*
This research was supported by the Air Force Office of Scientific
Research under Grant AFOSR-68-1415.
THE MEASURABILITY OF A STOCHASTIC PROCESS
OF SECOND ORDER AND ITS LINEAR SPACE
Stamatis Cambanis*
Department of Statistics
University of North Carolina at Chapel HiZl
Institute of Statistics Mimeo Series No. 832
June, 1972
THE MEASURABILITY OF A STOCHASTIC PROCESS
OF SECOND ORDER AND ITS LINEAR SPACE
Stamatis Cambanis*
1.
THE MEASURABILITY OF A STOCHASTIC
PROCESS OF SECOND ORDER
Let T be a separable metric space and
B(~)
the a-algebra of Borel
sets of T, and let X , t€:T, be a real stochastic process on the probabit
lity space (nfI?).
X ,t6,T, is called measurable if the map (t,w)+Xt(w).
t
is B(T)xF-measurable. A process Y , t~T, on (n,F,p) is called a modification
t
of Xt,t€T, if P{Xt=Yt}=l for all t in T. Xt,t€T, is of second order if
E(X~)<+= for all t in T, and then its autocorrelation R is defined by
~
R(t,s)aE(XtXs) for all t,s in T.
It is clear from Fubini's theorem that
if a second order process Xt,t€T, has a measurable modification then R
is B(T)xB(T)-measurable.
That the measurability of R is not sufficient
for the existence of a measurable modification of Xt,t€T, is demonstrated
in Remark 2.
It is thus of interest to find a condition which along with
th~measurab1lity of
cation of Xt,t€T.
R would imply the existence of a measurable modifi-
This question is answered in Theorem 1, where in fact
necessary and sufficient conditions are given for a second order process
to have a measurable modification.
A remarkable consequence of these
conditions is that the existence of a measurable modification of a second
order process is a second order property.
*This research was supported by the Air Force Office of Scientific Research
under Grant AFOSR-68-l4l5.
2
The proof of Theorem 1 is based on the necessary and sufficient
conditions for a process (not necessarily of second order) to have a
measurable modification given in [5], which are expressed as follows
(here the terminology of (6) is followed).
Let M be the space of all
real random variables on (O,F,p) with the topology of convergence in
probability, where random variables that are equal a.e. [P) are considered identical.
denoted by [t].
Then
if the map from T to
range [5,6].
If t is a real random variable, its class in M is
Xt,t~T,
M
has a measurable modification if and only
taking t to [X ] is measurable and has separable
t
Moreover, thei measurable modification can be taken to be
separable and also progressively measurable, the latter if T is an interval and a nondecreasing family
For a second order process
~
Ft,t~T,
of sub-a-a1gebras of F is given.
Xt,t~T,
we deri0te by H(X) the closure
in L2 (O,F,P) of the linear space of the random variables {Xt,t~T} and
we call it the linear space of the process.
We also de11.ote by R(K) the
reproducing kernel Hilbert space of a real, symmetric, nonnegative
definite function K on TxT.
It is well known that R(R) consists of all
functions f on T of the form f(t)aE(;X ),
t
t~T,
for. some tEH(X), and that
the map t+E(tXt) defines an inner product preserving isomorphism between
H(X) and R(R) [16, p.302].
THEOREM 1.
correlation R..
Let Xt,tET, be a real, second order process with autoThe following are equivalent.
(i) Xt,tET, has a measurable modification.
(ii) R is B(T)xB(T)-measurable and H(X) (or R(R»
PROOF. (a) We first show that (ii) implies (i).
~
is separable.
It suffices to
verify the conditions of [5,6]; the construction of a measurable modi-
3
·e
fication is the same as in [5] or in [6].
Since convergence in L (O,F,P) implies convergence in probability,
2
the separability of H(X) as a subset of L (0, F,P) implies its separability
2
Thus its subset {EXt]' t€T} is separable in M.
as a subset of M.
complete the proof it suffices to show that the map X:
..
X(t)=[Xtl is measurable. 'Ihe.m.ttic
~,n€M,
fIil~on U
T
+
M defined by
p(~,n)·E
defined by
metrizes the topology of convergence in probability.
To·
( l+I~-nl
I~-nl) '
Thus for the
measurability of X it suffices to show that X-l(B)€F for every set B in
M of the form B={Y€M:
{t€T: p([X ], YO)
t
p
([Xtl,
Yo)
.
