(tt.)· 0, for all
Proof.
of A(T)
E:
L
2
< -
and
co
u T • T.
n-l n
0,
E:
N}
If
of elements
E:
N, define
• t(t,w)IT(t) I£S:k(s,B)<n}(t),
n
,
r ~) • Jlt(n)(tp)IPv(dt)
fit (n) (t,w) 12 ~
ill.
.!.:..!..
For each n
is the autocorrelation function of {t t: t
T
process defined
T, ~ E(IIC:(·,w)II:> < -.
n
En)(t,(a)
n
t E:
t (. ,tIJ)
Choose a non-decreasing sequence . {T : n
such that v(T)
." _ .
n
where k
~urabk .2~~!!l stochastic
and s
n
• f(r ~».
n
€
T}.
(4.10)
Let
I
Using the faccsthat
v (Tn) <. and LeJI1I"I'..a 3.3, we have
22
s
n
f(Jlt(n)(t.w)fPv(dt~
•
1
'T
• C(p.2)![Elt(n)(t.w)1 2 ]p/2v (dt)
T
• C(p.2)! [Elt(n)(t,w)1 2 ]p/2v (dt)
T
n
£
2
C(p.2)nP / v(T ) < -.
n
(4.11)
Using Schwarz' 8 inequality. Lemma 3.3 and (4.11), we have
E[r:(w)]2.
E(~lt(n)(t'W)JPV(dt»)2
• !fErlt(n)(s.w)(Plt{n)(t,w)I P ]v(d8)v(dt)
TT
~ " ([E It (n) (t,CAl) 12t']l/2[E It (n) (t.w) 12P l l / 2 ) v(ds)v (dt)
(II t
• C(2p.p)E
(n) (t,w) IP\I(dt») 2
T
• C(2p, p)s2 < _.
(4.12)
n
From (4.10) aDd the dominated convergence theorem, we have
flt(n)(t.CoI)(P~(dt) +
T
Ilt(·.w)II P
f [rn <w) J
as n +
a.s. [Pl
Ge.
a.s. [P]
P
If
n+ -.
+ E [j It ( • ,!IS) II:]
(4.13)
a.s. [1], for all n
r (w) • 0
n
the conclusion of the lemma is trivially true.
r
+
is no loss .of. aenerality in assuming that . r n (W)
EN.,
+0
0 < s
In view of this and (4.11), we have
E
N. then
Thus we assume that
(w)
0 a.s. [P]. for 80me n E N. Since r (CrI)
n o n
o
n
Thus
n
+ a.s.
[P}. there
a.8. [Pl, for all
< -, for all
n
E
N.
The rest of the proof is similar to that of the proof o£·Lemma 2 of
[25].
lor each
n
C
N. denote by
{CAl E Q: r
n
F
n
the set
(w) > e(l - 8)0 }.
-
n
The proof, of the 1. . . will be complete if we can show the existence of en
23
a
P(F) > e. for some
n the conclusion of the lemma is false, then, by (4.13), s
>
0 and aO < B < 1 such that
this fact along with the condition PCF) >
n
Ilt(·.w)11 p ••,
£
as
aD
n
+ -;
a.s. [Pl.
L
p
It is easy to verify that, for each n
F
::>
n-
n: Ir n (w)
{w ~
- as
n
N and any a
f
-
a,
and
I -< aSsn }.
o<
Therefore, using Chebyshev's inequality, and the facts
E(r 2 (w» < C(2p,p)s2 (aee (4.11) and (4.12», we have
n
t
on a set of positive probability; which contradicts
t(·,w)
the fact that
n
For, if
and Lemma 3.2 imply that
£
-
> O.
£
s
n
<-
and
n
P(p' ) > 1 - P{w ( n:·
Ir
n -
n
-
(~)
aps
I
a8
n > 1}
(1
n
2
E<!rn(N) - aanl )
> 1 -
-~-=-~...:::..-
222
a 6 &n
(4.14)
P(F) > £, follows from (4.14) by taking a'
n 2 1/2
2a > C(2p,p) and 1 > a > JLC(2ptp) - ~ + a]
•
Now the proof of the fact
a
and
such that
Cl
LEl-2!..-\ 4. 3.
1!!'
E(II(.,w)ll;p)
Proof.
<
o<
-
(
~
8
k-l k
0
aD, for all r
Let {T : keN}
k
such that v(Tk)
(P); then
satiaf: tt-;.£ ~T),.,theses of
{(t: t t: T}
< -
and
~ kflSk (w)
(W»r t
t
uT
£
.!h!m
N.
• T.
II t (. ,w) II:
. P
4.2.
be a sequence of disj oint elements of A(T)
~l k
Ilt(·,w)llrp
~a
a.s.
Let
Sk(w).
a.a. [P]
[p]
f
~
1(t,w)IPv(dt) a.8.
aa n" -.
Hence
i
as n'" -.
TherQfore/ the
proof of the lerema will be co=p1ate if ws can show that
24
n
r
lim,E( t skew»~
n- . kllll
<~.
Using Tonelli's theorem, the generalized Holder
inequality and Lemma 3.3, we have
E(Sk(w»r.
f ···f
T ,· .T
k
k
E(I~(tl,w)IP••• It(tr,w)IP)v(dtl) ••• v(dt r )
• J ... J E[(lt(tl,w)lpr)l/r"'(I~Ct
Tk "'Tk
~
f "'f
[f(ltCt ,w)l
l
T "'T
k
pr
k
r
,w)lpr)l/r]v(dtl)···v(dt r )·
.
r
)]l/ ".[EClt(t ,w)I Pr )]l/r v (dt )"'v(dt ).
1
r
r
• C(pr,p)[f c(lt(t,w)I)Pv(dt)]r < -,
(4.15)
Tk
for all
r,k
E:
N.
Using (4.15), the ganeraU.zed Holder inequality and
Lemmas 3.3, 4.2, .we get
n
• [C(pr,p) HE( t (skw»)]
r
k-l
< -.
/
This completes the proof.
THEOREM 4.1.
1!!
J!
l!..!.
Gaussian mes!l1ra ~
:e.; ~. [llx l;lJ(dX)
J
p
ill r.?
O.
< -,
!.2!.
25
Proof.
The proof follows from Lemmas 4.1, 4.3 and 3.1.
Now we proceed to prove Theorem 4.2.
First we will prove Lemma 4.4,
which essentially carries the burden of the proof of Theorem 4.2.
LEMMA 4.4.
{~t: t
Let
~
a(t)
~~
(T}
measurable Gaussian stochastic 2rocess
!2.!. ill
• E(tt) • 0,
t € T.
(a)' If 1 < p ~ 2 ~ ~ (. ,111)
a
1
(b)
> 0
fee
·l~
p) <
te·,w)
f(e
~roof ~
ill, !h!.!!
there exists .!!l
M.
Let k
) <
E:
c..
L1 !.:.!..Ql,!h!m. there exists..!!!. a > 0
2
£
(1211 F; (. ,111 ) Iii
~~
L .!:..!.
2
(lllIE;(·,w>II P
~!!!!!.
If P
£
0».
Then, using the same arguments as used
N.
to prove inequality (4.15), we have
f(I1 F; (t,l1I) IPv (dt»k ~ C (pk,p HI fq t (t,w) Ip)v (dt) lk
T
T
(4.16)
•
From (4.16) and the definition of C(pk,p), we get
ll~{E(llt(.,w)II:P») ~ al(k)b~(P)'
(r(k
r (1<1 + ~)
r (k
where a (k).
1
Since 1 < p
~
+ 1)
and
( 'If • E( II ~ (. ,ll»
p
r(
+
2
II:) )
1
.-
)
2, it follows that
o<
for all large k
E:
N.
