,
,
Ie
This research tJae paniaZZy supfJol'ted by the Nationa't Science Foundation
under' GranJ GU-'2069.
~ EQUIVALENCE OF PRCeABIUTV ~lI£S
Q.wus R. BAKER
Depa.I'tment of Statistics
University of Nonk Carolina at Chapel. Hill
Institute of Statistics Mimeo Series No. 701
July, 1970
ON EQUIVALENCE OF PROBABILITY MEASURES
Charles R. Baker
DepaPtment of Statistics
University of Nonh Caro."ina at Chape'" Hi."."
Let
sets, and
H be a real and separable Hilbert space,
III
and
two probability measures on
ll2
equivalent (mutually absolutely continuous) 1f, for
1J 1 (A) == 0 <=> 1J2 (A) ==
tained in this paper.
be Gaussian.
o.
f
the Borel a-field of H
(H,r).
III
and
1J2
are
Aef,
Several sufficient conditions for equivalence are ob-
Some of these results do not require that
III
and
112
The conditions obtained are applied to show equivalence for some
spec1fic measures when
H is
L [Tl.
2
This r6seca>Ch UJas paniaZZy supponed by the Nationa'" Science Foundation
under Grant GU..2059.
ON EQUIVALENCE OF PROBABILITY MEASURES
Charles R. Baker
University of Nol'th CQ1'OZina at Chapel. Hill
I NTROJU:TION
Conditions for equivalence of probability measures on the Borel a-field of
a Hilbert space have been the subject of much research during recent years [1] [9].
For two Gaussian measures
lJh lJh
either
(mutually absolutely continuous, denoted by
orthogonal
(PI.l lJ2),
lJI
lJI"" pz)
and
lJz
or else
are equivalent
PI
and
lJ2
are
and general necessary and sufficient conditions for
equivalence have been obtained (e.g., [5]) •
. When one or both of the measures is not Gaussian, few conditions for equivalence are known.
Moreover, even when both measures are Gaussian the general
conditions for equivalence are often difficult to verify, requiring one to prove
existence of a Hilbert-Schmidt operator with prescribed spectral properties.
In this paper, several conditions for equivalence are given.
conditions are stated in terms of sample function properties.
Most of these
Several results do
not require that both measures be Gaussian; when the two measures are Gaussian,
the sufficient conditions given here will often be easier to verify than those
previously obtained.
This r8searah b1as ptrPtial'Ly supponed by the National. Scienae Foundation
undeP Gront GU-2069.
2
IEFINITIOOS RID ProBLEM STAT&£NT
Let
and Borel a-field
and
aIr
N are
into
H.
r.
Let
measurable mappings (e.g.,
rxf
A,BEf.
S-l(A)
a
E
for all
A E r)
of
S
n
air measurable mapping of n into H will
Following Mourler [10], a
able space;
by
be a probability space, and suppose that
(n,a,p)
be called a "random element" in
AxB,
(.,0)
H be a real and separable Hilbert space with inner product
H.
Let
(HxH,rxr)
be the usual product measur-
is the smallest a-field containing all measurable rectangles
HxH
is a separable Hilbert space under the inner product defined
( (~1'~2)' (Il'X2»
Define the map
= (ASl'%l) +
H~H
n~
(S,N):
HxH
(AS2 '%2)
by
.!t'~ ,11'%2
for
H [11] •
in
(S,N)(w)
= (S(w),N(w»;
Sew)
{w:
this map is
6/r x r measurable, since for A,B in r
{w:
(S,N)
Hence
defined by
(S(w),N(w»
induces from
11 ,N[C). p{w:
S
induce measures
11
= P[S-1 (A)].
PS(A)
where
11S
II ~[AxB)
is
aIr
:!
C},
•
on
(HxH, fxr)
C in
~
A, B in
f;
rxr.
H,
and
\.IS,N[f
f(!,I)"
H,
e.g., for
M. f
so that
-1
(A) J
=
denote any fixed element of
at
:!'
f
is
rxr Ir
we define the measure
11v+N
~
A in
by
1!v+N(A)
~
r,
11S ,N" \.IS 8 ~,
H.
measurable.
