This research was supported by the Air Force Office of Scientific Research
under Grant AFOSR-68-l4l5 and by the Office of Naval Research under Grant
N00014-67A-032l-0006.
SOME ZERO-ONE LAWS FOR GAUSSIAN PROCESSES
by
Stamatis Cambanis and Balram S. Rajput
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 783
November, 1971
SO}IE ZERO-ONE LAWS FOR
GAUSSIA..~
PROCESSES
I
by Stamatis Cambanis and Balram S. Rajput
University of North Carolina at Chapel Hill
ABSTRACT
A number of interesting zero-one laws ou path properties of
Gaussian processes are derived -by using
a zero-one law for
Gaussian processes which extends a result of G. lCallianpur aDd
H. C. Jain.
"
--
:
1
This research was supported by the Air Force Office of Scientific
Research under Grant AFOSR-68-14lS and by the Office of Naval Research
under Grant NOO014-67A-0321-0006.
-.
1
1.
INTRODUCTION.
A zero-one law for Gaussian processes was recently
proven by G. Kallianpur [6] and extended by N. C. Jain [5]; namely for
a Gaussian process of function space type satisfying certain conditions.
every measurable subgroup of the function space has probability zero
or one.
It is pointed out in Theorem 1 of this paper that the assump-
tions under which this zero-one law is derived can be considerably
relaxed.
A number of interesting applications of this zero-one law
are given in Theorem 2. where it is shown that with probability one or
zero the paths of a Gaussian process have most of the important function
properties.
The proof of the theorems are collected in Section 3.
In Section 4
a simple proof is given for the expression of the Radon-Nikodym derivative of the translate of a Gaussian measure.
This expression, which
is needed in the proof of Theorem I, is known but no 8traightforward
proof seems to be available in the literature.
2.
ZERO-ONE LAWS FOR GAUSSIAN PR.OCESSES.
Let
{t(t,CI),t
a real Gaussian process defined on the probability space
E:
T}
be
(O,F ,P) with
mean function sero and covariance function R(t,s); T is any set and
F is the smallest a-algebra of subsets of n with respect to which the
functions· H: (t,CAI) , t
E:
T}
of . F with respect to ·P.
are measurable.
Denote by
"F the completion
It is the purpose of this paper to derive
classes of "F-measurable events whose P-probability i8 either zero or one.
Now let
alJaoet surely
U(X)
X be a set of real functions
[P]
x on T which contains
the paths of the Gaussian process .t(t,CAI).
the a-algebra of subsets of
Denote by
X gen.rated by sets of the fora
2
{x
E:
X: [x(t1 ), ••• ,x(tn )]
dimensional Borel set).
n}
( T and Bn is an nl
n
Then the transformation +: (n,F,p) + (X,U(X»
(B
(t ,. ~ .. t
defined by
~
is measurable, and induces a probability measure
stochastic process {x(t),t (T}
on
(X,U(X».
The
defined on the probability space
,
(X,U(X)t~)
R(t,s).
aDd
~learly
is
Denote by U(X)
by H(R)
the completion of U(X)
~
with respect to
the reproducing kernel Hilbert space determined by the
covariance R.
(Cl)
Gaussian with mean zero and covariance function
The following conditions will be assumed:
X is a linear space of functions under the usual operation
of addition of functions and multiplication hy real scalars
(C2)
H(R) c X
Both conditions (el)
following CAses:
(2)
(1)
and (C2) are easily seen to be satisfied in the
X • RT , the set of all real functions on T;
T is a measurable set on the real line, t(t,w)
is a measurable
process, and X is the set of all real measurable functions on T;
(3)
T isa compact metric space, the paths of (
are a.s. [P]
con-
tinuous on T, and X. C(T), the set of all real continuous functions
anT.
THEOREM 1.
If conditions (el) and (C2) are satisfied and G is
a U(X)-measurable subgroup of X, then P (G) • 0 or 1.
'This theorem generalizes a zero·one law proven in [6] for U(X)measurable modules over the rationals and for U(X)-meaourable groups
and extended to U (X)-measurable groups in [5].
In [6, S] Theor_ 1 is
proven under the additional aS8umptions.that T i8. complete separable
a.tric space aDd R is a continuous function on T
~
T.
This Assumption
on
R~
which is equivalent to the mean square continuity of
C~
is quite
restrictive, while the assumption on T 1s satisfied in most applications.
For r-mcasurable events we have the folloving
COROLLARY 1:..
Let
tains the paths of
and (e2).
