Download Cambanis, Stamatis; (1971)The equivalence or singularity of stochastic processes and other measures they induce on L_2."

..
Department of Statistics~ University of North Carolina~ Chapel Hill~
North CaroZ':;.a 27514. This research was supported by the Air Force Office
of Scientific Research under Grant AFOSR-68-1415 and by the Office of Naval
Research under Grant N00014-67A-0321-0006.
TH~EQU1 VALEiJCE
ORS INGULARITYOF STOCHASTI C' PROCESSES
Aim OF THE HEASURES THEY INDUCE ON
L
2
by
Stamatis Cambanis *
Depa1>tment of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 759
July~
1971
THE EQUIVALENCE OR SIHGULARITY OF STOCHASTIC PROCESSES
AND OF THE l·jEASURES THEY INDUCE 011
L2
Stamatis Cambanis*
ABSTRACT
It is 'shown that the proposition:
"Two stochastic processes are equi-
valent or singular if and only if the measures they induce on an appropriate
L
2
space are equivalent or singular respectively", is not true in general,
and sufficient conditions are given for its validity.
For a wide class of
square integrable martingales it is shown that this proposition is valid and
a number of results are obtained which generalize known results for the
Wiener process.
>+
Department of Statistics~ University of North Carolina~ Chapel HiU~
North Carolina 27514. This research was supported by the Air Force Office
of Scientific Research under Grant AFOSR-68-1415 and by the Office of Naval
Research under Grant N00014-67A-0321-0006.
1.
INTRODUCTION
Problems in detection theory and in classification theory or pattern
recognition reduce to studying the equivalence or singularity of the probability measures induced on the probability space by the stochastic processes
under consideration; and in computing the Radon-Nikodym derivative in the
former case.
These processes induce probability measures on appropriate
L
2
spaces, and the original detection or discrimination problem is often treated
on this Hilbert space where a powerful structure is available
(see for
instance [16, 17, 1]).
It is the purpose of this paper to study the validity of the proposition
stated in the abstract.
It is shown that this proposition is not true in
general (Proposition I and Theorem 4); specifically, two stochastic processes
may be singular and yet their induced measures on
4).
L
2
be equivalent (Theorem
Under appropriate continuity conditions it is shown that the proposition
is valid (Theorem 2).
The second order stochastic processes for which the
proposition is shown to be valid are those that are continuous in probability
or smooth, a notion introduced in [4].
Weakly, as well as mean square,
continuous processes are smooth and a further class of smooth processes is
given in Section 3.
It is shown there that all square integrable martingales satisfying
certain conditions are smooth stochastic processes (Theorem 5).
Also their
linear span, their reproducing kernel Hilbert space, and the range of the
square root of their covariance operator are characterized (Theorem 6); these
spaces playa significant role, especially in the Gaussiom case.
These
results generalize well known facts for the "Uener process and, in the case of
the reproducing kernel Hilbert space, for mean square continuous processes
with orthogonal increments.
2
Finally, the relationship between the reproducing kernel Hilbert space
of a stochastic process and the range of the square root of its covariance
operator is established in the most general case (Theorem 3), generalizing a
known result for mean square continuous processes defined on compact interva1s.
This relationship is of primary importance in the Gaussian case.
2.
THE
RELATIOI~SHIP
BEHJEEi',1 THE EQUIVALEiKE OR SINGULARITY OF STOCHASTIC
PROCESSES AHU OF THE HEASURES THEY
Ii~DUCE
OIJ L •
2
Throughout this section the following nota,tion is used.
are two probability measures on
stochastic process
{~(t,w),
and covariance functions
r (t,s),
1
R1(t,S)
tET}
respectively;
R,
DEFINITION 1.
P
and
1
with respect to which the measurable
is second order with mean, autocorrelation
mO(t)=O,
generated by the random variables
the real line
(n,F)
Po
rO(t,s),
RO(t,s)
F is the
and
m (t)=m(t).
1
a-algebra of subsets of
{x.(t,w), tET},
and
n
T is any interval on
open or closed, bounded or unbounded.
N is the class of
measu~es
the a-algebra of Lebesgue measurable subsets of
the Lebesgue measure,
v
~
v
T)
on
(T,B(T))
(B(T)
is
which are equivalent to
Leb, (i.e. mutually absolutely continuous) and
satisfy
fT
r.(t,t) dv(t) < +
~
,i=O,1.
00
(1)
N is a nonempty class of measures, as it is demonstrated by the following
construction.
