It This re8eaPCh was partia'Lty supported by the Nationa'L Science Found- under Grant GU-20S9. 1 2 Department of Mathematics and Department of Statistics. Department of Statistics. GAUSSIAN STOCHASTIC PROCESSES AND GAUSSIAN MEASURES lt by Balram S. Rajput 1 Stamatis Cambanis 2 Department of Statistics University of North Carolina at Chapel: HiZZ Institute of Statistics Mimeo Series No. 705 August, 1970 .'" , -
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ItThis re8eaPCh was partia'Lty supported by the Nationa'L Science Found-
ati~ under Grant GU-20S9.
1
2
Department of Mathematics and Department of Statistics.
Department of Statistics.
GAUSSIAN STOCHASTIC PROCESSES AND GAUSSIAN MEASURESlt
by
Balram S. Rajput1 Stamatis Cambanis2
Department of StatisticsUniversity of North Carolina at Chapel: HiZZ
Institute of Statistics Mimeo Series No. 705
August, 1970
.'" ,
-
GAUSSIAN STOCHASTIC PROCESSES
AJID GAUSSIAN MEASURES*
Balram S. RajputDepartment of Mathema.tics and Department of Statistics
University of North CarolinaChapel Hill, North Carolina 27514
and
Stamatis C&mbanisDepartment of Statistics
University of North CarolinaCha.pel Hill, North Carolina 27514
,
*This research was partially supported by the National ScienceFoundation under Grant GU-2059.
1. INTllJOOcrION
Gaussian stochastic processes are used in connection with problems
such as estimation, detection, mutual information, etc. These problems
are often effectively fo~ulated in terms of Gaussian measures on appropriate
Banach or Hilbert spaces of functions. Even though both concepts, the
Gaussian stochastic process and the Gaussian measure, have been extensively
studied, it seems that the connection between them has not been adequately
explained. Two important questions arising in this context are the follow-
ing:
(Ql) Given a Gaussian stochastic process with sample paths in a Banach
fUftCtion space, is there a Gaussian measure on the Banach spae;e which is in
dueedby the given stochastic process?
(Q2) Given a Gaussian measure on a Banach function space, is there a
Gaussian stochastic process with sample paths in the Banach space which in
duees the given measure?
•The purpose of this paper is to explore questions Q1
B.Y part (l.ii) it follows that for every t (Ac there exists Nt ( B(X)
•
such that peNt) = 0 and
N Clim t ~ (t,u) = y(t,u) for all u £ Nt'N*co n=l n
N N co
Since {ik w(t,u)}koo_l is a subsequence of {E $ (t,u)}N-l'n=l n - n=l n -
c cit follows that for every -t ( A , Nt .£ Bt • Hence for all t E: A and
,.
Y(t,u) = Y' (t,u) = E $ (t,u).n=l n
It now follows from (0,1] x X = {A x Xl u HAc x X) 0 B} u HAc x X) nBc},
Leb(A) • 0 and (Leb x p)(B) =0, that (Leb x lJ){(Ac x X) nBc} =(Leb x lJ){(O,l] x X}. For all (t,u) ( {(AC
x, X) nBc} we have t £ AC
and u (B~ and thus Y(t,u) = Y'(t,u). It follows that Y(t,u) = Y'(t,u)
e..e. [Leb x lJ] on [0,1] 1C x. IIProof 2t Theorem 3.3. Define the probability space (n,F,p) by
n • X, F =. B( X), P = lJ and denote the identity map 'between n and X
e·
by I: !w • u, where u = w. Define X(t ,w) by
{ aon A x n
X(t,w) •uo(t) + Y(t,I-l(w» AC x n,on
,
16
where A and Y are the same as in Lemma 3.2. Since u (t) and Y(t~u)o
are B[O~l])( SeX) measurable. it follows that X(t.w} is R[O.l])< F " ..
measurable and hence (n.F.p; X(t,w), t £ [0,1]) is a product measurable
stochastic process. It also folloW's by parts (l) and (3) of Lemma 3.2
that for all ct fA.