.2! ill.
(4.18)
a (k) < 1,
1
Now the proof of this part follows from (4.17) and
0 such that a 1b i ~) < 1.
,Using similar arguments as in Part (a), we obtain
(4.18) 'by choosing (11
Proof
b l (p) •
(4.17)
>
~(k'; 1~(E(llt(.'I1I)llik»)~ a2(k)b~k(l),
for each 1t
f:
Nt where
a 2 (k).
!.~k++l{~~
and b (1)
2
(4.19)
.1l'.E(llt(·,w)11 1).
,
26
Again we have
0 < '2(k) < 1, for all large keN.
Thus the proof follows
from (4.19), by choosing a 2 > 0 such that a b 2 (1) < 1.
2 2
THEOREM 4.2.
_
Let u be a Gaussian measure on L, 1 < P < 2•
.r.. -2.
(a) .y. 1 < p ~ 2, then there exists,.!. ~ > 0 ~!h!!
f
~lIxli
e
p p(dx) < -.
L
Proof.
Let
{tt: t
~
.!!!£h ~
> 0
£
which induces the measure p on L,
1 -< p
p
lor all t
of Lemma 4.4.
f(e
<
-
Let
2; and let a(t).
) < -.
From Part (a) of Lemma 4.4, we have
p
<
-
11 ,( J 611r;C
(61\0
e
e
{w:l
611 aII P~ ( ca + J e611 r; ( • ,trJ) II ~P(d~))
(
e
L
l
o
.~
f
,w)ll p
ISlIt<.,(a)ll ·
P(dw) + e
P(dw) pJ
It(·,CIt) II<l}
( 611 e II p) (e~
< a
{w:Ilr;(·,lJl)lI~l}-
+ E[e
a.s. [P}.
{w: Itc·.w)II~l}
611 t C• ,CIt) II:])
Again let t(t,w)· t(t,w) - aCt); then, by
~~o
.A
Using this and Lemmas 3.1, 4.1, we conclude
alltc.,w)ll p
p(dx) • Ere
]
1; C, ,CAl) ~
"-
~.
~lIxllp
Proof
E(~t)'
aI' where a 1 is the S8me as in Part Ca)
t(t,w)· t(t,w) - aCt); then, in view of Lemma 4.1,
Define
aII t ( • ,«II) II:
Je
L
£!J!l.
Lp , a.s. [Pl.
~
<
-
T. ,We give the rest of the proof separately for each part.
~
Proof
tC.,w)
-;frxn2
Ie
l(dx) < -.
L1
T} be a measurable Caussian stochastic process
p. 1, ~ there exists.!!l
.!!
(b)
L~a
I
< ••
4.1)
Prom. Part (b) of Lemma 4.4, we have an (\2 > 0
such that
(4.20)
~hoose an
&.
> 0
Lemma 4.1, e ELI'
such that
(1 +
Write n(,.q)
positive real number
,.
f;
J
{W:11 (W)< I}
21! I) Ill)
for
~
~ (,'!;1'
1(;(·,w)11 1 ,
ti~ .~~ t~... f',~, 1n vi~r.l cf:
a~':.d
1st H b,o a
s8t1£f7t~~
..,
exp{ ',.r...
:' •./~·,.,·
""',
. ..j . ;;
.;.. '1,·,\:1""
<
~'
i',) I 11 ]'P(.1.·)
J
"llI
M•
(4.21)
•
t
..
.~
27
This completes the proof.
Let
f
itt: t
T}
€
be a measurable Gaussian stochastic process and let
be a real valued measurable function on T.
We obtain, in Proposition 4.1,
·necessary and sufficient conditions, in terms of the mean, autocorrelation
and covariance functions of
{~t:
t
that almost all sample functions of
where
I
~
P
<~.
T}
€
and the function
{f(t)lttI P : t
f, in order
Ll ,
Although this proposition is important in its own right,
it is also used in the proofs of some of the later
€
T} belong to
results.
In Prop-
osition 4.2, we prove the existence and the uniqueness of the covariance
operator (see Definition 4.2) of a Gaussian measure
~
on L.
p
Note
that, in view of Theorem 4.1 and Corollary 1 of [1, p. 112], the mean
value of
~
exists.
However, this fact 1s included in Proposition 4.2
mainly to show that the mean element of
class
c~rresponding
~
is equal to the equivalence
to the mean function of any measurable Gaussian
stochastic process which induces the measure
~.
For the proof of Proposition 4.1, we will need the following
elementary lemma.
LEMMA 4.5.
1!E. . ft t :
t.€
T}
J?! .!.measurabl~
second order process ~ ~
..
28
!,
functjon
~
1
p/2
K
~
P < 00;
(.,.)
E:
E:
~
2
k P/ (.,.)
f:
~
.!!. ~
L1
covariance function ~.
onlI
J..!.
10(,) IP
~
L
l
f:
1'1'
Proof.
t
1
autocorrelation function
The proof follows from the fact
k(t,t) ~e2(t)
for all
L spaces, 1 ~ P < -, and the inequality
P
h(t)lrv(dt) ~ flg(t)lrv(dt) + flh(t)lrv(dt),
T, Minkowski's inequality for
+
flg(t)
T
where
T
T
0 < r < 1.
PROPOSITION 4.1.
1!!
!!.!.!. measurah!! .Qto.U!:.lf!.!!!!. !tochastic
{tt: t (T}
defined .2!!..!!?1!!!. probability space
correlation !_u.nction
k
~
f:
f(·)le(·)IP
L
1
4£
~ ~ function
1
into
! . ..1!! !
!. .!!!!! ill
p c:
.!,~­
~.!. ~
[1,-2...
~
.!.:.!.. ill if ~ only.tL f('~KP/2('t') ~
1
(equivalently
f(')k P/ 2 (.,.)
£
L ).
1
In view of Lemma 4.5, it is sufficient to prove that
Proof.
.f(·)lt(.,w)IP
f ( •)
L
F"P2
covariance function
vl'I.lu-ad measurabl~ functipn.!!:!'.!!
f(·)lt(·,w)IP
(0,
process
Ia(. ) Ip f:
E:
L
I
a.s. [P]
if and only if
f(')KP / 2 (.,.)
and
L •
1
2
f(.)K P / (.,.)
First assume that
we wi.ll prove that
f(·)/t(·,fAl)/Pe: L
Gaussian stochastic process
l
~l) (t,w)
f(')IO(')I P ( L
8.8. [P);
1
a.s. [Pl. Define the measurable
and
• If(t) 11 / P t(t,(o).
By our
assumption, we have
f[E(t(l)(t,w) - c(t(I)(t,w»}2]P/2v (dt) < •
(4.22)
.T
and
fJE(~(l) (tj.fAl»/Pv(dt)
T
Using (4.22) and ~ 3.3, we h:.w~
J
< -.
(4.:l!\)
29
fE(I(l)(t,w) - E«((l)(t,w»IP)V(dt) <~.
~emma
3.3
From this, (4.23) and
fE(I~(l)(t,w) \P)V(dt)
, we conclude that
< GO; i.e.
T
It(l)(.,w)IP. If(·)IIl;(·,w)lp
Conversely, let
f!
L
1
f(·)IH·,w)IP
a.s. [Pl.
L a.s. [Pl, we must prove that
l
2
f(·) lec·)IP and f(')KP/ (.,.) f! L • Since almost all sample paths of the
l
measurable Gaussian stochastic process t(l)(t,w):: If(·)ll/Pt(t,w) belong
E:
to Lp ' it follows, from Lemma 4.1, that the mean function If(·)ll/Pec·)
of {~~l): t € T} belongsto L p ' This implies f(·)lsc·)IP (Ll • To see
that
f(')KP/ 2 (.,.) ~ L , we proceed as follows:
1
Let m be any integer> p; then an application of Hinkowski's
inequality, Theorem 4.1 and Lemmas 4.1 ,
c( fI~(l) (t,w) -
3.1 give
f(t(l) (t.w» IPV(dt»)
T
~ ~ (m)(E(I/I.;(l)(.,W)\1
r-O r
From (4.24) and
Lemma 3.3,
f
£
notation f(x)
L
q
)r&-r)(II(f(t(l)>>1/1'Il) <GO.