If
F
£
X be a set of real functions on
is such that
measurable subgroup of
which con-
a.s. [Pl, and satisfies conditions (cl)
t(t,,,,)
F
T
F • • -l(G) , where G is a U(X)-
X, then P(F). 0 or 1.
By applying Corollary 1 we obtain a. number of zero-one lava for
path properties of a Gaussian process.
THEORFH 2.
Let' {t (t,,,,) , t e T}
be real separable GeuBoian procells
with zero mean on the probability space
on the real line.
(n,F,p), where T is an interval
Then with probability zero or one the paths of
t(t,tIl)
are
(1)
bounded on T
(2)
free of oscillatory discontinuities on T
(3)
of bounded variation on f!Nery cOllpact subinterval of T
(4)
continuous on T
(5)
uniformly continuous on T
(6)
absolutely continuous on T
(7)
(satisfy) a Lipshitz condition of order a, 0 < a
(8)
differentiable on T.
~
1, on T
The zero-one law on the boundednes8 of the paths is known [7].
AD
important property for Which we have not been able to prove a zero-one
law is the measurability of the paths.
However, under the
add~tional
assUDlption that the process is product measurable, it is shown in (12)
that its Paths belons to
Lp ' 1
~
P < . , with probability zero or one
and also necessary aDd suffieient conditions for the two alternatives
4
are derived.
Sharper results to the effect that with probability one
the paths are either continuous or very irregular (the latter defined
in a precise way) are known for mean square continuous stationary
Gaussian processes [1] and for mean square continuous Gaussian processes
[11].
In the course of the proof of Theorem 2 it Is shown that for a
separable process the sets of paths that have each of the stated properties are measurable, and also explicit expressions for these sets
are given.
This result is of independent interest.
It should be
remarked that for the measurability of the sets in (I), (2), (4) and
(5) separability relative to closed intervals suffices, while for the
measurability of the sets in (3) and (6) to (8) the separability relative
to closed sets is needed.
However, in the particular case of Caussian
processes, the measurability of the sets in (6) to (8) can be shown
assuming only separability relative to closed intervals.
Additional zero-one laws can be obtained by using the techniques
employed in the proof of Theorem 2.
For instance, it is easily seen that
with probability zero or one the paths of a separable, zero mean Gaussian
..
process:
every
ever
€
(i)
>
vanish at •
0 there exists
if
T· [a,.)
(f vanishes at •
0 < N(e) < .... such that
If(t)1
< E:
if for
when-
t > N); (ii) have uniform right (left) limits on T; (i1i) are
uniformly right (left) continuous on T; (tv) are uniformly right (left)
differentiable on T..
It should be noted that we have not been able
to prove the more interesting zero-one laws that are obtained by deleting
/
the word "uniformly" in (i1) to (iv).
The problem of finding necessary and sufficient conditions for the
two alternatives in the zero-one laws of Theorem
2 is wide open at present
,
5
and seemingly a difficult one.
Sufficient conditions for a number of
path properties to hold are known. especially for stationary Gaussian
processes, and some are very
3.
PROOFS.
clo8~
to being also neces88ry.
The notation and assumptions introduced in Section 2
are used here.
P:ROOF OF THEOREM
!.
This theortD is proven in (6, 5) with the
additional assumptions that
and that
R(t,s)
T i. a complete deP8rable metric space
is a continuous function on TXT.
These assumptions
are used in two places in the proof given in (6, 5).
First they are
u~ed
in concluding the v81idity of (i) and (ii) aD
stated in Section 4; i.e., that
p11 - P0 (- p)
and that the Radon-Nikodym derivative of p•
,iven by (1).
if and only if •
~
with respect to p0
H(R)
1s
However, a8 explained in Section 4, (i) and (ii) are
valid in ,eneral with
DO
restrictive assumptions on either T or R.
Secondly, Lemma 6 as stated and proved in (6] employe. the
separability of H(R), which is implied by the above .entioned assUllptiODS on T and R.
If the statement of this Lemma 6 of [6) is
.odified to read as foll0W8~ "If g
such that for every x
~ X
and a
~
is a U(X).....usurable real fuuct:f.on
H(R), S(x
+ .) •
Sex)
then
,(x) • constant a. e. [p)", tben it is proven as Lemma 6 of [6J, the
separability of H(R)
is' not needed, and this modified l81lD8 is
precisely what is needed in the course of the proof of this theorem.