~
p(Y'YO)~r},
where YO€M and r
> O~
-1
Since X (B)=
r}, it suffices to prove that the real function
on T is B(T)-measurable for all YO€M.
co
Let {~n}n=l be a complete orthonormal sequence in H(X) (which exists
because H(X) is separable).
Then for all t€T we have
co
X
0:
t
L
a ('t)~
n=1 n
n
in L2 (O,F,P), where a (t)
E(~ X).
Thus a € R(R), and in fact
n
n t
n
co
{a}
n no:1 is a complete orthonormal sequence in R(R). If for every t€T
0:
we let
X(~)
...
rn~l an('t)~n' then x(:)converges to
thus in probability.
Let y
t
D
X
t
in L2 (O,F,P) and
X - y.o and y(N)o: X(N)_ YO for all tfT.
t
U
t
t
Then y(~) converges to Yt in probability, i.e., p([y(:)], [Yt)
N+
co.
+ 0
Dropping the index t for simplicity we have
ly(N) 1:_
IYI
l+ly(N) I· l+IYI
'"
~llilL- s
(l+ly(N)I)(l+IYI)
Thus
IYI
l+IYI
ly(N)_YI.
l+ly(N)-YI
as
4
·e
It follows that for all te:T,
Note that every
~ction
in R(R) is either a finite linear combination of
the functions {R(t,.}, te:T} or a pointwise limit on T of such functions.
Hence, since R is B(T}xB(T) - measurable, R(t,.} is B(T} - measurable for
all te:T, and it follows that every f in R(R} is B(T) - measurable.
sequently y(:) (til) is B(T}xF - measurable.
E
r Ix(N)_y I 'j
l (i> 0
Con-
By Fub111i's theorem
is B(T) - measurable, and thus so is p([X ], YO}' which
t
l+IX t ~!Ol.
completes the proof.
(b)
We now show that (i) implies (ii).
follows from Fubini' s theorem and (i).
The measurability of R
In order to prove the separabi-
lity of H(X} we first assume that R is uniformly bounded on T:
R(t,t}
~
foralltinT.
C<+oo
We will show that this implies the uniform integrability of the family of
random variables {X , te:T}.
t
I
IX I>a
Indeed we ha.ve for all a>o,
IXtl dP
=
In
I{IXtl>a} IXtl dP
u
t
~
~
[P{IXtl>a}.
R(t,t}]~
R(t 2t)
a
Thus
lim
a-+-
oo
and {X ' te:T} is uniformly integrable.
t
o
5
.e
Now (i) implies that {[X ], teT} is separable in M.
t
Thus there
exists a countable subset M' of {[X ], teT} such that for every t in T,
t
[X ] is the limit in probability of a sequence in
t
L (n,F,p), since M' is uniformly integrable
2
M',
and hence also in
[14, p. 57].
H(X) equals the L (O,F,P,) closure of the linear span
2
It follows that
of M' and, since
M' is countable, H(X) is separable.
We now consider the general case and define for N
TN
=
{teT: R(t,t)
S
=1,
2, ••• ,
N}.
Since R is measurable, TNeB(T) and by (i) tXt' teT } has a measurable
N
modification.
It follows by what has been proven that the L (O,F,P)
2
closure of the linear span
e
is separable.
TNtT.
of the random variables {X ' teTN}, HN(X),
t
Since X is of second order, R is finite valued and thus
t
It follows that H(X) is the L (O,F,P) closure of UN: l HN(X) and
2
thus H(X) is separable.
0
Thus a B(T)xB(T) - measurable, symmetric, nonnegative definite,
real fUnction R on TxT is the autocorrelation of a measurable process
if and only if R(R) is separable.
REMARK 1.
The mean m and the covariance C of a real second order
process X ' teT, are defined by met)
t
[X -m(s)]) for all t, s, in T.
s
= E(Xt )
Then R(t,s)
and c(t,s)
= m(t)m{s)
= E([Xt-m(t)]
+ C(t,s).
In
connection with (ii) of Theorem 1 it should be noted that
R is B(T)xB(T) - measurable if and only if m is B(T)- measurable
and C is B(T)xB(T) - measurable.
e
The "il''' part is obvious.
The "only 1f" part is shown as folloY1s.