P
t
we have
Jlf(t~KP/2Ct,t)V(dt)
(4.24)
< -;
'I'
i.e. f(')KP/ 2 (.,.)
L'!t
p
€ ~l'
(EL *)
p
for
This completes the proof.
and x
L.
€
p
!f(t)x(t)v(dt)
From
now
on. we shall use the
for the sake of brevity.
T
The proof of Proposition 4 will partly depend on the following lemmas
LEMMA 4.6.
~
.
!
~~ symmetr~~
from T)( T into
!
non-negative definite measurable function
satisfying
Ir:.p / 2 (t,t)V(dt)
< •
"
t
T
where 1
~ p < -.
!2!:.!!£h
f
(4.25)
j
(=L*), def:J.ne (pointwi...!!.)
99
L
£
(Sf)(s) • !K(s,t)f(t)v(dt).
T
(4.26)
30
Then S is a bounded linear onerator from L into L. Further. it is
-I:.
-!I.. ....P.
-symmetric and non-negative definit~ ~ ~ sense!h!! l(Ss)· g{Sf) ~
f(Sf)
0. for all f
_
_ _-> _
Proof.
all
t
g
~ L9 •
Since K is symmetric and non-negative definite. we have,for
stt € T.
(4.27)
From (4.25). (4.26). (4.27) and the fact that
f
~
Lq • ·it follows that
I(Sf)(s)IP ~ KP/2(Sts)[!Kl/2(t.t)f(t)v(dt)]P
T
~ KP!2(s.s)[ II Kl/ 2 1I p Ilf Ilq]P.
IIKl/2 11 • ({KP/ 2 (t.t)v(dt»I/p.
where
p
T
II Sf lip ~
Thus (4.28) shows that
(1IKI/2I1p)2(
S maps
L
Therefore
Ilf II q )
(4.28)
< • •
into Lp continuously; the rest is
q
obvious.
Following Kuelbs aDd Mandrekar [16} we make the following definition:
DEFINITION 4.1.
Let K be a symmetric non-negative definite measurable function from
TXT
Lq
into R satisfying (4.25).
into LP defined by (4.26)
1s called the kernel of
S then K· K'
S.
The continuous linear operato= S froc
1s called an S -operator.
_
_P~
Note that 1f K and
K'
The
function K
are two kernels of
a.e. [v x v].
PROPOSITION 4;2.
_
elel'lH!n_t
.
e
€ L,.. I'M a ~~iS:ce
__
!:..... - .-
S -Op0rator
_'IL.
f (9) •
S from
- -
J f (x)l1 (ebe)
I.
R
--
L
to L such that
-S ....R.
(4.29)
31
g(Sf) •
f
L
for all
--
f,g
p
q
Let
{tt: t
which induces the measure
covariance function of
T}
£
be a measurable Gaussian stochastic process
on L.
~
Let
p
{tt: t (T}
e and K be the mean and the
respectively.
a.s. [Pl, it follows, from Proposition 4.1, that
to
belon&
Since t(·,w) ( L p
SC·) and Kl/2 (.,.)
Lp • In view of Lemma 4.6, K determines an Sp operator from
S.
Denote this operator by
fL
f2(x)~(dx)
<
-
IIfl1
2
By Theorem 4.1, we have
IlxI12~(dx)
f
qL
P
P
P
Using (4.31), Fubini's theorem and Lemma 3.1, we get
<
J [f(x)
L
- f(O)][g(x) - g(e)l~(dx)
P
m
(4.31)
110.
.
J f(x)~(dx)
L
ana
(4.30)
L •
€
Proof.
[f(x) - f(9)][g{x) - g(e)]~(dx),
• fee)
p
ffK(s,t)g(s)f(t)v(ds)v(dt) • g(S£).
TT
This proves (4.29) and (4.30); the uniqueness of
e
and
5
follow from
an application of the Hahn-Banach theorem and from (4.29) and (4.30).
DEFINITION 4.2.
Lete
and
5 be the same as in Proposition 4.2.
clans corresponding to
~~
0 and the operator
The equivalence
S are called respectively the
and the covariance operator of the measure
~.
i
/
32
IS
A ZERO-ONE LAW FOR GAUSSIAN MEASURES ON FRiCHET SPACES AND APPLICATIONS.
Recently, Kallianpur (Theorem 2 of [13]) has proved a zero-one law for
Gaussian measures defined on certain function spaces.
This result 1s sub-
sequently slightly extended by Jain (TIleorem 1 of [10]).
In the first result
of this section (Theorem 5.1). we obtain. using the above mentioned result
of Jain and Kallianpur, a zero-one law
separable Fr'chet spaces.
f~r
Gaussian measures defined on
~
Specifically, we show that if
is a Gaussian
measure on a separable Fr'chet space with mean element zero and if G is
subgroup of
to
pl.
X and belongs to 8(X)
then
~(G)
(the completion
o~
Sex)
relative
is either zero or one.
Kallianpur (Theorem 3 of [13]), using his result mentioned in the
previous paragraph. has obtained a zero-one lew for zero mean Gaussian
. stochastic processes to the effect that if {tt: t ( [a.b]}
is a continuous
sample path (and hence continuous covariance. see Leurna 5.4) zero mean
Gaussian stochastic process,
on
[a,b]
and p
f
1s a real valued Borel measurable function
is auy positive real Dumber, then either
f(·)IC(·,~)IP ( t l ([a,b], teb) a.s. [P] or f{·)lt(·,w)I P f tl(rs.b], Leb)
a.s. [Pl. This result is a generalization of an earlier zero-onel~~ due
to Shepp [22], who proved the above result for a Wiener process defined on
the Banach space (in the sup-norm) of all real valued continuous functions
on
[a,b]
and for
p. 2.
Another generalization of Shepp's result was
given by Varberg (Theorem 3 of [?8]), who showed that Shepp's zero-one law
holds for any zero mean Gaussian stochastic process with continuous covariance.
It should be notcHl that 'Varberg does net require the continuity
of almost.all
seJllP~e
paths for the validity of his
follows that lallianpur's
res~lt.
From this. it
33
zero-one law mentioned above neither implies (contrary to his claim) nor is
implied by Varberg's zero-one law.
law similar to that of Kallianpur
{~t:
process
on
t
If I IE~tIP.
E
T},
for
1S P
Using Theorem 5.1, we prove a zero-one
for any measurable Gaussian stochastic
<~.
under the integrability condition
Thus our result (Theorem 5.2) includes, as particular
esses, all the three zero-one laws mentioned above, for the case
1
S
P <
w.
In addition, we give necessary and sufficient condition, in terms of the
autocorrelation function of the process
itt:
t
~
T}
and the function
f,
for the occurrence of the two altematives.
Recently, Kallianpur [13) has shO"..m that the Karhunen-Lo4ve (K-L)
expansion of zero mean continuous sample path Gaussian stochastic processes
converges uniformly on
or one.
a
compact interval T of R with probability zero
Sharper result to the effect that this probability is indeed one
is also known [11].
However, no zero-one law for the uniform convergence
of the K-L expansion of a mean square continuous Gaussian stochastic process
In Theorem 5.3, we prove a zero-one law for .the uniform convergence
is known.
in
t
on any Borel subset T of R of a certain random series.