By separability of the stochastic process
{t(t,~),t
c T}; we
.'
mean separability relative to closed sets of the extended real line
[2, p. 52), unless otherwise specified.
Tbia type of separabUity is
equivalent to the following condition:. ''1here ex_Rts a set 00
f:
F
6
with P(Qo>· 0 and a countable dense subset S of T such that for
every
CI)
n - no
E:
and every t
E:
eo
{s}
1 in S converging to t
n~
(2, p. 59).
T - S
there exists a sequence
such that
H:n t(o n .w) • ~(t.w>"
n
This equivalent condition for separability will play an
important role in some of the proofs.
Iknrever, it will be clear from the
following proofs that the weaker kind of separe.bility relative to closed
inteTValq of the extended real line is sufficient for scme of the zeroone laws.
PROOF OF
TH:tQ!~.
If
is the set of
'I
property is catisfisd we will show that. F
Proof 'of
Q.l•. Let
F· {w
E:
E:
CI)
r
n for which each
E:
and P(P'). 0 or 1.
n: Buplt(t,w) I <
-te}, and
uT
F' • {w
•
Then
F'.
€
.
€
Q: supl~(t,w)1 < +-}
uS
. '
u
n {w
N-I tE:S
E: Q:
It(t,w)1 < N}.
-
F, and because of the l!Iep~rability of t
t
P
€
F
and
P(F) • PCP'). Also if we take X. aT and define
G' • {x c X: sup Ix(t)1 < +-}
tcS
•
then G'
•
u
n {x
N-l tcS
E:
X:
Ix(t) I -<
N}
is clearly a U(X)-measurable subgroup of X and.,' ••-I(G').
Corollary 1 implies that P(7')· 0 or 1.
Note that separability
relative to closed intervals only is used here.
Proof of (2).
Let p . {w
discontinuities on T}.
E:
Q:
t(t,w)
It is sbown in [9, p. 61] that
Similarly, if X· aT and G· {x·E: X: x(t)
discontinuities on T}, tben G E: U(X).
and
is free of oscillatory
Pc F.
is free of oscillatory
Since G 18 a subgroup of X
r ••-l(G), Corollary 1 implies. P(F) • 0 or 1.
7
Proof of (3).
intervals such that
t(t,w)
Tk
•
Tk+l c: T and
C
k~lT
k .. T.
If
F - {w
0:
€
T},
is of bounded variation on every compact subinterval of
then
•
•
•
k~l N~l n~l nn(t~;Tk){w
F -
where
nn(ti;T )
k
o
S
< ••• < t
l
Let
P'
{bk}k_l)'
P' c F
Then
p'.
and
•
{w c
n:
t(t,w)
variation on every bounded subset of
S},
i8 shown as in the Proof of (4) that
r
take
X: x(t)
X· aT
n
{ti}i_O
in
T
k
be liven by the same
i8 augmented so as to include all end points
•
o
• b ,
k
n
.
~ N} ,
P with Tk replaced by Tk n S (here the separating
expression as
set
n:
€
is the class of all sets of points
ak - to < t
such that
n
i:l1t(ti,w) - t(ti_l,w)I
we have that
G'. - {x
on every bounded subset of
5}
E:
U(X)
€
•
{~}k.1
and
is of bounded
By the separability of
c rand
per) -
P(F').
t
it
If ve
is of bounded variation
and since
G'
is a group and
F' • .-lCG'), Corollary 1 implies P(l')· 0 or 1.
o
Proof of (4).
such that
Tk
C
Let
•
{Tk}k-l
Tk+l c: T and
is .continucus on
T
•
be a sequence of cOilpact intervals
~lT
k - T.
Then, noting that a function
if and only if it is Uniformly continuous on every
T , we have
t
p.' {fit € Q: t(t,w)
•
is continuous on T}· n lk
pI
where
P .-{w c Q:t(t,w)
, k
co
-; n
is continuous on
•
u (w
n-l mal
€
0:
I
•
Ttl
sup
It(t,w) - tCt' ,w)I,.'!.!}
t.t'«Ttc.
n
t-t' I~llm
•
•
n u n {w ( 0:
u-l .-1 t€Tk
•
n u n E . (T ),
0-1 .-1 tcTk n,_ k
••
t
8
Let
P' •
Fi
€
.-1
s
CD.
Fi·
n
F' • {~ ( n: t(t,w)
k-l k
F for all k, and F'
U(X)-measurable.
Then clearly
T
F.
E:
If we take X - R
{x € X: x(t)
G'
Since
Corollary 1 implies
S}.