Define
g E L (T,B(T),Leb),
1
v
on
g(t) > 0
(Tf> (T))
by
a.e. [Leb]
[dv/d Leb](t) = f(t)g(t),
on
T,
and
where
3
= {
f(t)
Clearly
v€N
v€N,
For every
x(·,w) € L (T,B(T),v)
2
v
(H =L (V) , C(H
v .
2
»,
nV ,
subsets of
v
and also
is finite.
the measurability of
x(t,w)
and
= L 2 (V)
i=O,1.
Also the map
where
a.s. [Pi]'
v
B(H)
is the
(1)
imply that
T:
(n,n-+
a-algebra generated by the open
defined by
T w =
x(·,w)
(2)
is measurable [15], and thus the probability measures
induce probability measures
v
~i
P., i=O,l,
~
on
en, F)
defined by
on
(3)
for all
v
B€B(H ).
It follows by (1) and (3) that
E [
i
II ull 2V ]
=
H
.
2
Ilx(·,w)11 V dP.(w)
n
H
~
J
Hence [13] the mean elements and the covariance operators of the probability
measures
v
~i'
i=O,l,
are defined as the elements
u
v
i
v
€ H
and as the
bounded, linear, nonnegative, self-adjoint. trace @lass, operators
which satisfy
v
E.[<u-u., v>
~
1
HV
v
<u-u , w> ] =
i
HV
4
v
v, weH •
for all
It is easily seen that
integral operator with kernel
Ri(t,s)
and that
[15].
{x(t,w), teT}
if the stochastic process
v
].Ii
then the induced measure
s~
~
is an
It is also shown in [15] that
is Gaussian with respect to
P.
~
is Gaussian.
A remark should be made about the need to consider the family of measures
N. What is really needed is that the probability measures Pi induce probability measures
interval
T,
].Ii
on some space of square integrable functions on the
and that
].Ii
have covariance operators.
vEN,
for this induction to be possible is that
square integrable functions on
that if
x(t,w)
x(t,w)
\leN
V
H
•
is Gaussian with respect to
necessary condition.
choice of
T is
in which case the space of
In particular, it is shown in [15],
Pi'
\lEN
then
is also a
If the Lebesgue measure satisfies (1), then the natural
is clearly
\I
= Leb.
This is the case if, for instance,
is mean square continuous with respect to both
is a compact interval.
For example, if
A sufficient condition
x(t,w)
Po
and
PI and T
However the Lebesgue measure is not always in
is wide sense stationary with respect to
Po
N.
and
PI and T is an unbounded interval, then the Lebesgue measure is not in N,
but every finite measure equivalent to the Lebesgue measure belongs to
N.
Before we consider the relationship between the pairs of measures
we prove that the equivalence
and
of the measures
and
].I~
does not depend on
(~)
or singularity
(~)
\I€N.
THEOREM'l. If anyone of the following relations is satisfied for some
\I€i'J:
then it is satisfied for all \lEN.
For the proof of Theorem 1 we need the following
5
LEr~~
1 [4, Theorem 2J.
functions in
LZ(V)'
vEN, {f~(t)}~=l
Let
be a complete set of
""
{n k (w)}k=l
and the random variables
be defined by
(4)
a. e. [P.].
1
Then the subspace
random variables
v
{fk}k
in
Hi(X,{f~}k'V) of
v
{nk(w)}k'
L (Q,F,P ),
2
i
does not depend on
L (V); it is denoted by
2
which is spanned by the
vEN and on the complete set
H(x,smooth).
PROOF OF THEOREM 1. For vEN and Hv = LZ (v) define
Then
T
GV is a a-algebra of subsets of
Gv
(defined by (2»,
c
F.
Q
and because of the measurability of
depend only on the values of the measures
the completed
and prove that for all
G~1
=
G~1
on
11 i
We will denote by
a-algebras with respect to the- measure
p., i=O,l,
1
v,AEN
, i=O,1.
(5)
is the smallest
If
algebra of subsets of
v
It is clear from (3) that the measures
a-
V
H with respect to which the linear functionals
v
}""
v
v
{Fk(u)
= <u,f v
k > v k=l are measurable, and since nk(w) = (Fko T)(w), it
H
follows that
GV = FV(n), where FV(n) is the a-albebra of subsets of
generated by
FV(n)
c
F.
Because of the measurability of
x(t,w)
and
(4) ,
Thus it suffices to prove
F~(n) = F~(n)
, i=O,l.
Because of the symmetry, it is enough to show
Q
(6)
and this for
6
i=O.
for every
A
j,
A
v) = HO(x,{fk}k'
HO(x,{f~}k'
Since by Lemma 1
'V
n j E HO(x, {fk}k' v)
A),
it follows that
and thus for some constants
{ajk}k'
00
I
a jk n~ (w)
k=l
where the convergence is in
n.A(w)
which implies that
A
=-
FO(n)
L (Q,F,P ). Hence along some subsequence
2
O
is
J
F~(n) - measurable for all
v
FO(n).