From this
theorem, it easily followa that the seriea representations, obtained by
,
Cambanis and Masry'in Theorems 5 and 6 of [2], for weakly continuous measurable stochastic processes
uniformly in
addition that
all
t
€
'X.
t
{'t:
t ( T},
t E 'X}
a Borel subset of
i, converge
on T with probability zero or one, provided we assume in
tet:
t
€
'X}
is separable and Gaussian with E(tt) • O. for
It is worth pointing out that the series representation given
in Theorem 6 of (2] coincides with the
{l;t:
'X
K-L
expansion when the proce68
is mean square cr.lntinuous and T is a compact
interval of
R.
Therefore Theorem 5.3 also gives a·zero one law for the uniform eonvergence
in
t
on 'X of the
K-L expansion for the ,zero mean mean square continuous
34
Gaussian process pTovided we assume that the process is separable and measurable~
which, as is well known. are not restrictive hypotheses.
nlroughout this section. the following notation and terminology are
fixed:
Y will denote the separable Freebet
The letter
(locally convex
complete and metrizable) space of all real valued continuous functions on
R
endowed wi th the topology of uniform convergence on compact subsets of
R (see [15, p.8l]).
U(Y)
The notation
will denote the a-algebra of
subsets of Y generated by the sets of the form
.
(y(rl) •••••y(rn »
€
B
(n)
}.
where
n
£
Borel subset of the n-Euclidean space
C[a.b]
{y €
N. rl••••• r n
~
For any
R(n).
Y:
(n)
R and B
a. b
£
is a
R. ths notation
will denote the Banach space (in the sup-norm) of 411 real valued
continuous functions on
[a.b].
The following leDlllla. which is an immediate consequence of Skorokhod's
theorem [24] mentioned in
LE..~
14.
is recorded here for ready reference.
5.1.
1.U !
l>..!.! separa.ble !!!!Feb
.!~l!{mJ ~
- -in-X.
spac!•
.!!!.4.
f Ilxll r",(6c). £.o_t !::t.,c'£i. r
X
~
J?.!..! ~sien.
~ 0. wh~re
l.L:.ll
mea~u.!!.
!?!l
denot!!,!!!.!.
DClm
The proof of Theorea 5.1 wiU partly depend on Lemmas 5.2 and 5.3.
"hich we prove first.
LEMMA 5.2.
!!'.!. ~~gebra ill!l
jl,l p .100] •
is egU<11 ~
BeCl)
(an4. therefo~.
.!!!.2. egua~ £.2. !!!l;.!!!.ll!2!. ~
definitions
!9..!!!!!!:!.
J!!. !J!l
~
. Be(t».
Prcof.
., €
y*.
By the definition of
Be<T).
(the conjugate space of Y). then
it is
.,
cno~gh
to show that if
is measurable relAtive to
35
and
B(R)
Fix an
U(Y).
F
y*; then, for all
t:
y
E
Y. F(y) '. !y(t)A(dt),
R
where
A is a regular Borel measure (possibly negative valued) on
with compact support (see [IS. pp.126,121]).
we can choose
subsets of
write
a, b
R such that
E
fba
y(t)A(dt), for all
[a,b].
y
€
the conjugate space of the Banach space
fb y(t)dg(t), for all
a
y
bounded variation on
A has compact support,
A assigns zero measure, for all Borel
R that are disjoint from
F(y)·
Since
(R,S(R»
Y.
In view of this, we can
This implies that
C[a.b].
'nterefore
F
C[a,b] ••
€
F(y).
Y, where g is a real valued function of
€
[a,b).
Since
y's
are continuous, we can write
(5.1)
l{y) - lim
D,"+OO
for all
of
y (Y.
where
'P. relative to
definition of
t.
ult,n - a
and
S(a)
+
b-a
k(-),
k • O,l,.u,n.
n
U(Y),
The measurability
now follows from (5.1) and the
U(Y).
LEMMA 5.3.
1.!!. X be J! separable Frechet space. !:!£. J!. E.!...! .Q.aussian measure
~
!.
~!h! me~ ,,~lue
Proof.
p.21S]).
Let
•
-
be a topological isomorphism of
Then Y:: .(X)
define the semi-norm P
n
'-
Y.
Let
y € Y and let v(y)
,: 11+ .p(;)
and
111
and on
by
Pn (y).
I.
N}
is non-decreasing and generates the
on
Y
n
E:
t E: [sup
-n,n 1 ly(t)
be an arbitrary but fixed element of
Y
is continuous from
\.11
(C[-m,m),8(C[-m,m]»
and the definitions of
-
(see [15,
For each
y
be the restriction of
liZ· lIl·,-l; then
X into Y
is a separable Freebet space.
{pn :
sequence of seminorms
topol08:Y of
of J!. existA,.
Pm
and
}.IZ
into
C[-m,m].
"
we have'
€
N,
Then the
N.
Let
[-m,~l; then the map
Now set
are Gaussian measures on
respectively.
and
to
n
lI1-).l••-1
(y,8(Y»
From this, Le1lll18S 3.1 and 5.1,
36
f_Pm(Y)~l(dY) • J_Pm(~(Y»~l(dY).
Y
where
Y
II ·/1
Ilzll~2(dz)
J
C[-m,m]
C[-m, m] •
denotes the norm i.n
{pn :
Since m was arbitrary and since the sequence of seminorms
1s non-decreasing and generates the topology of
i,
~1
Denote the mean value of
is an iSO'lDOrphism).
by
8 ;
1
and let
a
We assert that
e ..
~l
F be an arbitrary member of X*,
definition of
a
l
N}
€
+-1 (6 1)
exists.
(recall that
is the mean value of
see this, let
n
it follows, from (5.2)
and Proposition 10 of [I, p.llll, that the mean value of
•
(5.2)
< -,
1'••-1
then
€
~.
To
Y*. By
the
and by LetDDa 3.1, we have
(5.3)
Equation (5.3) proves our (\83erticm.
LEMMA 5.4.
Let J:!.
.!!!!
!?!..! CauRs~~.!.!!!!..2!!.
It(s,tL· fyy(s)y(t)~(dy).
.!.!. ~ ~ ,!t1ement.2!
Proof.
y* and since
1!!. aCt) • fyy(t)~(dy)
!h!1! ! .!!!! .! .!!!.
continuous, and
!
~.
Since. for every fixed
~
{!,B(Y».
r
€
R,
the map
is Caussi,3n, it follows that
y 1+ y(r)
e and
k
belongs to
are
well
defined
real valued function..
From Lt'JmIl8 5.3, un1.queness of the mean value of
and from the fact that
a(t) ~ Iy'y(t)~(dy).
value of ·P.
and therefore
e
€
Y.
Thus
e
we have
that
~
e i . the mean
18 cont1nuouo.
To prove the
37
continuity of
k
on
on the set 6· {(s,s):
•
element of
A.
to
t
[t - 1,
it is sufficient to show that
R x Rt
R}.
S!
For every y
-1
~1. ~o~
t + 1], 8(C[t - I, t + 1)).
88
n
~
(tn ,t')
n ~ (t,t)
u {t':n
n
-.
E
Y,
£
denote by
~:
+ 1]; then the map
Crt - I, t + I}. Set
such that
Let' (t,t)
;
Let
is continuous
be an arbitrary but fixed
~(y),
the restriction of
~1
then
{Ct ,t'):
n
n
is Gaussian on
DEN}
(C[t - I,
be a sequence from RxR
we will show that k(tn ,t')
n ~ k(t,t)
Without loss of generality, we can and will assume that
N}
c
-
[t - I, t + 1].
y
is continuous from Y onto
n * -;
as
k
{tn:ncN}
Using Lemmas 3.1, 5.1 and the dominated
convergence theorem, we have
lim k(t ,t') • lim
n n
n-+w
n-+eo
f
yet ) yet') p(dy}
n
n
Y
f
• lim
yCt ) yet') P1(dy)
.....
n
n
n
~[t-1.t+1[
• J
y2(t)P1(dy) • k(t,t).
e[t-1,t+1]
THEOREM 5.1.