We
6
·e
have m(t)
= E(Xtl n)
for all t in T, where I is the indicator function.
the projection of I €L (n,F,p) onto the subspace H(X).
n 2
Then met) = E(Xt~) for all t in T and ~€H(X), and thus m€R(R). Since
Denote by
~
R is B(T)xB(T) - measurable, m is B(T) - measurable (see part (a) of
the proof of Theorem 1) and C(t,s)
= R(t,s)
- m(t)m(s) is B(T)xB{T) -
measurable.
RE}~RK
for t
~
2.
= [0,1]
Let T
s in T.
=1
and R(t,s)
for t
=s
in T and R(t,s)
Since R is symmetric and nonnegative definite, there
exists a probability space (n,F,p) and a real process X , t€T, on it
t
with autocorreLatfun R. R is clearly B(T)xB(T) - measurable, but since
the values of X are orthogonal in L (n,F,p), E(XtX ) = 0 for t ~ s in
2
t
s
T, H(X) is not separable and by Theorem 1, Xt,t€T, does not have a
measurable modification.
~
1.
This can be also shown without using Theorem
Indeed, assume that Xt,t€T, has a measurable modification Yt,t€T.
Then
= II
R(t,t)dt
=
1 <+00
o
implies that
Ily~
set in L (T)
= L2 (T',
dt <+00 a.e.[Pl.
If
{
o
2
in L2 (T) a.e. [P], where
E(~~)
~n = 0 a.~.
=
2
n=l
~n = Jly t $n (t)
00
is a complete orthonorma 1
~n $n(t)
dt a.e. [Pl.
Then
o
=
J1Jl R(t,s) $n(t)
o
0
(Pl. and thus
y~ dt) = 1.
o
measurable modification.
contradicts E<J
}
n n=l
B(T),' Leb) then
Yt
i.e••
$
It
~n(s)
dt ds
J\~ dt - Ln:! ~ = 0 a.e.
f~llows
=
0
(Pl which
that Xt,t€T, does not have a
=0
7
REl1ARK 3.
·e
For Gaussian processes it can be easily shown that (ii)
implies (i) without relying on the results of [5]; this is done in [15,
p. 44].
COROLLARY 1.
function on TxT.
Let R be a symmetric, nonnegative definite, real
If R(R) is separable the following are equivalent.
(i) R(t,.) is B(T) - measurable for all t in T.
(ii) R is 8(T)xB(T) - measurable.
PROOF.
It suffices to show that (i) implies (ii).
Since R symmetric,
nonnegative definite and real, there exists a probability space (n,F,p)
and a real process Xt,teT, on it with autocorrelation R.
It is clear
from part (a) of the proof of Theorem 1 that the separability of R(R) and
(i) imply the existence of a measurable modification of Xt,tET, and thus
(ii).
tit
This result can be shown in a simpler way without using an associated
process.
00
Indeed, i f {an}n=l is a complete orthonormal set in R(R), then
it is easily seen that R(t,s) = Ln:l an(t) a (5) for all t,s in T. Now
n
(i) implies as in part (a) of the proof of Theorem 1 that every an is
B(T) - measurable and thus (ii) holds. 0
COROLLARY 2.
A second order process X , teT, which satisfies any
t
of the following conditions has a measurable modification (in(iii) also
progressively measurable).
(i) X ' tET, is weakly continuous on T.
t
(ii) T is an arbitrary interval and X ' tET, has orthogonal increments.
t
(iii) T is an abitrary interval and X ' teT, is a martingale.
t
PROOF. (i) Since T is separable and X weakly continuous on T,
t
H(X) is separable [16, p. 272].
By the weak continuity of Xt ' R(t,.)
8
is continuous, hence B(T) - measurable, for all t in T.
.~
The conclusion
follows from Corollary 1 and Theorem 1.
(ii) It is knoliO that H(X) is separable [8, p. 110].
Also, that
Xt has left and right L (n,F,p) limits on T and that except on a countable
2
subset of T, X
t-
= Xt = Xt +'
This implies the measurability of R and the
result follows from Theorem 1.
(iii) Define the function F by F(t)
= E(X 2t )
for all t in T.
By the
martingale property, with respect to the nondecreasing family F , tET,
t
of sub-a-algebras of F , we have for all sSt in T, E(X X ) = E[E(X X IF )]
t s
t s s
2
E[Xs E(Xt/F)]
= E(Xs ) and thus
s
E({X -X }2,
t
s
=
F(t)-F(s).