11 +
from
S N
p{w: (S(w) ,N(w»
therefore is
""
Sand N
is continuous relative to
Define
rlr
E:
f
-1
(A)}.
by
f (!) , =
v
~I;
""
measurable and for
= ~[f-l(A)].
~
Hence
P,
f : H+H
v
""
fv' ,.is the section of f
Further,
f.
measurable, and induces a measure
11S+N[A]
Let
PS,N
N(w) E B}.
N are independent if and only if
f: HxH
HxH
E:
E A} n
respectively, on
• 11S(A)~(B),
the norm topologies of
SiN
a measure
P
~,
Sand
Consider the map
{w:
(S(w),N(w»
and
S
=
E AxB}
A in
r
3
Two measures
if for
Aff
exists
A in
~l (A)
~I
and
~2
=0
<==> ll2 (A) = O.
on
(H,f)
are said to be equivalent
They are orthogonal
r such that lll(A)· 0,
~2(A)
(~l.l ~2)
(llI"" ll2)
if there
= 1.
The problem considered in this paper is that of obtaining sufficient con-
11q and
llS+N.
are stated in terms of the measures
lJv+N'
ditions for equivalence of
...
Several of the conditions obtained
...VEH.
4
CoVARlNa O'JERATORS: GAussIAN
Ell sew) 11
. Suppose
2
<
lID;
rtPsuREs
Rg in H such that (!!!sf!!)" E(S(W),S), (RSS'y')
an operator
.{S(w)-!!!s.y.),
s,!
Cl
of H and
E(S(w)-SS'S)·
The operator R is a "covariance" opS
erator; i.e., it is linear, bounded, non-negative, self-adjoint, and trace-class.
If
for all
!!s
then there exists (10] an element
EIIN(w)
11 2
<
defined by
moreover,
RgN
[U J.
RSN
5
(S,.9)
then
Sand N have a "cross-eovariance" operator
(RSNu,v). E(S(w)-18 ,o!) (N(W)~,'y) for all s,.! in H;
s
RS~~~ for a bounded linear operator V, with I IVII :S 1
RSN : H+H,
lD
CD,
in H.
is thus trace class, and ~S" RSN *'
is said to be a Gaussian element, and
is a Gaussian random variable for all S
operator
If
llS
and
llS+N
or orthogonal [2,31.
denotes adjoint.
a Gaussian measure, if
in H.
Rg and a mean element mg, and Ell S (w) 11 2
~
*
where
llS
<
lID
then has a covariance
[10] •
are Gaussian measures, then they are either equivalent
A number of conditions for equivalence have been given by
various authors; the conditions most useful for our purposes are the following
[5]:
LE""" 1.
(a)
(b)
llN"" 1lS+N
!Bs -
~
if and only if
k
is in the range of ~:2;
R + = ~~~ +~, where W is a Hilbert-Schmidt operator that
S N
does not have -1 as an eigenvalue.
In dealing with the equivalence of Gaussian measures,
~
often needs the
following results on the range of square roots of covariance operators [13],
[141 :
l.fMt.\\ 2.
(1)
Suppose
R1 and R2 are covariance operators. Then
range (Rl~) c range (R2~) if and only if the following (equivalent)
conditions are satisfied:
5
(2)
(a)
R lz .. R2lzG for
l
(b)
R
l
(e)
(R
III
l
R2'rQR2lz
!!,)!)
G linear and bomded;
for
S k(R
Q linear and bomded;
1!,l!)
for all 1! 1n H and some finite scalar k.
2
range (R lz) .. range (R lz)
l
2
if and only if the follarl1ng (equivalent)
conditions are satisfied:
R IrG,
2
(a)
Rl'r
(b)
R • R2!rQR !r
2
1
III
for
G linear and bounded with bounded inverse
for Q linear and bO\l1ded with bounded inverse.
6
In most applications H is
terval T of the real line.
L [1]
2
(Lebesgue measure) for some compact in-
In such cases.