Is continuous on
G'·
similar way that the set
Is
Sk· Tk n S, and let
nun E
(St)' where
n-l
S€Sk n,.
we obtaln in a
F'..-1 (G'),
is clearly a group and
P (F') - 0
or
1.
Clearly
'k - Fk COo'
shown that separability implies
S}
Is continuous on
F
C
k
Then
'k:
F
k
It will be
E:
F
and
P(Fk ) • P(Pk) and thus .p £ 1 aDd P(P). PCP') • 0 or 1.
In order to prove Pi - Fk cOo' it suffices to 8how that
IlJ £
Fi - 00
w£
n, since
flYery
s
implies
Fi'
Sk' w
E:
S,8'
€
•
Is - 8' I
'
k - 00'
F
n)
(depending on
~:-
For f!Very fbad
8uch that
It(s,faa) - t(s' ,w)1
then
be arbltrary but fixed.
and the fact that
£
for
E~n, m(Sk); i. e. such that whenever
E:
Sk
w
Fix any
there exist8
t ( T - Sk
k
. Let
Fk •
c.I E:
~ 00
IlJ
~ 2~
•
It follows by separability
that . there exists a sequence' {8 }i 'in
i
Sk
such that
lim
1
8
1
- t
Then there exists integer I
J t - s1 1
If
8 (Sk
,
.
Is - s1 1 ~
i.e., ~
t c T
f:
8
81
'8 -
~~
t
f:
the proce88
, then, for
-
It -
i
~
~
It(ei'~)
I
2i
then
c
T}
and
- t(s,w)1
~ ~+
2; -; ·
'1
It (t,w) - t (8,~) ~;,
t
t
En ,2m(Tk ) c En ,2m(Sk)'
(t(t' ,~) - t(t,III),t'
then
I,
i
sl ~
I
~~ •
8I ~ "iii1 + 2m1 . ;1
- t(sl,w)1 +
Clearly
i
It (t,~) - t (si ,~) I
I + It
.
Sk with
t
En ,2m(Sk)'
.. t (t,~)
such that whenever
and
~ It(t,~)
It(t,w) - t(e,w)1
Thu8 whenever
~~
It -
with
lim t (sl ,~l
1
and
I
Since for fixed
is separable, it followQ
9
Hence
CIJ
~ BD, 2m(Sk)' w ~ n0
t
imply
t
En ,2m(Tk). It is thus seen that for every n there exists m
(depending on n) such that for all t € T, CIJ € E:,2m(T ). Hence
k
....
t
CIJ €
nun E
(T ) · F
and the proof 1s complete.
k
n-l .-1 t€T n,m k
CIJ €
This proof is presented in some detail because it is typical of
the use of the aeparsbi1ity of the process in provins measurability of
path sets and because a stmilar argument is used again in this paper.
It should be remarked however that this proof uses separability relative
to closed sets.
As
pointed out in the Proof of (5), this zero-one law
on path continuity can be proven under the weaker kind of separability
relative to closed intervals.
Proof of (5).
T}.
F -' {w ~ Q: t(t,w)
Let
It follows fr~ the proof of (4), that
. This proof uses separability of t
is uniformly continuous on
F~
F
and P(F). 0 or 1•
relative to closed sets.
We now
give" proof emp10yiaa the weaker kind of separability relative to closed
intervals.
It should be clear then, that uDder the same weaker separa-
k appearing in the Proof of (4) can be
s1Jllilarly shown to belong to f', which proves the zero-one law (4). Let
bUity assUliptiona, the sets
F
r' .' {w c n: t(t.w) is uniformly continuous on
•
•
n
u
n-lm-l
F and F c ·P·.
f:
s,s'e:S
{w e:
n:
It (s,w)
-
t (s' .w) I :i.~}
rs-s' 1~1/1ll
.~
Then' F
•
n
S}
Fix any w e: F' - no • Then t ( .•wJ
is
uniformly continuous on S, . and since S is dense in T. it can
be
/
uniquely extended to a uniformly continuous function
Now let
t
E:
T - S be arbitrary but fixed.
Then
i (. ,w)
on T.
10
t(t,w).
t
and the separability of
lim t(s,w) • t(t,w)
lim t(s,w) • lim t(a,w)
SE,S
SE,S
s-+t
s-+t
relative to closed intervals implies that
Hence t(t,w)· i(t,ro), i.e., t(t,~)
[8, p. 505].