It follows from this relationship, as in [8, p. 110] and in (ii), that
H(X) is separable and R is B(T)xB(T} - measurable. 0
REMARK 4.
Let X ' tET, T an arbitrary interval, be a real separable
t
process of second order with autocorxelation R. If X is mean square
t
aR(t s) a 2R(t,S)
differentiable on T anda~'
etas
are locally Lebesgue integrable
in t and in t,s respectively, then with probability one the paths of
X ' tET, are absolutely continuous on every compact subinterval of T.
t
This is shown in [10, pp. 186-187] with the additional assumption that
,
the mean square derivative X of X has a measurable modification, which
t
t
is always satisfied because of Theorem 1. Indeed, since Xt is mean
square differentiable on T, it is mean square continuous on T.
Thus
H(X) is separable and the continuity of R implies the measurability of
2
2
a R(t,S)
Since a R(t,S)
i s t h e autocorr~
~~i on 0 f X't and s i nce
atas
atas
H(X') S H(X} , the conclusion follows from Theorem 1.
=
9
We conclude this section with a property which is useful in con-
.~
nection with problems involving conditional probabilities; such as for
instance the existence of jointly measurable densities (see [9, pp. 616-617])
and properties related to metric transitivity (see [17, Ch. IV. 8]).
A a-algebra is called separable if it is generated by a countable class
of sets.
A sub-a-algebra F' of F is said to coincide mod
o with
the
a-algebra F if for every set E in F there is a set E' in F' such that
P(E6E')
= O.
Let F(X) be the sub-a-algebra of F generated by the random
variables {X ' t€T}. It is kno~m that if X is continuous in probability
t
t
on T, F(X) coincides mod 0 with a separable a-algebra. Corollary 3
generalizes this result (and in fact, as it is clear from [6], it is
valid for any process with values in a compact metric space).
e
COROLLARY 3.
If a real process X , t€T, has a measurable modifit
cation, then F(X) coincides mod 0 with a separable a-algebra.
PROOF.
Since X ' t€T, has a measurable modification, {[X t ], t€T}
t
is a separable subset of M.
M'
Thus there exists a countable subset
{[Xtl, t€S} of {[X ], t€T} (S is a countable subset of T) such that
t
for every t in T, [X ] is the limit in probability of a sequence from
t
a
M', and thus X is the a.e.[P] limit of a sequence from {X ' t€S}. If
t
t
F' is the sub-a-algebra of F generated by the random variables {X , t€S},
t
then F' S F(X), F' 18 separable and F(X) coincides with F' mod O. 0
2.
ON THE SEPARABILITY OF THE LINEAR SPACE
OF A SECOND ORDER PROCESS
The linear space H(X) of a second order process Xt ' t€T, plays an
important role in the structure of such processes and in a variety of
problems in statistical inference.
If H(X) is separable then X admits
t
10
series representations and also integral representations (Theorem 2)
that can be effectively used in problems such as linear mean square estimation.
Also the separability of H(X) is the only condition needed for
a second order process to have the
instance [11]).
Hida-Cram~r
representation (see for
It is thus of interest that a measurable second order
process has a separable linear space.
H(X) is known to be separable when
the process X , teT, is weakly continuous [16, p. 272], has orthogonal
t
increments [8, p. 110], or is a martingale (Corollary 2.(iii».
In
Theorem 2 necessary and sufficient conditions are given for H(X) to be
separable in terms of integral representations of X •
t
Before stating the theorem we mention a few basic facts about random
measures, that can be found for instance in [7, 16].
measurable space.
~
Let (V,V) be a
A random measure Z on (V,V) is a countably additive
map from V to L2 (n,F,p); i.e., whenever A is the disjoint union of the
= In:l
Z(An ) in L (n,F, P). (Here we consider the case
2
where Z is defined on the entire a-algebra V). To each random measure Z
sets AneV, Z(A)
on
V there corresponds a finite signed measure
E[Z(A)Z(B)], A,BeV.
~
~(AXB)
=0
on
VxV by
~(AxB)
=
is symmetric and nonnegative definite on the
measurable ractangles of
if
~
VxV.
A random measure Z is called orthogonal
whenever A and B are disjoint, and to each orthogonal
random measure there corresponds a finite nonnegative measure v on V
by v(A)
= E[Z 2 (A)],
AeV.