S and N are random functions
corresponding to measureable stochastic processes
functions belong almost surely to L [T].
2
stochastic process
(St)
(St)' (Nt)'
It is easy to verify that a measurable
with sample functions a.s. in
surable function; one uses the facts that
whose sample
(S ,S)
is
L [T]
2
B/BIR]
(B[R] :: Borel sets of the real line) for all.9 in L [TJ,
2
sir mea-
is a
measurable
r is the
and that
smallest a-field such that all bOtmded linear functionals on
L2 [T]
are
riB [R]
measurable.
In many signal detection problems (see, e.g., [6], Chapter 20). one observes a sample function
X(w)
the sample function is "noise"
S(w)+N(w»,
If
llSiN
J.
belonging to
(X(w)
L [T]
2
and must determine whether
=- N(w» or "signal-plus-noise"
(X(w)
=
these two possibilities being mutually exclusive and exhaustive.
l\i'
there exists a statistical test such that the probability of
making an incorrect classification is zero.
However, if
IJS+N""~'
then the
probability of deciding correctly when signal is present can be mity only if
the probability of deciding incorrectly is unity when signal is absent.
alence holds in virtually all practical signal detection problems.
Equiv-
Thus condi-
tions for equivalence are of interest to aid in constructing valid mathematical
models of signal detection problems.
7
EwIVALENCE (a.mITIC14S FOR
JJS,N" JJS •
In this section it 1s assumed that
THEOREM I.
PlmF:
If
JJv+N'"
...
For A in
~ a.e.
INIEPENIENT 5, N
dJJS (X) ,
then
~.
JJS+N"'~•
r,
(*)
Suppose
JJv+N'" lJN
,.,
~ lJS+N{A) .. 0,
~ (A) ..
o.
.1 ~,
lJS+N
a.e.
from (*).
Hence
dl!s(;!).
Also,
lJS+N'" JJ •
N
since the set
A!
~(A)" 0 ~ JJv+N(A) .. 0
Then,
,.,
lJS+N(A). 0
:=>
lJ.!"'N(A) • 0
1.1v+N J. ~a.e.
,.,
(Note that
a.e.
dlJ (!)
S
a.e.
dJJs(.t)
dl!S(!)
=l>
does not imply
can vary
satisfying
with ,.,
v.)
Although the above result assumes that
lJS,N" JJ
S
• JJ ,
N
it can be applied
to yield conditions for equivalence when
Sand N are not independent.
example, suppose that
N
l
N .. N +N ,
l 2
transformations inducing measures
loiS N .. loiS • 1.1N'
, 2
2
where
JJ
are
aIr
measurable
If 11....
• 1.1
• 1.1N
N2 •
~wl,N2
Nl
2
one can use the above result to determine if lJN"" lJN
N1
and
and N
2
For
lJ
2
and
and
~
If both these equivalences hold, then by the chain rule for Radon2
Nikodym derivatives one must have lJS+N -~. This simple modification should be
lJS+N ,.., lJN
useful in many applications, especially in practical signal detection problems,
where the noise usually cODtains an additive Gaussian component that is independent of the sianal and of the remainder of the noise.
8
e
In order to apply Theorem 1, one must first determine sufficient conditions
for
-
liV+N - liN'
~everal such
!f:H.
~
conditions are known when
is Gaussian,
and are utilizecl to obtain the following corollary.
CDraJ.Mv.
P
s Ql~.
~ is Gaussian with
Suppose that
Then
liS+N'"
(1)
% f: range
(2)
E" S (w) 11
'1i
('1.\iJs)
2
0,
&I
and
PS,N·
if any of the following conditlO1lS is satisfied:
a.e.
< .,
II~II
!9s
dp {;!);
s
raDge
£
(~Js),
~.
and
"NlfWI1fJz
with
W trace-
class;
(3)
H is
L [T]
2
for a compact interval
T.