1!! X be.! separable Frechet. space and 1£!
.2!!. (X,S(X»
~ ~
element .!!!.2.• ..1.!! £
.l!. .!?!..! Gaussian measure
E.!..! sub-group
(relativ~.!2.
vector addition>.. of X which belongs.!Q. 8eX) (the completion.2!. B(X)
relative !.2
Proof.
,V;
~ .1!.,{Ql
.!!. either
~ ~..2!!!.'
Since, in view of LeDllD& 5.3, the mean element of p exists,
there is no amb,igu1.ty in the statement of ths Theorem.
be the same as defined in the proof of Lemma 5.3.
Let
Extend PI
in the natural way and denote the extended measure also by
G c SeX), , it follows that G· BuD,
•
where B (S(X)
Pl.
and PI
to' (Y,8 (Y»
Since
and D is contained
38
~-measure
in a
M of SeX).
zero set
plete metric spaces and since
S(Y)
SeX»~
and
is a one to one measurable (relative to
•
X into Y.
map from
[19, p.21J. that the sets H' •• (8)
Clearly
0'· +(0) • H' u
in H'.
Since
PI (M') • p<+
0' ~ 8(Y),
Thus
nt,
the function
UCl)
Since the mean eleme:lt of
PI).
we have that
it follows that
p{K),
0' E
lyy(t)PI(dy). 0
for all
t
k(s,t). I y y(s)y(t)Pl(dy)
belong to
'IJ
8(Y).
and therefore contained
i8 the completion of
From this and Lemma 5.2, we have that
relative to
~(D).
D'.
(+CH».
'BOO
where
it follows. from Theorem 3.9 of
and H' • +(H)
where
-1
X and Yare separable com-
Since
8(Y)
J,ll (M') • O.
relative to
(the completion of
U(Y)
is the zero element of
~ R.
.of
Now the proof of
is continuous.
k are continuous functions on R (see [13, p. 200]).
t
k!.!
T}
€
· '.£!l'~ 1!Fcb~.bilitI .!!a~ce
_-
· func:tt.,,,
...
-k.
P E [1,-).
Let
..-........
from T
into
g
...1Y.u..
l
~l
E:
~hQi).e
t c T ' (such a
~t~
m
meg fUflct~ !
ITI fee) II G(t) IPv(dt)
~•
0
f(·)Il;(·,I'.I),t P • L
Proof.
(0, FIi')
v~lu~d rM'~3u~able
e·
.!.!. .f(_)kP / 2 (.,.)
measu1'cble Q.a l J8sian stocnaatic E..t:~c7SS defined
1::0 n Teal
function on
-f -------,,-----
Assume~.
· 'instance, ~q
h
0'
and the elements of the reproducing kernel Hilbert 8p3ce
U(Y)
~ '{~t:
or
X.
Further, from Lemma 5.4,
the theorem follows from Theorem 1 of (10] observing that the subgroup
belo.nss to
Pr
~illt
Ib.!n. ,!i.ther
00
(this
a
-T,
and let
-
!h!. £e.f!!J ill
f(·) 11;(-,1'.1) IP e L
l
.!:.!.:.ill.
~ thas.!. alternatives oecur llccordinl
2
.2!".f(-)kP/ (.,.) • L :
1
an element
g
exists. because
R by
<
autocorrelation
of
L
l
suc:h that
"
i3
a-finite).
get) > 0
for all
Define the/function
39
S(t)l/p
k(t,t)1/2
if
~
k(t,t)
1
(5.4)
h(t) •
g(t)l/p
Let
n(t,w). h(t)r;(t,w);
then
with E(nsn t ) · h(s)h(t)k(s,t)
follows that.
tnt:
t
E:
and f(n
I TIE(n(t,w»2)p/2v (dt)
t
i£
T}
).
is
.
Let
p
1.
measurable and Gaussian
h(t)8(t).
(5.4), it
From
• IT[h2(t)k(t,t)]P/2v(dt)~fTg(t)V(dt)<••
From this and from Proposition 4.1, we have that
he ~ L.
~
k(e, t)
~(l)(t,w). n(t,w) - h(t)8(t)j
n(',w)
E:
{tel):
then
t
L
a.s. [P]
p
t (T}
and
is a
measurable Gaussa1n stochastic process with mean function zero and covariance
function • h(s)h(t) [k(s,t) - 8(s)8(t)].
•
.
{t~l): t
E T}
{~rt(l) :
E
t
element of
belong to L '
p
T}
Further, almost all sample paths of
From this and Theorem 3.2, if follows that
induces a Gaussian measure
p 1s the
element of L
ze~o
p
p on Lp'
(see the proof of Proposition 4.2).
We will prove that the probability of the set D· {w
'. fTu(t) 11;(1) (t,w) IPv{dt) < ClIt}
Moreover, th e me"''''
_.
~
n:
is either zero or one, where
is product measurable,
Tb~n. as foll~~s fro~
G i8 a sub-linear manifold (and hence a sub-group) of
t
E
T}
'l'heorea'3.l.
be the measurable Gaussian stochastic process defined
(L .B(L )"p). as constructed in the proof of
p
f
=0
p
Then, as is shown in the proof of Theorem 3.1, there exists a
p-null set go such that
~(·,x)
< -}.
•
Let G
p'&se 210 on [13),
.p
on the probability space
•
D E F.
E
Let' {~t:
fTo(t)lx(t)IPv(dt)
Ih(t) Ip
denote the set' {x
Lp '
L:
Ifl't\1
a(t)·~
for all x
€
go'
CI(t)!t(x,t)I P v(dt) < -}.
T
follows that
G'
.
E
B(L).
p
te·,x). x a.e. [v],
for all
x.
go~
,
and
Let G'
denote the set' {x E Lp :
Since a(·)t(·,·) is product maaaurable, it
From the definitions of G and Gt
,
the facts that
40
~(·.x) • x
a.e. (v],
it follows that
for all x
4 0,
o
G' n g~ ~ G ~ G'
and (·,x) : 0,
and G· (e' n g~)Ug1'
for all x
~
0 ,
o
where OC denotes
o
Thus e! B(L ), the
the complement of go and 0 1 is a subset of g.
o
p
completion of B(L) relative to J,l; therefore, by Theorem 5.1, ... (G) is
p
either zero or one.
We assert that
P(D)
is either zero or one.
Indeed,
we will show that P(D). 1, if ... (G). 1. and P(D)· 0, if ... (G)· O.
let
p(G). 1.
e. (G'
Since
n gC) u g
o
l'
it follows that
... (0' n Oc) • 1.
0
n.
.But rl(G' n Oc) c ,leG) (see (2.1) for the definition of
e 1 • peG' n gC) • P[r-l(G' n gC)] S P[T-1(G)] • P(D).
o
.
!at
... (G). O.
0
First
therefore
i.e. P(D) • 1,
Now
"
Again, using the fact that G· (G' n g~) u
nl ,
we have
peG' n gC)
• O. Since G'. (G' n gC) u n and since p(n). 0, it
o
0
0
0
follows that" p(G') • O. From this and from the fact that G.=. G'.· we
have P(D). p[~l(G)] s p[r1 (G')] • p(G') • o. Thus P(D). O. Using
the fact that P(D)
is eith9r zero or one, and noting that a(t).
and t(l)(t,w). h(t) [t(t,w) - a(t)],
Ih(t)IP
we hav~ that
P{w en:" ITtlt(t)I I~(t,w) - 6(t)I P ]v(dt) <.}
~
is either zero or one.