Let H(Z) be the closure in L (n,F,p) of the
2
linear span' of {Z(A), AeV}, and let
h2(~)
be the Hilbert space of real,
V-measurable functions on V with inner product <f,g>A2(~)
=f f
f(u)g(v)d~(u,v)
(of course A (~) consists of equivalellce classes. of functions1 A
Z
(~)
two functions f and g considered identical i f <f-g,f-g> A
.
.= 0) •
Z
There is an inner product preserving isomorphism between
A2(~)
and H(Z),
....
11
denoted by ++, such that I
'e
A
++ Z(A), A€V, and integration of functions
in A2 (ll) with respect to Z is defined by I;
=
I
f(u)dZ(u), where f++l;.
V
If Z is orthogonal, there is an inner product preserving isomorphism
between L2 (v)
= L2 (V,V,v)
and H(Z}, denoted again by
++,
such that
I A ++ Z(A), A€V, and integration of functions in L (v) with respect
2
to Z is defined by I; = ff(U)dZ(U), where f ++ 1;.
V
THEOREM 2.
Let Xt , t€T be a second order process.
(i) If H(X) is separable then for every finite measure space
(V,V,v) such that L2 (v) = L (V,V,v) is separable and infinite dimen2
sional, X has a representation
t
=
Xt
Iv f(t,u) dZ(u)
for all t in T
where Z is an orthogonal measure on V with corresponding measure v
~
and f(t,.) € L2 (v) for all t in T.
sentation, H(X) is separable.
Conversely, if X has such a repret
(ii) If H(X) is separable, then for every measurable space (V,V)
and every finite signed measure
II
on VxV which is symmetric and non-
negative difinite on the measurable rectangles of VxV, and such that
A (ll) is separable and infinite dimensional, X has a representation
t
2
=
Xt
fvf(t,U) dZ(u)
for all t in T
where Z is a random measure on V with corresponding measure
f(t,.)€A (ll) for all
2
H(X) is separable.
PROOF.
e
(ii).
t
in T.
II
and
Conversely, if Xt has such a representation,
(i) being a particular case of (ii), we will prove only
We start with the second claim.
If Xt has such a representation
then Xt€H(Z) for all t in T, hence H(X)~H(Z) and the conclusion follows
12
'e
A2(~)
from the isomorphism between H(Z) and
the latter.
We now prove the first claim.
<Xl
{~n}n=l
separable and let
and the separability of
Assume that H(X) is
be a complete orthonormal set.
Then for
all t in T
<Xl
I
=
a (tH
n= I
in L2 (n.F.p), where
orthonormal set in
nn
co
= E(Xt ~).
n
a (t)
n
A2(~).
~
Since
Let {f} I be a complete
n n=
is finite, lA€A2(~) for all A€V.
Then
lA
III
L A (A)f
n=l n
n
in A (lJ), where
2
= <IA,fn > ... JAJvfn(V)d~(U'V).
A (A)
n
Throughout the proof we will write <.,.> for <.">A (lJ). Thus for
2
all n, An is a finite signed measure on (V,V). We also have
L
n=l
...
...
2
A (A)
n
Hence
<Xl
Z(A)
defines a function from
= L
A (A)F,;
n=l n
n
V to L2 (n,F.p) (the convergence being in L2 (n,F,p».
We will show that Z is a random measure with corresponding measure
The latter is clear since for all A,B€V we have
<Xl
E[Z(A)Z(B)]
= L
nal
A (A)A (B)
n
n
...
a
ll(AxB).
~.
13
disjoint sequence of sets in
V.
Then
K
co
{~(A)-L ~ (~)}2
= L
0=1
k=lO-1<.
0
co
00
= L {L An(~)}2
n=l k=K
co
~ 2(
.. L
n=l
00
~
:U
n k=K-1<.
00
00
~(U ~
=
k=K
since
Uk: K ~ ...
r/J
as K -+
We now s'how that for every g in A
2
in L (n,F,p).
2
=
gdZ
(~)
2
0
r
n=l
,
J
<g~ f > E;.
n
n,
This is true for indicator functions by definition of Z,
and therefore also for simple functions.
L
x U Au)~
k==K
K-+co
Thus Z(A) = Lk:l Z(An)'
00.
JV
)
Since H(Z) is defined as the
(n,F,p) closure of the linear space of {Z(A), A€V}, it follows by
the isomorphism between
A2(~)
{lA' A€V} is dense in A2(~)'
and H(Z) that the linear span of
Thus every g in A2(~) is the A2(~) -
limit of a sequence of simple functions
Iv gdZ
co
{~}k~l.