X(t,lil) =u (t) + 'f <ljl ~ I-1 (w) - u > • (t)o n-1 non
I»
a.e. (p] and also in L2 (O,F,P). Since the random variables {<"u - U >} 1• n 0 n=
on (X.8(X),lJ) are jointlY' Gaussian. so are the random variables
{<.n,I-1 (w) - uO>}:=l on (o,F,p). Hence the convergence in L2(O,F,P)
of the series C3.13), along with C3.12), imply that X(t,w), t e [0~1] is
a Gaussian stochastic process.
Now uo(t) € L2[0.l] implies uoCt) € L2([0,1] )( 0). Also
y(t.u} £ L2([0,1] x X} implies Y(t,r-l(w» £ L2([O,1] )( 0). It .fol10ws
that X(t,w) € L2([0,1] x 0) and by Fubini's theorem we obtain,1 1
+1» > I jx2(t,w}dtdP(w) = E(jx2(t,w)dt} ;n 0 0
i.e., X(t,w} satisfies (II). Thus X(t,w), t e [0,1], satisfies both
(Il ) and (II) and by Theorem 3.1 it induces a Gaussian measure lAX on
(X,8(X)}. It will be shown that lAX = lA. For this it sUffices to show
l!X(B) • \J(B) for any open sphere with radius £ > 0 and center any
w € X. Let B = {u € X: Ilu - wll < d. Then we have by (1.2), (-3.12)
and (3.13)
\JX(B) = P{lA) f 0: IlxC' ,w} ':"' w(·} II < d
= II {u e X: IIu - t <4l •u - U >. -wII < £ }_ 0 n-l non
Since the support of II is sup (ll) = tio +"ii"P{.n}' we have
17
~x(:a) • ldv ~ sp{~ l: -II'll + t <~ ,v>. - wi I < e}n 0 n=l n n
== lJ {v f; -;p"{ $ }: IIu + v - vii < £}n 0
ll'l ll{U E: Sup(l1): II'll - vii < d
= ~{'ll E X: Ilu - ..,11 < d == u(B)
which completes the proof. I I
4. SOME GENERALIZATIONS AND EXTENSIONS
In this sect1.on the folloving generalizatiorB and extension$ of the'
results presented in Sections 2 and 3 are considered.
4.1 From [0all to any Borel meas'urable subset T of the real line.
All the results presented in Sections 2 and 3 for the case vhere the
parameter t,
takes values on T == [0,1) clearly hold for any compact
subset T of the real line. Moreover it is easily seen that all the results
of Section 3 also hold for every Borel measurable subset T of the real
line. Th~ only use of the compactness of the interval [0,1] is made in
the proof of Theorem 3.2 s in concluding that the continuous, orthonormal
and complete functions 4>n(t) in L2[0,1] are uniformly bounded. However,
what is essential in the proof is clearly the uniform boundedness on T
of each fUnction in a complete orthonormal set in L2(T,B(T),Leb), and this
can be always satisfied by an appropriate choice of a complete orthonormal
set even if T is a set of infinite Lebesgue measure.
4.2 On the induction of a Gaussian measure on an appropriate~ S'Dace by
a Gaussian stochastic 'Drocess.
The two questions raised in the introduction have been ansvered in
the affirmative for the function spaces e[O,l] and L2[O,1] in Sections 2
and 3 respectively. It should be remarked that the only case where an
affirmative answer is obtained under restrictive assumptions, and by no
(IV)
18
means in general, is the important case of inducing a Gaussian measure on
L2[0,lJ from a Gaussian process X(t,w), t ( [O,l)~ the restrictive
assumption being condition (II). We now turn our attention to this case
and ask whether assumption (II) is essential to inducing a Gaussian
measure or not. It turns out that the need for assumption (II) is solely
due to the specific way in which we attempted to provide an answer, and
that an affirmat~ve answer can indeed be given in the general case with
no restrictive assumptions whatever. Specifically, it is presently shown
that every. product measurable, Gaussian stochastic process X(t,w), t ~ T,
where T is any Borel measurable set on the real line, induces a Gaussian
measure on an appropriate Hilbert space of square integrable fUnctions on T.