Note that (5) implies that every set
F =
where
A.DC.,
~
~
and
Clearly
zero.
and thus
j,
C.
~
F
c.EF.
G
V
is a subset of a
can be taken disjoint from
and in this case
A is of the form
in
G
p.
set of
measure
~
A. (if not replace by
C .'VA.)
~
~
~
This will be used in the following.
~
We now complete the proof of the theorem by proving the following proper ties
v
v
].10
Pl/Gv
«
P \)
OIG
V
if and only i f
v
]..11 J. lJ O
Pl/Gv
Pi/Gv
where
v
v
A
Proof of (7).
FEG
V
A
]..11 J. ].10·
,
P l/Gv
i f and only if
v
V,AEN ,
]..11 «
A
A
i f and only i f
and only if
Since
POIGv
J.
Pl/Gv « POIGv-
i f and only i f
denotes the restriction of
show that for any
]..11 'V ]..10
i f and only i f
]..11 «
]..11 'V ]..10'
v
].10
P.
~
Pl/GIi. «
J.
(7)
POIGA
(9)
POIGv
Pl/GA
to
GV •
i f and only i f
J.
POIGA
(10)
Clearly (7) and (8)
A
].11 «
Assume
F = T-l(B)
and (9) and (10) imply that
v
].11 «].10
for some
A and hence,
]..10
For notational convenience we will put
v
Then
PO(F)
=0
v
\)
if
].11 J. ].10
Pi/Gv
be such that
and let
v
BE0(H ).
(8)
= Pv.•
PO(F)
~
= O.
and (3) imply
7
v
= 0,
~O(B)
v
P~«
Conversely assume
Then, by (3),
v
(3),
~l(B)
PO(F)
= 0,
. Iies
1mp
= O.
PO(F)
Since
PO(D ) = 0,
O
= 0,
)
and by (3,
Pl(F)
P~
B€o(H
and let
where
= O. Thus Pvl « Pov •
v
v
~~(B)
be such that
)
v
= T-1 (B)€G.
F
Hence
Pl(F)
=0
= O.
and by
v
~l«
which implies
Proof of (8).
v
v
P1 «p 0
=0
~l(B)
hence
~O·
Because of the sYmmetry it suffices to prove that
A
A
PI «PO.
A
F€G,
we have
which implies
Pl(F) $ Pl(AOUD O)
A
A
proves that Pl « PO.
= AOUC O'
F
= O.
Pl(DO)
and thus
v
v
P l « Po
Assume
A
v
=0
PO(F)
= 0,
F€G
AO€G,
where
Now
Pl(AO)+Pl(D O)
$
and let
i.e.
be such that
C~DO€G
implies
Pl(F)
= 0,
v
Pl(AO)
and
=0
which
Then t here exi sts B~-l3(Hv)
~
suc h t hat
v
v
v ~
and by (3), Pl(F) = 1 = PO(F c)
~l(B) = 1 = ~O(B).
If F = T-l(B), then F€G
v
v
which proves that P l « PO.
Proof of (9).
v
Assume
v
Conversely assume
v
~l L
PI
~O·
v
F€Gv ,
Pl(F)
= 1 = PO(F~).
Since
some
v
BO' Bl€J(H ).
It follows by (3) that
-1
T
-1
(B nB ) = T
1 O
v
~l(Bl~BO)
(Bl)nT
v
L
v
Po
that
ioplies
Pl(F)
A
Ai€G , Ci€F
Po (AO)
and since
c
=
F
= T-l(B l ),
~~(Bl) = 1
0 and by (3),
which implies
v
v
~l L ~O
= ~~(BO).
v
~i(BlnBO) =
for
But
O.
Hence
•
Because of the symmetry it suffices to show that
A
A
PI L PO.
= 1 = PO(F'~.
where
we have
(B ) = FnF
O
v
= 1 = ~O(BO~Bl)'
Proof of (10).
PI
-1
such that
Oc
F = T-l(B )
O
Then there exists
PO.
L
and
v
v
PO. Then there exists F€G
such
v
F, FC€G
we have F = AlUC l , Fe = AOUCO'
Assume
Since
Pi(C i )
AlnAO =~,
v
PI
= 0,
L
i=O,l.
A
A
PI L PO.