From this and the fact that f(·>lac·)IP (L1 , we have that either
f(e>I(;(·,w)I P E: L
8.8. [P]
or f(e)Il;(·,W) p ~ L 8.S. [1']. "The rest
l
1
of the proof will follo" if we can shu", that fee) II;(e.w) IP E: L a.8. [P]
1
2
P
if and only if f(e)k / (.,.) eLl' but this follows from Proposition 4.1.
lWWUC. 5.1.
The following two points are worth mentioning regarding Theorem 5.2.
(a)
The proof of Theorem 5.2 depends heavily on Theorems 3.1, 3e2 and
S.l. Since Lp ' 0 < p
< 1,
is not a locally convex space (and hence i8
not a Fr6chet space) except in the rather cninteresting case when the set
of values taken by v
1& finite (Gee [15, p.S3» and since we do not hnv(l
the analogues of Theorems 3.1 and 3.2 for Lp ' 0 < P < 1,
our methode do
•
41
not give a zero-one law analogous to TIlcorem 5.2, for the case when 0
(b)
<
p < 1.
In Theorem 5.2, we have proved the zero-one law under the assump-
tion that
fTI£(t)IIS(t)!Pv(dt) < w;
If fTlf(t)llo(t)IPv(dt)
fTI£(t)IIO{t)IPv(dt) • w.
~{w
from Lemma 4.1, that
and we did not consider the case when
€
c
w,
then, it follows,
ft1f(t)llt(t,w)IPv(dt) .~} > O.
n:
(but are unable to ptove) that 'this probability is indeed one.
We believe
In one of the
following corollaries, we show that this is the case for a particular choice
of
{l;t:
T}.
t €
COROLLARY 5.1.
~
~
{~t:
function
E
~
T}
be.! measurable Gaussian sto.chastic process
l!lli! covariance ftmction
.!lli!..!!:.! p
K,
€
~
[law).
Then
p ~ ill .2!. t(· ,w) ~ L p ~ ill, .!ill! these alternatives occur according.!!. KP/2(.,.) € L .2!: KP/2{·,·).4 L
either 1;(. ,w)
•
t
L
E:
r
1
COROLLARY 5.2.
~!.
ill!!.
n:
{X
n
E
~
N}
2. sequence .2f jointly Gaussian random variables
n) - 6(n) and E(XmXn )" k(m,n).
of real numbers and let p E: [la ClO ) .
E(X
(a)
Assume that
nrl
lanI16(n) IP <
CXI.
~
{an:
n
E
be.! sequence
N}
Then either ·n~J. Isnllxnl P <
or n~l Ian II Xn Ip •
~.Ill ~ these sltarna:ives occur
according.!!. n'"
E1 ·1 an IkP/ 2 (n, n) < co -or E1 Ia IkP/ 2 en, n) • _
!!.:.!..
ill
CIO
..::n~·
(b)
CIlI.
n~~
Assume -that -!L
X 's are
Ellan Ile(n)I P • _GO
- independent -and ..
n_c
~ ri~l·larillxriIP Proof.
~ill.
The proofs of Corollaries 5.1 and 5.2 (a) follow
from Theorem 5.2.
. <Ian Ilxn IP:
CD.
n
£ N}
•
Thus we need only prove 5.2 (b).
immed~ately
Since the sequence
of random variables is independent, it follows. from
the classical zero-one law (see for example [8, p.201]), that
CIO
42
p{w ~
n!l lanllxn(w)IP -~}
n:
This probability
is either zero or one.
is necessarily equal to one; for otherwise it will contradict
For the rest of this section. we assume that
R,
A(T)
is the class of Borel subsets of
a-finite measure on
(T,A(T».
T
T
and
is a Borel subset of
v
The hypothesis that
4.1.
Le~a
1s any non-negative
L _ Lp(T,A(T),V)
p
is
separable is also in effect.
THEOREM 5.3~3)
Let
itt:
t
E:
be.!. ~ ~ measurable Gaussian stochastic
T}
8pac~
. ·process defined .2!!..!. probability
_a.s. rp';
and let
~
-
--
Let .. {c : n
-
E:
n
N}
{g:"n
E:
n
~ ~
(P.,f,p)
such that
be any s{"..:9!tence of elements of
N}
-
--
-
-
.!!!!.
L
£
e
=L*.
p
L
0
seQuence of Gaussian random variables defined
'new) - IT ~(t,w)gn(t)v(dt). (see Theorem~. -Le t
sequence of
1;(',111)
continuous functiC!!!!.en.
n
{f:
_,:;;,n
1.. !!!!m. ~
E:
_N}
B!.
hI.
any
series
CD
I
r-l
(5.5)
f (t)c; (w)
r
r
converges uniformly
E!. £
..2a.
! .!:!!.!l
probability
~.2. 2.!.Q!!!:..
addit:f.on to the above hypotheses .!S. ~~ that the series
stochastic process tnt:
forml>; in 1. .2!!. T
Proof. Let
in t
on
Tj
D
l
t
~
E:
probability
r
tnt:
t
E T}
~­
~ 2!:~.
then
CD
f's
converges £2
(5.5) converges
be the Cil-set where the series in (5.5) converges uniformly
j
CD
D • n u
0
l m-l n-l j>i>n
Since
~.!!.
T},
.!n.
ll.!!l
are continuous on
o
t
T,
8ubset of it in the expression of
E:
{wdl:
T
T
D •
1
r
r-i+l
f (t), (111)1 SlIm}.
r
r
can be replaced by any countable dense
It follows that
D c F.
l
Let
1I
•
43
be the Gaussian measure induced by
{'t:
t
E:
on L
T}
P
(see Theorem 3.2).
Let
I fret) JTx{s)Sr(s)v(ds)
converges uniformly in t on T}
r-I
and
{x
G' •
1
where· {t :
s
r
E:
L :.
fret)
p
r-l
s
€
J ~(s,x)g (s)v(ds) converges uniformly in
T
r
is the measurable Gaussian stochastic process defined on
T}
- the probability space
(L ' 8(L )' IJ),
p
p
as constructed in Theorem 3.1.
Then
G is a linear manifold (and hence a sub-group) of L(I and rlCG l ) •
l
(see (2.1) for the definition of 1). Now the proof of the fact that
P(D )
1
on T}.
t
D
1
is zero or one follows by repeating the arguments used in the proof
of Theorem 5.2 replacing D by
Dl , G by Gl and G' by Gi.
For the proof of the last part of the theorem it is sufficient to show
that if the series in (5.5) converges uniformly in
on T with probability
t
one and if
T,
t €
{n : t E: T} denotes its limit, then n(t,w) • n(t,w), for all
t
off a P-null set F.
Let A be a countable separating set and F
o
be the P-null set appearing in the definition by separability of
Since, for each
t
E:
T,
converge.s uniformly on
t
j
E:.
Ft •
j
A.
the series
T
E:
T}.
converges in probability and since it
F
tj
such that n(tj.w) • n(tj,w)
off
Let G . be the P-null set where the series in (5.S) does not converge
Set
and
t E:
~
t
with probability one it follows that, for every
there exists a P-null set
uniformly to. n(t.w) •
•
{n : t
t
T},
• F u G u (~
0
~
F
Then
)
tj
o
j
(A e • the complement of
there exists a sequence
and n(s, w )
nOD
F
net , w)
0
0
convergence of the series in (5.S).
a8
n
{s:
n
+ ~
A);
n
E
N}
Let
then, by separability
in A such that
(see [6, p.59]).
n~,lA)o)· is continuous on
T.