Let (n,f,p) be ~ probability space. In this section by (Il ), (I2 )
and (II) we mean conditions (11), (12) and (II) with the interval [0,1]
replaced by a Borel measurable set T of the real line. It follows by
Theorem 3.,l and Section 4.1 that a stochastic process X(t,lIJ), t ( T,
satisfYing (Il ) induces a Gaussian measure on L2
(T,B(T),Leb.) if (II)
'is satisfied, i.e., if !r(t,t)dt < +~.T
Consider a measure v on (T,8(T» such that
!r(t,t)dv(t) < +T
That such measures v exist tollows from the following particular choice:dv
Define vo on (T,8(T» by [d L~b](t) =f(t)g(t), where get) ~ 0,
.....,
t(t) •
r(t,t)
1
tor
tor
o ~ r{t,t) < 1
1 ~ r(t,t)
It is clear that vo
satisfies (IV) and is a finite measure. The folloving .
theorem can be proved in the same wrq &8 Theorem 3.1.
19
Theorem J!.l. If the stocha.stic process (n,F,p; X(t,w), t E'. T)
satisfies {II}' then for every measure v on (T,B(T» satisfying (IV),
X(t,w), t ( T, induces a Gaussian measure ~x on (X =L2
{T,B(T),v),
B(X» with mea.n m (X and covariance operator S generated by the
kernel R(t , s ) .
IIenee, even though a product measurable Gaussian process does not
necessarily induce a Gaussian measure on (HLeb =L2
{T,B(T),Leb.), B(~eb»'
the necessary and sufficient condition for the latter being (II), it always
induces a Gaussian measure on every (H =L2
(T,8{T),v),B(H» with vv v
satisfying (IV). For instance, a wide sense stationary process X(t,w),
t € (--,+-) :: R, satisfying (II) does not induce a measure on L2
(R,B(R} ,Leb. ),
but it induces a Gaussian measure onL2(R,8{R),v) for every finite measure
v. Also a harmonizable Gaussian process X{t,w), t E'. R, does not necessarily
induce a Gaussian measure on L2
(R,B(R) ,Leb.), but it induces a Gaussian
measure oJ; every L2
{R,B(R),v} for v a finite measure. For a more concrete
example consider the Wiener process +W{t,w), t € [0,+-) =R ; then
ret,s) = min(t,s) and even though W(t,w) does not induce a measure on
L2
(R'+ ,B(R+) ,Leb.) it induces a Gaussian measure for example on
+ + dv ]() -tL2(R ,B(R },v), where v is defined by [d Leb t =e •
Remark 4.1. Denote by N the set of measures v on (T,B(T» which.satisfy (IV). A meaningful choice of v in N assigns positive measure
to all Lebesgue measurable subsets, vith positive Lebesgue measure, of the
set To = {t € T: r (t,t) .; O}. That such measures v exist is demonstrated
by the measure v. The construction of v makes also clear that thereo 0
exist v € N which are absolutely continuous with respect to the Lebesgue
measure with, moreover, ~d ~b. ](t) ; 0 a.e. [Leb.] on 'lo; 1.e., there
exist v E: N Which are equivalent to the Lebesgue measure on (To,BCTo».