It follows that
Pl(A l ) = 1 =
0
We now consider the relationship between the equivalence or singularity
of the pairs of measures
(PO,P )
l
and
it suffices to do this for a fixed
v€N.
script indicating the dependence of
H,
v ' v
(~O' ~l)'
v€N.
In view of Theorem 1
Thus in the following the super~i
and
Si
on
v€N will be omitted.
8
The proof of the following two propositions is contained in the proof
of Theorem 1.
PROPOSITION 1.
(1)
PI « Po
(2)
PI
(3)
III .L
'V
implies
Po
implies
Il O
implies
]ll «
III
'V
Il
O
Il O
PI .L Po
In general the inverses of (1), (2) and (3) of Proposition 1 are not
true.
This is not surprising because of the following.
algebra of subsets of
defined by (2).
the measures
Since
Il
T is measurable,
Thus i f
F is larger than
PI
on
'V
Po
Theorem 4.
G.
G be the
which is the inverse image of B (H)
Q
depend only
i
Let
GcF
under
a-
T,
and it is clear from (3) that
.. on the values of the measures
G it may happen tha t
Pl.L Po
Pion
G.
on.F and ye t
That this is not an hypothetical situation, it is proven in
Thus Proposition 1 cannot be improved unless restrictive assump-
An appropriate assumption would be that
tions are made.
larger than
G, specifically that
F is not essentially
Gi = Fi , where the subscript i
completion with respect to the measure
P., i=O,L
1.
denotes
The implications of this
assumption are stated in Proposition 2 and it is shown in Theorem 2 that a
large class of processes satisfies this assumption.
PROPOS ITION 2.
If
Gi
= Fi ,
if and only i f
then
i=O,l,
(1)
PI «
(2)
PI
Po
if and only i f
III
(3)
PI .L Po
if and only if
III .L 11 0
'V
Po
DEFINITION 2.
processes:
III «
'V
Il
11
O
0
Define the following classes of real, measurable stochastic
9
Sl is the class of continuous in probability stochastic processes
{x(t,w), t€T}
such that
S2
Leb«
v
and
x(',w)
€
a-finite measure
L (v)
2
v
on
(T,8(T»
a.s.
is the class of continuous in probability, second order stochastic
processes
.
S3
for which there exists a
{x(t,w), t€T} •
is the class of smooth, second order stochastic processes
{x(t,w),
t€T}, defined in [4, Theorem 4].
S is the union of Sr S2
and
p
S3'
It is shown in [4] that the weakly continuous, and therefore the mean
square continuous
proces~are
smooth.
Further classes of smooth second order
processes are given in Section 3.
THEOREM 2.
{x(t,w), t€T}
If with respect to both probabilities
Po
and
PI'
belongs to the class $, then (1), (2) and (3) of Proposition
2 are valid.
PROOF.
For
x(t,w)
x(t,w)
take
any complete set in
as in (4).
in
v
L (v)
2
Denote by
Sl
a measure
v
on
to be any measure in
(T,8(T»
N.
Let
is given.
For
{fk(t)}~=l be
00
and the random variables
F(n)
the
{nk(w)}k=l
a-algebra of subsets of
be defined
n generated by
(11)
This is shown in [3,11] for the class
for the class
S3
$1;
in [11] for the class
S2;
in the same way as (6) or as in [11, Theorem 3] •
shown in the proof of Theorem 1 (between (5) and (6»
follows by (11) that
G = F
i
i
that
G = F(n).
and thus Proposition 2 applies.
0
and
It is
It
10
Before we proceed we need the result stated in Theorem 3.
L (n,F,P )
O
2
be the subspace of
tET}
and let
covariance
RKHS(R )
O
RO(t,S).
spanned by the random variables
be the reproducing
form
RKHS(R )
O
f(t)
= EO[~
~ernel
It is well known [14] that
isomorphic with corresponding elements
and that
x(t,w)
for all
the integral type operator from
t€T
HO(x)
and
HO(x)
and some
to
L (V)
2
~
HO(x)
{x(t,w),
Hilbert space of the
and
RKHS(RO)
RO(·,t)
consists of all real valued functions
x(t)]
Let
E HO(x).
are
respectively,
f
on
T of the
Let also
with kernel
A be
x(t,w).
Since by (1),
A is a Hilbert-Schmidt operator.
THEOREH 3.
fE range
i.e.
(1)
If
fERKHS(R )
O
then
f€ range
(S~).
Conversely, if
(S~) then f is equal a.e.[Leb] on T to a function in RKHS(RO);
every equivalence class in range
(2)
range (A)
(S~) contains a function in
= range (S~).