~
is the measure induced
by a measurable mean-square-continuous stationary stochastic process
~,
with rational spectral density
(a)
< .,
and
there exists a rational spectral density function
(Rs!,2,)
S
kJ:. ~(A) I!(),) 1
2
Fourier transform of
It,
J!!s (
(b)
u, u(t}=O
......
with
range
H is as in (3),
for all .y in
d)'
J
for
~().)
H
io
such that
(,g
is the
t4T) ,and some finite scalar
la().)
.......d)' <
_CIt
(4)
Ell s (w) 11 2
CD,
(~Js).
l1i
is induced by a measurable mean-square continuous
stationary stochastic process with a spectral density function,
~,
and
a.e.
(l)
(2)
Let
From Lemma 1,
{A}, {e}
n
-n
;!
in range
k(1\/>
<=-
lJ;rtN - ~.
be . the non-zero eigenvalues and an associated set of
orthonormal eigenvectors of
~.
Then
9
and (2) follows, since
(3)
Sew)
€
range
("N%)
almost surely.
According to Hajek [4],
implies that
R "
O
~Js~1f,
W trace-class.
range
(RgY,») S k f:~~{A)I~(A)12 dA
<Ro%) , and this last condition
RO~G
for
tion
implies that
range
O
~%QGGfcQ*~ 1f,
Ro~ a ~%Q,
and QGG*Q*
(Rg~)
implies (Lemma 2) that
G bounded; while the representation R •
trace-class implies
Its ..
By Lemma 2, the condi-
Q Hilbert-Schmidt.
~1f~1f
c
S1f
R
..
with
W
Hence
is trace-class, so that (3) follows
from (2).
(4)
This result is due to Kelly, Reed, and Root [15], and can be proved
quickly as follows.
By Lemma 2,
if there exists a finite scalar
»
in H.
X is in
range
k such that
(~~)
if and only
/\! ,JIj/\2 S k\/ ~lhJlj/,
\
all
The condition given in (4) implies that
a.e.
so that :!
f
range
(~%)
a.e.
dJJ s (:!).
The conditions cited above can be used to detemine equivalence when
N1+N 2 , where JJN N
l' 2
dis(:ussed previously.
= llN1 •
1JN ,
2
N
=
1s Gaussian, as
10
Dr. T.T. Kadota has kindly provided the author with a preprint of a forthcoming paper [16].
for
H .. L [0,b],
2
In that paper, part (1) of the above corollary is proved
b <
ClO,
under the additional assumptions that
tinuous covariance function and
~
liN
is strictly positive definite.
has con-
11
1!S,N = lJS •
When the assumpt:f.on that
and
1J8-foN
~
suppose that
is not implied by
~
integer and some scalar
range
RS+N •
(~l/p)
a.e.
k.
dP(w),
S
k~2/p +
only if
k· - Ai-lIp
example,
St(~)
l:lS+N
implying
-
2k"Nlip + I
J. ~
if
Sew)
a.e.
d1!s(x).
1!v+N -
As a counterexample,
.~l/PN(W).
~
a.e.
for
p. 1
d1!S(!).
p
a positive
or
2.
Sew)
€
One has
".,
From Lemma 1 and Lemma 2,
1!S+N.L
l1i
if and
has zero as an eigenvalue; this will occur if and
for sOlIe non-zero eigenvalue
H is
_«20+1)2/4).2
for any integer n.
t;at
is not valid, the equivalence of
is then Gaussian. and for
Fwlf[k2~2/p+2k"NIIp+I]~If.
only if
11l
1!v+N -
is Gaussian and
~
L [O,1],
2
J:
~
Ai
of
~.
As a specific
is Wiener measure, and
I: Nv(~)dvdu ~ -«2n+1)2/4).2[RNNlt(~)'
12
S, N: GALssIAN rtAsue
IgeraNT
~,
In this section, it is assumed that
).lS'
and
are Gaussian.
).ls-tN
previously I'lOted, this implies the existence of covariance operators
and ~
).l (S+N)
iN' J!!g'
and mean elements
and
l!s+N;
~,
we assume
"!Uti"" O.
for
r
As
RS'
Let
denote the Gaussian measure defined by
(f(l"%) .. ~).