20
ReFla.rk 4.2. t.l'heorem l~.] states tha.t for every \l ( N, a. product
measurable, Gaussinn stochastic process X(t,w), t ( T, induces a Ga.ussian-
Sup(lJ)=m+sp{, } =m+'Jf(S)," \l,n n "
of the range ~(S) of the opera.tor"
lithe closure in H\l
on H\l =L2(T,B(T),v). Let S" be the covariance operator
{.p } its eigenfunctions corresponding to its nonzero",n nThe support of ~ is
"
mea.sure lJv
of lJ and\l
eigenva.lues.
where R{S)v
S. Hence the Hilbert spa.ces (H) and the induced Gaussian measures" "(~ ), as well as the supports (m + R(S », depend on the choice of v inv v ~
N. Some interestill@; questions arise in this connection. First, whether the
equivalence or singularity of the Ga.ussian measures induced by two product
measurable, Gaussian stocha.stic processes depend on v in N. This problem
will not be considered here. Secondly, how does sup(lJ,,), or R(Sv)' depend
on v £ N. Note that X(',w) - m(') (f(s) a.s. for all ,,( N, andv
therefore it would be interesting to know whether there exists a minimal
R(S) fof' some \l (N. Even though an affirmative answer to the latter\l
question does not seem in general plaus i ble, the following remarks can be
made.
(i)
d\li[d Leb](t)
Let "i ( N, i =1,2, be such that vi « Leb. and let
• t. (t).1
~owever, an inclusion relationshipc: R{S )."1
does not seem to hold in general, except ifand
and R(S )"2
R{S )"2
are strictly positive definite operators, in which caseand
between
R(S ) c RCS )."1 "2
(il) It Leb E N, then tor every ,,€ N such that ,,« Leb. and
,Ct) ~ c < +- on To we have 11.eb. cHand RCS ) c RCSLeb).
"
. ; ...21
However, there may exist v € N such that RCS ) c RCSL b)' For" e •
instance consider the Wiener process W(t,w), t € [0,1); then r(t,t). t,
Leb. E N, and if v is defined by [d ~~bJ(t) == 1P , 1 ~ P < 2, thent
" E N and by (i), RCs,,) • Hv ~ HLeb == R(SLeb.)'
(iii) If Leb ~ N then (1) is still applicable. However a measure
«N,
+ +measures "k' k· 1,2, ••• , defined on (R ,S(R » b,y_(t_~)2 ,
equal to 1 on [O,k) and to e 2k on [k,+ao). Then Leb.
" € N corresponding to a minimal R(S) does not seem in general to exist.v
For instance consider the Wiener process W(t,w), t ~
and by (i), R(S ) c; R(S ) for all k. Note thatvk+l ~ "k
€ [0,+-) and hence there is no "E N such that
"k E N, RCS ) =H ,. "k vk
'k(t) k 1 for all t
HE' H for all k." vk
4.3. Gaussian stochastic processes and Gaussian measures on Lp[O,l).
It is shown in Theorem 3.2 that if the stochastic process X(t,w),
.t £ [0,1], satisfies (12) and (III) then it induces a Gaussian measure on
(H2 == L2[O,l],B(H2». Note that (III) implies only integrability and not
square integrability of almost all sample paths of ,X(t,w). Hence the
question arises as to whether a Gaussian measure is induced on
(Hl == Ll[O,l),B(Hl » by X(t,w) if (11) and (III) are satisfied. The
answer to this question is shown to be affirmative in Theorem 4.2~
,This naturally raises the question of inducing a Gaussian measure
on L [0,1], 1 < P < +-. If the stochastic process (n,F,p; X(t,w), t E (O,l])P - 1
is product (8(0,11 x f) measurable and if flx(t,w) IPdt < +ao a.s. toro
1 ~ p < +-, then the map T defined by (1.1) with X. L [0,1] 1s measurable," p
and X(t,w), t E [0,1], induces on (X,S(X» a probability mea.ure lJX
22
(H = L [0,1),P P
defined by (1.2). The followin~ theorem can be proved as Theorem 3.1.