By uniform
Thus
44
~(s ,w ) ~
n 0
and
net 0 ,w)
0
n (s ,w )
n
0
~
as
n( t , w )
0
0
This completes the proof.
as
n
~
~,
it follows that
net o ,w0 ) •
~(t 0 ,w0 ).
45
§6
TltE CHARACTERISTIC FUNCTION AND THE TOPOLOGICAL SUPPORT OF GAUSSIAN
}>IEASURES ON
Let
be a probability measure on
~
support of
L.
P
is the smallest closed set
~
The topological support
x
L
p
(L
p
{L .8 (L
p
D
p
».
r.p
of
The topological
such that
~
D can also be characterized as the set of all
having the property that p(U).> 0 for each norm-open set
containing x
(D) • 1.
(see. from example. [J 9, p. 28]).
The measure
to be non-degenerate if every non-zero linear functional on L
p
as a random variable on the probability space
(Lp,B(Lp)'p)
l.t
U of
is said
considered
has a non-
degenerate distribution function on the real line R.
Let now
operator S
l.t
be a Gaussian measure on L
p
(see Definition 4.2).
e and covariance
with mean
In this section, we prove three results:
The first result (Theorem 6.1) shows that there exists a one to one correspondence between the set of all Gaussian measures on L
p
pairs
(x ~)
where x ELand S
P
and the set of
is an S -operator from L
P
q
into L;
P
this theorem also gives the general form of the characteristic function of \1;
the second result (111eorem 6.2)
sh~la
is equal to the algebraic sum of {8}
closure in L.
p
ditions for p
that the topological support of
and
l.t
S(L),
where -- denotes the
p
The third result (Theorem 6.3) gives five equivalent conto be non-degenerate on L.
p
NOY we state Theorem 6.1; the proof of the theorem will be given after
the proof of Lemma 6.1, which carries the whole burden of the proof of
Part (b) of the theorem
THEOREM 6.1.
(a)
1!! l:!. J?!..!
G~uss,ia~ ~£asure.£!!
~; ~.!h!
characteristic
46
.x.
function
of
is given h
~
if (0)- ¥(Sf)
for Every
f
£
operator £f
(b)
e
€
.X(f)
a
~
~
L , where
9
~
respectivell~
~
S is an S -operator from L
Land
(6.1)
.§. ~ the ~ and the covariance
Conversely,.! function X on
_ _~p -
,
e
-
--
p
into
-S -
-
function of .! unique Q.aussian measure JL .2!l
!!! respectively the
(6.1~,
of the form
.:e.
where
L, is the characteristic
-..E. - Moreover,.! .!lli!. !
and !h!. covariance operator.2!. JL.
~
LEMMA 6.1.
1!! !
(a)
from
TxT
~~~
defined
into
existR
~ ~
~M
~~.
!,
~
symmetric
!
and
non-n~ative
measurable Gaussian
ill,
Chooae
~n
element
y
into
process
{~t:
t e T)
R
e
Iv x v].
2
/kP/ (t,t)v(dt) <
00,
then
T
where k(s,t)· K(s,t) .. 9(s)O(tl..
(Ll(~)
(such a choice is possible,since v
T
~~
addition~assume that
t(·,r..») ( Lp(V) :: Lp(T,A(T),v) !.:..!.
from
~chastic
R.
'p£obab1lity snact:! _(&F liP) !E£h!h!!..!!!. ~ function ..
If in
Proof.
~nction
be..! measurable function!!.2!. 1: into
.!!!2.1!! covariance function·
(b)
definite measurable
such that yet) > 0
is a-finite).
Let
z
a.e. [v]
ba the function
defined by
~
k(tl,t)
if
0
{
. k(t,t)
if
k(t,t) > 1.
k(t,t)
~
I
s(t) •
Define a measure A on
(T,A (T»
follows that A is a finite
by the relation
~eesure,
mutually absolutely continuous).
and A and
y(t)z(t).
It
v are equivalent (i.e.
Further jk(t,t)A(dt) < -; this fact
T
along with Lemma 4.5 imply
(:~) (t) •
47
IK(t,t)~(dt) < -
(6.2)
T
J92(t)~(dt) < -.
(6.3)
T
Using 8eparability of
= L2 (T,A(T),A)
L2 (A)
that
(6.3) that
L (>.)
2
let
L (v)
2
6
E
2
into itself.
{e: DEN}
n
Let
1. •
2
..
1.
2
S.
Let
into itself with kernel
"
L (A)
2
be an i80morphism of
L2 (A).
be a Hilbert ba8is of
i. a R1lbert basis of
from
i8 a kernel of an S2-operator
JC
,
A, it i8 easy to 8how
It follows from (6.2) and
i8 a180 8eparable.
and
L (A)
and the definition of
Then
S
,.-so,-l; then
{a('m,n): (a,n)
E
S
ODtO
froD
1 : and
2
{en =.(en):
n E N}
is an S2-operatol'
B )( N}, where
A
lII'l
and <,> denotes the inner product in 1. • (nGre
2
n
are viewing t 2 . . L (N, A(N) , \ ), where A00 is the C1-algebra of all
2
.(m,n) • <e ,S(e»
11
we
N and
subsets of
is the COUDtin3 -.aauTe.)
\
(6.2) and the definition of
8,
i
that
Further, it foilova, from
is syEmetric and nou-D0sative
definite with
!
pl
th~ fact that
use
i8 .~etric aLd CDn-negntive definite permits us to
JCc~~vrov'8 GXt~n&ion theo~aA
{C : nEoN}
n
and
•
obtG1~
to
A
G£u~sian
defined on sow. probability apace with
f(C lin
t )
clear that
(6.4)
s{n,n) < -.
-
f(t • )f(t D ) • iea,D)
{t : n E N}
n
otochast1c procoS8
E(t ) .
n
e(n) :: ~. (8) (21)
(see, for example) [6, p. 72]).
18 meuurabla.
FrOlll (6.4), the fact that
It is
Ie
1.
2
and Proposition 4.1, it follows that almoct all s.ple pathaof thea proceeD
: {t : n c' N}
°
°
n
belong to
f._.
-Z
U.ina this fact and Theorem 3.2, we have that
{~: nEil} induces a Gaus8i&u lIeasure
Pr0polSition 4.2, that
ad
..
S
th~ BMD
respect.ively.
Cau•• ian _HUra on
Set
L (>.)
2
~
Further it follows, fra
..
aDd the covariance operator of II are 8
l I · ~...
with mean
on
1 •
2
..
Then it 18 euy t.o verify that
8
and covariance operator
S.
II
1. a
48
Using this,
Tlll~('lrem
stochastic process
3.1 and Proposition 4.2, we get a Gaussian measurable
ft t : t
T}
f
e a.e.
such that its mean function •
a.e.
(~
and
v x v
x
A].
Since
~
and. v
[~]
and its covariance function • K
Ax A
are equivalent, it follows that
are equivalent (see,for examp1e,[29, p. 40]).
follows that the mean function of
and the covariance function of
(O,F,P)
defined on a probability space
{~t: t
itt: t
This completes the proof of Part (a).
€
c T}
From this it
e a.e. [v]
is equal to
is equal to K a.e. [v x v].
T}
The proof of Part (b) is immediate
from Proposition 4.1.
A remark should be made about the proof of the preceding lemma.
having shown, in the above proof, that
on L (A)
2
e
f
L ().)
2
and
S
After
is 5 -operator
2
into itself, we could have evoked the known result (see,for
~ample,[27t
p. 156]), which states that for the pair
(a~S)
there exists
a Gaussian measure U on L (A) with mean e and covariance operator S
2
respectively. However, we have deliberately avoided the use of this result
and based our proof on the Kolmogorov extension theorem and Theorem 3.2,
in order to supply a proof of Part (b) of Theorem 6.1 for all
out
~seuQing
for'L (A)
2
L (v) withP
the validity of this theorerr. (i.e. Part (b) of Theorem 6.1)
(and hence for
L (\I».