A £
has covariance operator RS + ~
lJeS+N)
~
Finally, we assume that the range of
!!!s.
and mean element
is dense in H;
in cases where this is
not satisfied, one can obtain the results given in this sectioll by defining B
to be the closure of
range
(~).
We proceed to obtain some new sufficient con-
ditions for equivalence of
lJS+N
l.s+1A 3.
(~1;)}
Suppose
operator W,
PRooF:
and
liS{range
J!s
Then
By the corollary to Tbeorem 1,
from Lemma 1 this implies RS
range
.. 1.
(~~).
Since
Let .8::
liS
Rs"
~1;~%
for a covariance
~1;) •
range
£
~.
and
III
lls{range
'fwa?-'
~
~%)} ..
1
1l(S+N) ...
=0
W Hilbert-Schmidt, and
J!ls
~;
£
~-tf!s.
is Gaussian,
(S,S)
is a Gaussian random variable for all ,gER.
Define Y: '1"" R by
yew)
:II
Q
For any
~If
,QEA,
A£r, 11;J;[A]
:: {AS: ! ..
is a cODtinuous map.
forS(w)
~~J!
Moreover,
then y-l(A). {w: Sew)
E
for
4 range (~~)
seA} is an element of
y-l(A). {w: S(fAl)
~1;[AJ}
•
£
~t(A}}
U (/AI: Sew) (range
r,
since
if .QtAj
1f
(~+)). In either
13
case,
y-l(A)
£
lly
on
measure
6,
so that
(H,r>.
elr measurable and thus induces from P a
Y is
For any ,g
H we show that
in
random variable; first, note that there exists
Rs
Using
~~wa:,
=
almost surely and in
in
lJy
H,
THEOREM
range
2.
and J!lS
If
(~~)
PRooF:
~%~1t,
~%(I+Q)~~,
RS •
R
S
where
:>
range
(~%)} = 1,
(y'.Y)
(1\/)] •
1
-1
then
R 1t
S
implies
as an eigenvalue.
k
llS+N"" l1q
From Lemma l,
.my,
and mean element
if and only if
m
S
llS+N'" llN
E
S
range
(~%)
if and only if
-1
and
RS+N
=
as an eigen-
• a;G, G Hilbert-Schmidt; hence
1
satisfying
is Hilbert-Schmidt, so that
for range (~+N:iI)
Ry
Q is albert-Schmidt and does not have
where
(S,.Yn ) " (y,1d)
is a Gaussian r.v. for all y
implies, by Lemma 3, that
W trace class.
= ~%~If
Z == W+GV+V"'G*
one has that
~lt~n"~.
(~lf).
V is an operator of norm
not have
Hence
such that
is a Gaussian
= ~%my.
lls{range
lls[range
value.
L (n,B,p).
2
~%i'
is Gaussian with covariance operator
~~Fy~%,
RS =
W bounded, and !!!S =
{.Yn}
(y,,g)
~N
llS+N "" l1q
CI
RS%va;
[12].
if and only if
Now
Z
does
From Lemma 2, this is precisely the condition
k
:>
range (~2).
The results summarized in Lemma 2 can be used to determine whether
(~lt)
range
dll
S
(~)
•
c range (R +:)'
S
and thus
llS+N'"
~
whenever
llv+N "" llN
....
The following corollary lives two additional conditions for
U>Rot.1ARY.
llS+N'"
~
if
llS{range
(~%)}
= 1
a.e •
llS+N "" l1q.
and either of the two following
conditions is satisfied:
(a)
There exists no non-null .Y
in
H satisfyingG*l! = -V,g.
and
14
V*G*u •
R
SN
(b)
where
-U t
= RS1ff~J.j.
H is
Gt V are operators satisfying
~
T a compact interval t
(a)
teT, St
is independent of N
v
(I+GG*+GV+V*G*),y
=:
Q to imply
~
IIG*,y112 - 211G*MII IIVBII
(11,y11-IIG*»11>2
21Iv*G*ull Ilull
"...,
IIMII. O.