The relationship between the reproducing kernel Hilbert space of a
stochastic process and the range of the square root of its covariance operator,
established in Theorem 3.1, plays a significant role for Gaussian processes
as it is demonstrated in Theorem 4.
Theorem 3 generalizes a well known result
for zero mean, mean square continuous stochastic processes defined on a closed
and bounded interval (see for example [8]).
Theorem 3 that the linear manifold .range
the sense explained in the theorem.
Note that it follows from
~S~~) is invariant of vEN,
in
11
For the proof of Theorem 3 we need the following
LEf1MA 2[4, Theorem 3].
eigenvalues and the corresponding eignefunctions of
SO'
00
{~k(t)}k=l
let
be the versions of the eigenfunctions which are defined for all
Ak~k(t)
RO(t,s)~k(s)dv(s), and let the
by ~k(w) = IT x(t'W)~k(t) dv(t)
= IT
be defined
EO[~k~j]
= AkO kj ,
and for all
by
{~k(w)}~=l
~kEHO(x),
random variables
a.e. [Po]·
tET
Then
t€T
00
x(t,w) =
l:
k=l
!jJk(t) ~k(w) + w(t,w)
(12)
where the equality as well as the convergence of the series are in
L (G,F,P )'
2
O
and
PROOF OF THEOREr:J 3.
Note that
00
I ~~k
range(sJ) = {f =
00
in
L (v),
2
k=l
(l.i)
Let
fERKHS(R O).
I
a
2
k
<
+ oo}
(13)
k=l A
k
Then for some
~€HO(x)
and all
tET, we obtain
by (12),
00
00
f(t) =
00
L
k=1
00
Hence
L
k=l
2
ak<+oo
l:
ak~k(t)
k=1
2
00
a
k
= L EO[~
A
k=l
k
for all
~k
-]
A-t
k
~
tET'VT
EO[~2] < +
(14)
O
(15)
00
00
and
k~l ak~k(t)
converges in
which, because of (14), is equal a.e.[Leb] on
T
to
L2 (v)
to a function
f(t) •
Hence
12
co
=
f
L
ak~k
k=l
(l.ii)
in
L2 (V)
Conversely, let
(S~). Then f =
fe range
co
co
-1
L
with
k=l
L (n, F,P )
2
O
in
range(s~).
fe
and by (13) and (15),
Ak
~~k(w)
-1
Ak
~
2
co
L
k=l
~~k
in
converges in
co
<
+
Let
co.
~
=
\'
-1
L Ak
k=l
~~k
It follows from
LZ(n,F,P O).
co
= L
EO[~ ~k]~k
k=l
such that
f
that there exists a subsequence
in
f(t)
= Urn
k~
Nk
L
n=l
EO[~ ~n]
4J
n (t)
a.e. [Leb] on
T
N
k
= Urn
L
EO[~
k~
n=l
~n(tHn]
a •e. [Leb] on
and thus
EO[~
f
equals
a.e.[Leb[ on
T a function in
)' namely
O
RKHS(R
x(t)].
Let
(Z. i)
fe range (A).
Then there exists
a.e.[v]
(Z.U)
co
\'
with
L
-1
k=l
a.e.[Leb] on
Z
~ <
fe
T, where
f(t)
~
-1
= L Ak
k=l
= EO[~
ak~k
a.e.[v]
x(t)]
T.
It
in
L2 (V)
in
co
L
k=l
f(t)
L2 (n,F,P O)·
on
T,
i.e.
ak~k
= E[~
It
f =
x(t)]
follows that
A~
and
0
We now consider the case where
to both probability measures
~l
such that
a.e. [Leb] on
range(s~). Then f =
and as in (l.ii) we obtain
co
co
and
fe range(A).
+
T, hence
HO(x)
f€ range(SO).
Conversely, let
Ak
on
~e
k
follows as in (l.i) that
and
T.
Po
are also Guassian [15].
and
{x(t,w), teT}
Pl.
is Gaussian with respect
Then the probability measures
~O
It is well known that both pairs of Gaussian
13
measures (PO,P l )
(~O'~l)
and
are either equivalent or singular [9].
Necessary and sufficient conditions for
~l ~ ~O
are given in [16].
RCt,s),
and thus
~
PI
Po
are given in [18]
In the particular case where
So = Sl = S,
RO(t,s)
and for
= Rl(t,S) =
these conditions are:
i f and only if
m
€
RIaIS (R)
(16)
i f and only i f
m
€
ral1ge(S~)
(17)
The relationship between the equivalence or singularity of the pairs of
measures
(PO,P l )
(~O'~l)
and
is given in Theorem 4.