SE:S
S+t
is uniformly continuous and w E: V.
impli,es V
1-" and P(F) • p (V' ) •
€
similar way that the set G'.' {x
S}
is U(X)-measurable.
€
F' - no c V which
If we take X· RT we obtain in a .
It follows that
X: x(t)
Since C'
is uniformly continuous on
is clearly a group and V' • • -I(C'),
Corollary 1 implies P(F')· 0 or 1 which complete. the proof.
Proof of (6).
V •
{w
Let
€
..
n:
t(t,w)
•
absolutely continuous on T}
•
• n~l m~l k~l n:(ti~t1;T){w
where n:(ti,tiiT)
€
n:
Let also V'
t(t,tI)
n
be the subset of
F
€
and clearly
f
€
is absolutely continuous on S}.
This proves that F
Vor
{ti,ti}~.l
given by the same
Then F'
of t, it is shown as in the Proof of (4) that
to closed sets.
1
{(ti,ti)}~.l are disjoint and with total
expression as F with T replaced by S.
V' .' {w
k
i:l1t(ti,w) - t(ti,w)1 ~ nl
n:
is the clas. of all sets of points
in T such that the intervals
length ~;.
€
By the separability
f
and P(F). P(F').
f for every process t separable relative
€
a Gaussian process t, by using the already
established zero-one law (4), it can be shown that F
f
€
under the
weaker condition of separability relative to closed intervals.
applies also to (7) and (8».
Indeed if P{w
€
n:
t(t,(l)
on T} • 0 then F belongs to the completed a-field
On the other hand if P{tI)
clearly t
€
n:
t(t,w)
F,
(This
is continuous
and P(V). O.
is continuous on T}. 1, then
i8 separable relative to closed ssts.
Thus, since the
11
zero-one law on the continuity of paths is proven under the hypothesis
of separability relative to closed interval, the zero-one law of (6)
is proven under the same hypotheeis.
If
we take
T
X· R we have that
continuous on S} (U(X)
Gf
and since Gf
•
{x e X: x(t)
is ahsolutely
F' ••-1(G'),
is a group and
Corollary 1 implies P(F'). 0 or 1.
Proof of (7).
This zero-one law 1s shown as (4) and (6) by
noting that if F·' {~ «
order a
n:
on, T}, for fixed
~
•
0,
0
{~ ~ n:
n
F -
1, then
This zero-one law is shown as (4) by noting that
{w c
CD
-
< 0 ~
sup Ji(t,w) - ~(s,w)L < 5}.
tt's€T
It _ sla
-
N-l t€T
Proof of (8l.
satisfies a Lipshitz condition of
sup J~(t,C&l) - t(st..~lL < +oo}
t,s~T
It - s/a
tt's
n:
F • {~ €
t(t,~)
n:
t(t,w)
is differentiable on T}
CD
n
u
n-l .-1
{w
n
€
1
g: max {A ,6 ,6 } ~ n}
l
s, t,TCT
2 3
S<t<T
16-T1~l/m
where 6 1 A2 -
I
t~s,w) - t(t,~) _ (8,ro) - (T,ro)
s - t
s -
t
'
,t(\'lw) - (t.wL_ (T,W) - (S.t!!!lJ
T -
4.
MEASURE.
t
T -
S
'
63
-
I'(t
aw)
- (S.llI) _ '(t,w~ -
t -
S
-
(<t,w)l. :,'
T
I,
THE RADON-NIKODYM DERIVATIVE OF THE TRANSLATE OF A GAUSSIAN
In this section we discuss the properties of the translates of
Gaussian measures used in. the proof of n;eorem 1.
The notation and assumptions of Section 2 prior to Theorem 1 are
adopted here without further explanation•. Let us put Po • P and
. ..
"
12
L
the subspace of L (po)· L (X,U(X),p o )
2
2
2
spanned by the random variables {x(t),t ~ T}. It is well known [lOJ
Po • p
and denote by
that there 1s an inner product preserving isomorphism between
H(R). denoted by""", such that
x(t) ...... R(',t)
for all
L
2
and
T, and
t £
n ...... m 1f and only if
met) • !x(t)n(x)dp (x)
X
0
For m € X the transformation
for all
T :
11
t ~ T.
(X,U(X),p )
0
+
(X,U(X»
defined
by T.(X). x +. is clearly measurable and induces a probability
measure Pm OD (X,U(X». The stochastic process {x(t),t € T} on
the probability space (X,U(X),P m) is easily seen to be Gaussian with
mean function met)
and covariance function
R(t,s).