Thus
gkdZ
'f
=
lim
k-+co
==
limL
k-+co n=l
V
00
where
li~-+co
<8tc'
f >
n
~
is in L (n.F,p) and the result follows from
2
n
14
ClO
=
-e
<g-~~fn> ~n}2]
E[{!
n=l
<g_g ~f >2
= !
n
n=l
=
In particular we have
Now since
f (t,.) in
n
<g-gk~g-gk> k~
Iv fndZ = ~n
which implies that H(Z)
a 2 (t) = R(t~t) <+m
n
for all t in T by
= H(X).
for all t in T~ we can define
Ln=1
ClO
A2(~)
o.
ClO
f(t,u)
= 2
n=l
where the convergence is in
A2(~).
a (t) f
n
n
(u)
It follows from the property of the
integral just proven that for all t in T we have the following equality
IVf(t~U)dZ(U)
= La
n=l n
(tH
n •
Xt
which concludes the proof. 0
REMARK 5.
We assume throughout this remark that H(X) is separable.
Then it is clear that the first claim in (i) and (ii) is valid provided
the dimensionality of L (V) and
2
of the integers.
A2(~)
is no less than the dimensionality
Also, one can take (V,V) = (T, B(T»
or as V any
interval and V its Borel sets; in the latter case v may be taken the
Lebes gJe measure or one absolutely continuous to it, and
taken absolutely continuous to the
Lebes~lPe
~
measure on VxV.
may be
If a series
(respectively, integral) representation of X is known then one can
t
obtain integral (respectively, series) representations of X as indicated
t
in the proof of Theorem 2. These representations will be explicitly
obtained if one can find complete orthonormal sets in the spaces
15
-4It
L (V) and
2
A2(~).
If V is an interval and
V its Borel sets, complete
orthonormal sets in L2 (v) are given in [13] (see also [2]), and complete
sets in
A2(~)
are given in [3] (In [3] the case where V is the entire
real line is treated and the case where V is an interval can be treated
similarly).
If neither an integral nor a series representation of X is
t
available, the problem arises how to obtain explicitly. such a representation (in terms of the process X te:T, and its auto correlation R).
t
This problem is solved in [4] for weakly continuous processes X te:T,
t
and T an arbitrary interval.
REMARK 6.
Theorem 2 may also be stated in terms of integral repre-
sentation of the autocorrelation
a,
which for (i) and (ii) are
respectively
R(t,s)
=
Ivf(t,U) f(s,u) dv(u)
for all t,s in T.
R(t,s)
1
In [12] a second order process X , te:R
t
called OSCillatory if it has a representation
REMARK 7.
Xt
I
=
~
-00
e
itu
at(u) dZ(u)
= (-~,+~
) is
1
for all t in R
l
l
where Z is an orthogonal random measure on (R , B(R » with corresponding
measure v and a (.) e: L (v) for all t in T (this is a generalization of
2
t
1
a concept introduced by Priestley).
If X , teR , is oscillatory then
t
1
1
H(X) is separable, since L (R , B(R ), v) is separable. Conversely,
2
if H(X) is separable it follows by Theorem 1. (1) that for any finite
1
1
»
I
~
.
we have X =
f(t,u) dZ(u) for all t:ln T,
t
-~
1
1
where Z is an orthogonal random measure on (R , B(R » with corresponding
measure v on (R , B(R
16
r.
measure v and f(t,.) e L (v) for all t in
If we define at(u) =
2
-itu
1
e
f(t,u), it becomes clear that X , teR , is oscillatory. Thus a
t
second order process is oscillatory if and only if its linear space is
separable.
REMARK 8.
Some simple sufficient conditions for H(X) to be separable
are as follows.
If X , teT, is a linear operation on a second order
t
process Y , seS, with separable linear space, then H(X) S H(Y) and the
s
separability of H(X) follows from that of H(Y).
Also, because of the
isomorphism between H(X) and R(R), H(X) is separable if there is a
symmetric, nonnegative definite function K on TxT suCh that R(R) S R(K)
and R(K) is separable.
A sufficient condition for R(R)
K-R be nonnegative definite [I, p. 354].
c
R(K) is that
17
·e
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Stamatis Cambanis
Department of Statistics
University of North Carolina
Chapel Hill, N.C. 27514