This theorem demon-
strates that a detection or discrimination problem;., ,.,hich is defined on the
n,
probability space
functions
H
can be treated on the Hilbert space of square integrable
= L2 (v) ,
where powerful analytic tods are available, only when
case (iii) can be excluded.
S,
belongs to the class
This is the case, for instance, if the process
as it is shown in Theorem 2.
However, in general,
it may be that the two processes are singular and yet their induced measures
on
H
= L2 (v)
are equivalent.
THEOREH 4.
abilities
Po
PROOF.
measures
to
{x(t,w), t€T}
If
and
PI'
is Gaussian with respect to both prob-
then one of the following will always be satisfied
(i)
PI ~ Po
and
~l ~ ~O
(ii)
PI ~ Po
and
~l ~ ~O
(iii)
PI ~ Po
and
~l ~ ~o
In view of Proposition 1 and the fact that both pairs of Gaussian
(PO,P l )
and
(~O'~l)
are either equivalent or singular, it suffices
prove that case (iii) is possible.
This is shown by the following example.
14
Take
(n,A,p)
~
T = [a,b], -
We can find a probability space
and independent, zero mean and unit variance, Gaussian random vari~l(w), ~2(w), ~3(w)
abIes
+~.
< a <b <
properly included in the completion of
denotes the sub- a-algebra of
F(~1';2'~3)'
{xi(t,w), t€T},
d(t)~l(w)
for
aSt<c
d(c)~2(w)
for
t = c
d(t)~3(w)
for
c<tsb
i=O,l,
by
and all
wen
for all
where
and
d(t)
and
~
d(t)
0
m(t)
on
T.
Then
d(t) € L (v) ) •
2
PO' PIon
properly contained in
= Rl(t,s)
RO(t,s)
given
m(t) ,
v
Leb
~
t€T
and
Then Gaussian measures
F
i
~
0,
are measurable,
respectively.
J.I 0 ' J.I l
and
and thus
~
RKHS(R
).
O
f€ range (S~)
o
and
f~
Gi
is
Note that
It is shown in Lemma 3 that
d(t), one can always find a function
m
v€N
Take
are induced on
Gi = Fi (~l' ~3)
For this choice of
PI
~
3. There always exist real valued functions
such that
xl
t€T, d(c)
(so that Proposition 2 does not apply).
given
m € range(si)
{;j,j€J}.
is to be constructed it suffices
it follows from (16) and (17) that we have
LE~4A
all
o
and thus (16) and (17) apply.
RO(t,s), i.e.
such that
X
(n,F)
It is easily seen that
B(H».
j€J)
and
(note that if an example for a given
= L2 (V),
clearly
be
Wen
Also
induce Guassian probabilities
(H
t€T,
are defined and finite valued for all
a.e.[Leb]
Gaussian processes.
to take
F(~j'
where
A generated by the random variables
Define the stochastic processes
xO(t,w) =
F(;1'~3)
on it such that the completion of
RKHS(R ).
O
Po
m(t), t€T,
d(t) and
o
and
f(t)
defined for
15
PROOF. Let g€ RKHS(RO). Then, by Theorem 3.1,
there exists some
~
€ HO(x)
such that
g(t) =
EO[~
x(t)]
for all
t€T,
and by (12)
00
(18)
for all
t€T,
where
Note that for every function
f(t) =g(t)
a.e. [Leb] on
f(t), t€T,
f€ RKHS(R )
O
such that
and
T, we have
00
f(t) =
L
~~k(t)
k=l
for all
t€T
t€T '\, TO
all
[Leb]
and
T
on
n € HO(X).
Since by Lemma 2,
2
EO[w (t)] = 0
Leb(T ) = 0, it follows by (18) and (19) that
O
t€ T '\, TO·
f(t) = g(t)
have
and some
(19)
+ EO[n w(t)]
Hence, if we choose
to
for
f € range
g(t)
but not equal to
t € T '\, {to}
(S~)
f( t), t€T,
and
n
and
g(t)
f(t) = g(t)
on
T '\, TO
where
(for instance
to € T '\, TO)
we
0
It follows from (16), (17) and Theorem 3 that, in the case where
Rl(t,s),
for
equal almost everywhere
f(t O) ri g(t O)'
RKHS(R ).
O
for all
RO(t,s) =
necessary and sufficient conditions for (i) to (iii) of Theorem 4
are respectively
(i) ,
m
(ii) ,
m ~ range(SO)
(iii) ,
m
€
~
Rlms (R )
O
h:
RKHS(R )
O
and
h:
m € range(S~) •
Necessary and sufficient conditions for the general case
RO(t,s) ri Rl(t,s)
can be obtained in a similar way; they are not given here for space considerations.