It is also clear
that P is the probability measure induced on (X,U(X»
by the stochastic
.
m
process . {t(t,~) + m(t),t £ T} by means of the measnrable transformation
+m: (O,F,Po ) ... (X,U(X»
defined by
+mCca!)
• t (. ,fA») + m(.).
It is known
that
(i)
if m £
p and p
m
0
H(R), and
(i1)
if m £ R(R). the Radon-Nikodym derivative Pm of Pm with
respect to Po
is given by
Pm(x) • exp[n(x) -
(1)
where n ......
are mutually absolutely continuous if and only
11·11
a.e. [po]
L2 (p o ).
The validity of (i) is established in [4]; it can also be obtained as a
11
and
~i.lnI12]
denotes the norm of
straightforward application of the theorem proven in (3).
of (ii) is indicated in [13].
The validity
However, since the proof of (ii) is not
easily reconstructed from the references cited in [13]. an alternative
e·
simple proof is given here.
13
For any subset
S of T let
spanned by the set . {x(t),t
€
be the 8ubspace of L (po)
2
S}, let H(R,S) be the reproducing
L (S)
2
kernel Hilbert space of R restricted to S x S, and let U(X,S)
the a-algebra of subsets of
{x(t),t (S}
be
X with respect to which the functions
Clearly L (T). L2, H(R,T) • H(R) and
2
It is well known [10] that there ia an inner product
are measurable.
U(X,T) - U(X).
preoerving isomorphism between L2 (S)
that x(t) ++ RC· ,t), t
€
f(t) • !x(t)t(x)dp (x)
for all
X
S, aDd t
and
++ f
H(R, S), denoted by
++, such
if and only if
t« S.
0
Assume now m«
Then pm(x)
H (R) •
is U (X,T)-measurable and thus
there exists a cOUDteble subset S1· of T such that
measu·rable [2, p. 604].
Let
II E:
L2 be such that
Pm (x)
"++ m.
is U(X,SI)Than "
is
•
the L2 (llo)-limit of a sequence {"k}k-l
of elements of L2 (po) of the
Nk
form "k(x) - n-llt,n
t c:. x (t k ,n), where the c:.It,n ' 8 are real numbers and
and pm(x)
is U(X,S')-..measurable.
S be the set of those points
tit of
S'
which are such that the elements
x(tl), ••• ,x(t ) of L (P ) are linearly independent. Then S is a
2 o
k
countable subset of T, L (S). L2(S') and U(X,S)· U(X,S'), where
2
U(X,S)
that'
is the completion of U(X,S)
II C L (S)
2
and that Pm(x)
is
with respect to llo.
It follows
U(X,S)-measurable, which implies
that pm
(x) is also the Radon-Nikodym
derivative of 'p
restricted to
m
U(X,S)
with respect to p
restrieted to 11(X,S). Now an application
(
o
of the martinga:t.e convergence theorem [2) to the martingale
. (p:(x), U(X,Sn),l ~ n ~ .}, where
S• • S, .'·<at>;.l' p:~x) • p.(x),
14
1 -< n < ~ ' n
S • fs}u
k k.-1
and for
derivative of
to
U(X,S)
n
urn
and
1s the Radon-Nikodvm
,,-
Il(x, S)
'Jith
n
to
re~tricted
pn(x)
m .
rE"'spect
to
1.1
0
restricted
(or equivalently an appHcatton of Theorem 9.A of [10] to
{x(t),t
the stochastic process
5})
€
glves
(2)
r.
where
t
++
m
S
~ t?(S)
and
t
++
m ; m
S
S
being the restriction of
implies
met) • Jx(t)~(x)dV (x)
X
On the other hand, it follows from
L?!
....
met) • !x(t)n(x)dp (x)
X
It follows now from
(2) implies (l).
n, t
for all
t
E:
S.
0
0
~
L (S)
2
that
~ ~
m
for all
~
H(R)
t
(.
that
T
m to
S.
15
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[1]
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Continuity and Holder'. conditions for
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-
Fourth Berkeley Symp. &th.·Statist. Probe 2
Univ. California Pre•••
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23-33.
(2)
DOOiS, J. L. (1953).
Stochastic ProcHe88.
(3)
P'£WKAN, J. (1958).
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{ 4]
BA.JEK, J. (1959) •
Tr£M.
[5]
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a 8111p18 linear model in Gau.sian proee......
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(9)
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The ..,le fUDCtion rep1arity
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1Z
oi linear
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[12J
IAJPUT. B. S. (1971).
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