~eorem 4.2. If the stochastic process (o,F,p; X(t,w), t € [0,1})
satisfies (Il ) and (III) for p = 1 or1
(V) E(! Ix(t,w)/p dt)2/p < +-o
for 1 < P < +00, then the probability measure lJX induced on
B(Hp » by X(t,w) is Gaussian.
Theorem. 4.2 answers question ~ for the spaces Lp[O,l], 1 ~ p < +ClD.·
Theorem 3.1 isth~s obtained from Theorem 4.2 for p 18 2. Theorem 4.2
continuous to hold if' the set [0,1] is replaced by a Borel measurable
-set T on the real line.
The· study of question Q2 in the general Lp[O,l] space, 1 ~ P < +-,
appears to be considerably more complicated than in the case p = 2. The
results reported recently in [7] and [10] seem to prOVide the appropriate
structure to approach this question.
5. APPENDIX
We prove the following lemma which is used in the proofs of Theorems
2.1 and 3.2.
If the real random variables {~, n =l,2, ••• } aren
jointly Gaussian and lim E;n ,. f; a. s., where' E; is an a. s. finite random .n......-
variable, then t is Gaussian.
Proof. Since the sequence of random variables t converges a.s.n
to the a.s. finite random variable t, it follows that the sequence of
the characterist~
function of
converges completely to the distributioniu t-.ya2 t 2n n
• e
distribution functions of the f; 'sn
function of . t. Hence it we denote by l' (t)n
function at tn' n = 1,2, •• ., and by ret) the characteristic
t, we have
23
lim fn(t) =f(t) for all t € (-~,+.)n++011
Eq. (5.1) implies that
arg f (1) =u ~ U := arg tel)n n n-++OII 011
\, ..
and that 2-0
If (12") I = e n -:::t It( ~>1 < +011.n n-r-r-
Note that the lim!t
02
---+ +- and.nn-++-
If(i:2) \ is strictly positive; since if It(/:2) 1 - 0 then
{
l for t. 0Ir(t)l- lim If (t)! = , Which contradicts
n.....~ n 0 for t 'It 0
the fact that ret) is continuous in t. Hence
2 20-)0 <+011.n n-++OII 00
It follows from (5.1), (5.2) and (5.3) that
iu t - ~a2t2t{t) • e 011 n . tor all t € ( __,+00)
and thus ~ 1s Gaussian. II,
[1]
[3]
[4 J
[6]
[8l
[9]
[10]
24
REFEREnCES
C. R. Baker, Mutual information for Gaussia.n processes, SIAM J.Apple Ma.th., 19 (1970), 451-q58:
I. M. Gel'fand and A. M. Yaglom, Calculation of the amount of information about a random function, contained in another such function,Amer. Math. Soc. Transl. (2), 12 (1959), 199-246.
P. R. Halmos, "Measure Theory," Van Nostrand, Princeton, N. J., 1950.
K. Ito, The topOlogical support of Gauss measure on Hilbert space,Nagoya M.ath.·J., 38 (1970), 181-183.
G. Kallianpur and H. Ooda11'&, The equivalence and singularity ofGaussian measures, in M. Rosenblatt (ed-.), "Proc. of Symp. on TimeSeries Analysis," 279-291, Wiley, New York, 1962.
K. Karhunen, Uber lineare Methoden in der Wahrscheinlichkeitsrechnung,Ann. Acad; Scient. "Fennicae, Sere AI, No 37 (1947), 1-79.
J. Kuelbs, Gaussian measures on Banach space, J. Funct. Anal.,5 (1970), 354-367.
E. Mourier, Elements aleatoires dens un espace de Banach, Ann. d'Inst. H. Poincare, 13 (1953), 161-244.
~
W. L. Root, Singular Gaussian measures in detection theory, inM. Rosenblatt (ed.), "Proc. of Symp. on Time series Analysis,"292-315, Wiley, New York, 1962.
H. Sato, Gaussian measur~ on Banach space and abstract Wiener measure,Nagoya Math. J., 36 (1969), 65-81.