16
k
f e: range (S~)
It is shown in Theorem 3.1 that
equal
a.e.[Leb] on
T
to a function in
RKHS(R ).
O
implies that
is
f
By combining (17),
Theorem 2 and (16) we obtain for the mean function the following
COROLLARY.
{x(t,w), te:T}
implies
3.
Let
If with respect to both probabilities
is Gaussian and belongs to the class
and
then
PI'
k
m e: range(S~)
m e: RKHS (R ) •
O
A CLASS OF Sr·100TH SQUARE INTEGRABLE NARTINGALES
{x(t,w), te:T} be a real, measurable, second order stochastic process
defined on the probability space
functions
r(t,s)
Denote by
H(x)
included in
and
R(t,s);
The subspace
H(x)
(n,F,p)
T
the subspace of
{x(t,w), te:T}.
i,
with autocorrelation and covariance
is any interval on the real line.
L (n,F,p)
2
H(x,smooth)
so that
Pi=P).
is called smooth if and only if
spanned by the random variables
defined in Lemma 1 is always
N is as in Definition I for one
(in the present case
value of the index
te:T}
S,
Po
The stochastic process
H(x)
= H(x,
{x(t,w),
smooth) [4].
In the following the class of square integrable martingales will be
considered.
A stochastic process
matringale if
a.s.
for all
s
~
t
in
by the random variables
T,
where
{x(t,w), te:T}
is a square integrable
for all
and if
F
s
te:T,
E[x(t,w)IF]
is the sub-a-algebra of
{x(u,w),ue:T,u~s}.
F generated
If
is a square integrable martingale then we have the following.
exists a monotone nondecreasing function
2
E[{x(t) - x(s)} ]
= F(t)
= x(s,w)
Note that all stochastic processes
with orthogonal increments are square integrable martingales.
t€T}
s
F(t)
- F(s)
on
{x(t,w),
There
T such that
(20)
17
for all
s
~
t
in
T. [12].
Since
F(t)
function, the left and right limits
tET,
and the set
able.
F(t-)
and
F(t+)
D of points of discontinuity of
exist at every point
F(t)
is at most count-
It follows from (20) that the left and right mean square limits
and
x(t+ ,w)
exist at every point
points of mean square discontinuity of
x(t,w)
let
is a monotone nondecreasing
tET
x(t,w).
and that
It is also easily seen that
is not weakly continuous at the points of
F(t) f F(t-).
Then it follows by (20) that
t
n
t
t
and
~
E H(x)
D is the set of
D.
Indeed let
tED
x(t,w) f x(t-,w)
is not orthogonal to
and
in
x(t)-x(t-)
we have
lim E[x(tn )~] = E[x(t-)~] f
E[x(t)~]
n
which shows that
x(t,w)
is not weakly continuous at
tED.
The following
condition will be considered:
(Cl)
At every point of mean square discontinuity (tED) the square
integrable martingale equals either its left or its right
mean square limits
or equivalently,
(C2)
At every point of discontinuity (tED), F
equals either its
left or its right limit.
THEOREM 5.
If a square integrable martingale satisfies (el) then it is
smooth.
PROOF.
and
f(t)
= 0 a.e. [Leb]
assume that
f(t)
By [4, Theorem 4.1] it suffices to prove that if
f(t)
= 0,
T, then
f( t)
= 0 for all tET.
= E [~ x( t) ] for all tET, where
= 0 for all tET
Leb(T')
on
'V
T',
where
there exists a sequence
Leb(T')
{t }
n n
f E RKHS(r)
~
E H(x),
Indeed
and that
= o. Now let tET' . Since
of points in
T
'V
T'
con-
18
verging to
t;
in particular there always exist increasing (decreasing) such
sequences.
We clearly have
tim
f(t) = O.
n
n
and we obtain
If
f(t)
creasing sequence
tED,
=0
processes
(C2)
If
~
t
D,
it follows that
by (Cl), either
t
~
f(t) =
or
D by choosing an increasing or de-
respectively.
Thus
=0
f(t)
for all
tET
and
0
the theorem is proven.
REfvJARK.
n
as when
{t}
n n
= O.
F(t )
The proof of Theorem 5 applies to all second order stochastic
{x(t,w), tET}
For all
and
tET
x(t,w)
satisfying the condition
equals either
which are thus smooth.
x(t-,w)
the mean square limits
x(t-,w)
and
x(t+ ,w) exist
x(t+ ,w)
or
This follows from the fact that if (C2) is satisfied
then the set of mean square discontinuity points of
x(t,w)
is at most
countable [6, Lemma 1].