2
From Part (a) of Lemma 6.1,one can ask the question whether there
exists a Gaussian measurable stochastic process·
E (l;t) •
e (t),
(s,t) E
T )( T.
for all
t
E
T
ft t : t
E
T}
such that
and E (~s~t) - E (ts)E (tt) • K(s,t), for all
In a subsequent paper, we give a negative answer to the
above question by prOViding a counterexample.
For this example and other
related results we refer to [20].
----
-
Proof of Theorem 6.1.
.......
First
~e
prove Part (a).
it 1s clear (see for example [5, p. 289]) that
Let f
E:
L ; tben
q
49
122
ip(f)- 2(a (f)-p (f»
X(f) • e
where
p(f).
J f(x)p(dx)
L
and
a2 (f)·
f
L
p
f
2(x)p(dx).
The proof now follows
P
from Proposition 4.2.
Now we prove Part (b).
Let
K be the kernel of
Sj then K is
symmetric non-negative definite and measurable from TxT into R
2
with !KP/ (t,t)v(dt) < w. From these facts and Lemma 6.1, it follows that
T
there exists a measurable Gaussian stochastic process
probability space
(O,F,p)
{t : t ~
t
such that its mean function •
e
t(·,~) E
L
its covariance function • K a.e. [v x v]
with
p
T}
on a
a.e. [v}
and
a.s. [Pl.
From these facts, Theorem 3.2 and the proof of Proposition 4.2, it follows
that
induces a Gaussian measure p on L
p
covariance operator S with the kernel K.
the characteristic function of
that
P
is unique.
p
with mean
a,
and
Now using Part (a) we have that
is of the form (6.1).
Finally, we show
Let PI be another Gaussian measure with the same
characteristic function.
Then, for any
n
~
N and any
fl, ••• ,f
n
E
L ,
q
(f1 , ••• ,f ) of random
n
Therefore p1 and p coincide
the joint characteristic function of the n-vector
variables is the same under p1
on the cylinder sets of
and
p.
Lp ' and hence on 8 (Lp ) •
REMARK 6.1.
Analogue of Theorem 6.1 for
[26].
t
p
spaces was formulated by Vakhania
Similar result for real separable Hilbert spaces was proved earlier
by Mourier [18, p. 243].
Note that this result of Mourier follows by an
application of the Riesz representation theorem and Theorem 6.1.'
Theorem 4.9 of [19, p. 179],
L~a
6 of
£21,
S~e
also
p. 73} and Theorem 7.1 of [5,p.293].
In ,the next result we obtain the topological support of a Gaussian
measure
1.1
on Lp '
For the proof of this result, we will need a simple
50
fact. which states that if
function on
f
L with
P
h
is real valued non-negative continuous
h(x)~(dx) • 0. then
L
h(x). 0. for each x
P
belonging to the topological support of
The proof of this fact
~.
follows immediately from the definition of the topological
suppo~t.
THEOREM 6. 2.
Let .l!
!.
oper~tor
.. S (L)
_ ....9_
E!. l!. Gaussian measure
~
~
~
with
~
the topologi.cal support of l!. is
!. and covariance
e+
dertotp.s the closu.re 1.n L of the set S (L ).
-.P. - - Q
Pro~f.
Define the measure ~e on L by ~e(B).
--
every B
€
p
H(Lp>'
support of
~a
q
~(e
1s Gaussian on Lp with mean ze~o and
It is easy to verify that D is the topological
S.
if and only if
e+D
is the topological support of
~e
Thus we need only show that the topological support D of
to the set
=f
L
S(L
q
Lp
is a closed subspace of
~e
Therefore it is enough to show that
J.
=L
D· S{Lq ) ,
deuotcs the annihilator of D
(reopectivcly
(respectively of
XCf)
is equal
q
(see [12. p. 892]).
J.
~.
S(L ).
It is known that the support of
where D
+ B), for
~e
Then
covariance operator
S(L ). where
».
be arbitrary but
eif(X>~e(dX) • Ieif(X)~a(dX) • 1, since
D
P
-
~(Sf)
x € D, and lA (D) • 1. Since X(f) • e 2
e
- ¥(Sf)
.
. • 1. Thus f(Sf). O. Let u,v
e
€
fi~~d.
Then
f(x) • 0, for each
(by Theorem 6.1), we have
L ; then the generalized Cauchy
q
inequality
(u(Sv»
2
~
u(Su)'v(Sv)
can be shown to hold using stmilar arguments as used to prove the Cauchy
inequality for inner products.
Using·the fact
f(Sf)· 0 and (6.S).
.~
51
we have
f(Sg)
f('), it follows that
~--:-.l
f
~
5(L) cD.
q -
show that
g ~ L.
• 0, for all
S(L )i.
€
q
Let
S(L )~.
q
This shows that ni
Thus
S(L)
q
- ,k2 f (Sf)
X(tf) =/eitf(X)lJ (dx) • e
f
£
J.
By continuity of
f e
and
t
.. 1, since
2
e
D
q
c
-
5(L )~.
q
Now we
be any real number; then
f(Sf). O.
Thus
o.
1[1 - cos(t f(x»]lJe(dx) •
(6.6)
D
Let
~t(x).
1 - cos(t f(x»; then
is non-negative and continuous on D.
~t
From this, (6.6) and the fact mentioned before the statement of the theorem,
we conclude that Vt(x). 0. for all
. have that
all x
£
• 1, for all
cos(t f(x»
Hence f
D.
xeD.
t
~
Since
R and x
Therefore S (L ).L c DJ..
£ D.
q
-
t
£
D.
was arbitrary, we
Thus
f(x)· 0, for
The proof is now
complete.
REMARK 6.2.
It
is well known that
S(L2)
is equal to the closure in L2 of the
linear span of the normalized eigenfunctions corresponding to non-zero
eigenvalues of
S.
Using this fact and the Riesz representation theorem,
s result of Ito [9J (and also a theorem of Garsia et al [7J), where the
topological support of a Gaussian measure on a Hilbert space is obtained.
follows as a corollary to Theorem 6.2.
THEOREM 6. 3.
~
~
!.
~
a Gaussis1! measure
~
~
.!'!!!h covariance operator
S.
the followina .!!!. eg,uivalent:
,
(a)
! ..!!
(b)
S{L ) • L •
(d)
.!.h!. absolute value
g,
positive;
~.!!
f
€
Land
q
f(Sf) • 0
then f • O.
I
2
of
unity only.!!!E. f · O.
~. ch~rllcteristic funct:f.on
-
of l!. is
52
(e)
(U) > 0
\.I
for eve!!. .o~en ~ U of
L •
...P..
is non-degenerate on L.
-..2Proof. The proofs of all the implications except (8)
(f)
_\.I
simple and can be proved by using the
3~guments
of (a)
~
t
p
spaces (see
However, as is noted by Vakhania, the proof
(e) forms the fundamental part of Theorems 1 and 2 of [21) and
1s relatively difficult.
In the present case the proof of (a) -9 (e)
follows easily from Theorem 6.2:
Assume (a) is true; then an application
of the Hahn-Banach theorem shows that
the proof of theorem 1 of [27]).
support of
(e) are
of Vakhania [27], who
formulated and proved this result in the setup of
Theorems 1 and 2 of [27]).
~
II
open set U of
S(L). L
q
Thus, by
is the whole space L •
P
p
T.~eorem
(see the beginning of
6.2, the topological
Therefore lI(U) > 0, for every
Lp •
.... .
..
'
I
53
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<
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,I
e