We now proceed to characterize the reproducing kernel Hilbert space and
the range of the square root of the covariance operator of a square integrable
martingale.
These spaces play an important role in the Gaussian case, as it
is seen from Theorem 4 and the conditions (i)', (ii) , and (iii)'.
It will
be assumed that the following simplifying condition is satisfied:
(~3)
T
is of the form
x(O,w)
=
0
[O,a], a
<
+ 00,
or
[O,a),
a:S + 00, and
a.s.
A simple stochastic integral with respect to a square integrable martingale
.
will also be used here [5,7].
We define
F'
by
for all
and we denote by
m or
the usual way, m{(s,t]}
T.
For all
f E L (dF')
2
dF'
tET
the measure induced on
= F'(t)
- F'(s)
= F(t+)
(21) •
(T,B(T»
- F(s+)
the stochastic integral
by
for all
ITf(t)dX(t,W)
F'
s:s t
in
in
is a well
19
defined random variable in
L (n,F,p)
2
and has the property
(22)
THEOREI~1
(~l)
6.
For a square integrable martingale
satisfying
and (C3) the following are true:
(1)
H(x)
(2)
If
RKHS(R)
= {~(w) = JTg(t)dX(t,W),
E[x(t)] =
= {f(t) =
I:
°
for all
g(u)dF'(u)
where the integral in (24) is over
F(t).
{x(t,w), tET}
[O,t)
g E L (dF')}
2
tET, then
for all
for
(23)
tET
tET, g E L2 (dF')}
~
D and for
tED
(24)
with
= F(t-), and over [O,t] for tED with F(t) = F(t+); and
range(s~) = {f(t) =
t
J° g(u)dF'(u)
a.e.[Leb] on T, gEL (dF')} (25)
2
The results of Theorem 6 are known for the Wiener process, for which
F(t)
=
t •. The characterization of the reproducing kernel Hilbert space (24)
is also known for mean square continuous processes with orthogonal increments
[10].
As it is suggested by Theorem 6, a number of results established for
the Wiener process can be extended in an appropriate way to all square integrable Gaussian martingales satisfying certain conditions (like
~l)
and (C3».
Results in this direction will be included in a forthcoming paper.
PROOF OF THEOREM 6.
(1)
Let
L(x)
= {~(w) = ITg(t)dX(t,W),
g€L (dF')}.
2
It follows by (<3) that
x(t-,w)
= x(t-,w)-x(O,w) = ITIro,C)(U)dX(U,W)
€ L(x)
(26)
20
t€T.
for all
x(t+ ,w)
Similarly
L(x)cH(x)
Hence it suffices to show that
and
~ ~
~(w)
=
H(x)
imply
~
IT g(~)dX(U,w)
for all
= O.
and thus, by (el),
L(x)
€
~
Assume
for some
H(x)=- L(x) •
or equivalently that
€ L(x)
g € L (dF')
2
and
~ ~
H(x).
~
€ L(x)
Then
and by (26) and (22) we have
t€T:
=<
Since the set of functions
follows that
i.e.
~
= 0
(2)
g = 0
in
in
is dense in
{I [0, t) (u), t€T}
L (dF')
2
and by (22),
2
E[ ~ ] =
L (dF'),
2
II g II ~ (dF' )
it
= 0,
2
L (Q,f,P).
2
It follows by the definition of reproducing kernel Hilbert space,
(23) and (22) that
= E[x(t)~]
RKHS(R) = {f(t)
= {f(t)
where the integral is over
Theorem 3.1 and (24).
F(min(t,s».
to
[O,t]
t€T,
v€N,
Let
S
g€L 2 (dF')}
as explained in the statement
follows directly from
alternative proof of independent interest is the
An
i.e.,
= LL*,
v~Leb
where
and
=
R(t,s) = ret,s)
fTF(t)dV(t) < +
00.
Then it is
L is a bounded linear operator from
L (V) defined by
2
(Lg)(t)
and has norm
[O,t) or
for all
Note that, as it follows from (20) and (C.3),
easily seen that
L (dF')
2
g(u)dF'(u)
~€H(x)}
t€T,
The characterization of range (S~)
of the theorem.
following.
f:
=
for all
I
ILl
1
2
=
f:
g(u)dF'(u)
= fTF(t)dV(t)
<
+
a.e.[Leb] on
00.
0
g € L2(dF')
It follows by a straightforward
extension of a result given in [2, Corollary 2.c] that
and hence (25).
T,
range(s~)= range
(L)
21
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?Ol~
University of North Carolina
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AnaZ.~
[2]
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P.A. Meyer, ProbabiZity and
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22
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