(~J.j).
FU:t'ther,
11!:J11
~
IIul1
2
".",
11»11
2
2
+ IIG*uI1
+ IIG*»11
It is clear that
Q,
c
v
~
If
(RwJe) t
t.
or, by Lemma 2,
2
~
- 211G*ull Ilull
V*G*y
V*G*»
a
N
f!'W
-M
= -,g
I IG*,y1 I
a
11,y112 +
G*l!
~
and
V*G*)!
and
IIG~.Y112
11,y112 +
I'ttI
Ilyll
= -V,y
-Vy
IIlI
= (I lui I-I /G*uI1)2 ~
=:
-
0,
-y.
ok
range (R + ;?) :;) range
S N
only for y = Q,
and
~
then
=
- 21IG*,y11 11,y11
2
2
= Q,
(I+GG*+GV+V*G*)B
+ I IG*,g1 1 + 2(G*!:JtVM)
#IIt#
G*,y. -Vy and
GV+V*G*)y
for all
The LHS of this last equality is
with equality throughout if and only if
Hence, if
is measurable t and
~ 0, with equality throughout if and only if G*y
I IG*,y I I • IIMII.
e
CSt)
range (R + J.j) :;) range
S N
2
11»11 +IIG*,y112+2( G*B, v,y) = O.
,.,.,
lJS+N - lJ .)
N
From the theorem, it is sufficient to show that the stated condition
is necessary and sufficient for
for
= a;G t Ilvll S 1,
is induced by a measurable
mean-square continuous stochastic process,
PROOF:
S
(This condition is also necessary for
L (T)t
2
for all
~
R
together imply
so that the condition is also necessary for
(I+GG*+
range (RS+Nls) :;)
ok
range(~;?).
(b)
We show that the assumptions imply
pose not.
Ilyll
Then by Lemma 2 there must exist
= 1,
and
+ Je.Yn
S N
R
. . . ~.
X in L2(~,e,p) such that
L (O,13,P».
2
spanned by
Since
(Nt)
{(N,»), »E:H}
ok
range (RS+N 2) :;) range
{Yn}
in
ok
(1)/>.
Sup-
~%.Yn . . .
H such that
.y,
This implies the existence of a random variable
X = l.i.m.(»n,N)
= I.i.m.{-yn,s-ms)
(l.i.m.
is mean-square continuous, the subspace of
is identical to the subspace spanned by
for
L2(~,e,p)
{Nt' tfT}.
Moreover, this subspace is separable since it is isometric to the closure of the
range of
scalars
~%
{~M},
{under the map taking
k = lt2t ••• ,M;
M
~JeB
= 1,2, ...
into
(,g,N».
such that
Hence, there are
X
= I.Lm.
kfla~tk;
15
~:: kfl~k.
let
since
{Nt}'
tfT,
{~}
Note that the set
can be assumed independent of
spans a separable subspace of
M,
L (S1,a,p).
2
We now show that
EX(w)[St(w)-mg +Nt(w)] • 0 for all ttT. To see this,
t
one notes that EX(W)[St(w}-mst+Nt(w)]. -Umn E(l\t,S(W)-ms)[St(w)-l9S +N t (W)]
t
.. -Umn[Rs-Yn~S.Yn](t). In L2 [T], one has RS.Yn + ~S.Yn .. R/(GG*+V*G*)~~.Yn
-+
l\~~ (GG*+V*G*)!!
Since
~
• Q,
has continuous kernel
uniformly for all
.y and part (a) of this corollary.
from the assumptions on
R(t,u),
this implies that
[RS.Yn+~S.Yn](t) -+ 0
t in T~ To prove this, let RO(t,u) be the kernel of ~~.
Then
([~+~s9n)(t»2
<
1: ~
•
R(t,t)
~
Hence
[1: RO(t'.)[(GGOTVOG*)~~~)(.)doJ2
•
(t ,.)do
II (GG*+V*G*) ~~»n 11
II(GG*+v*G*)~~Ynl12
II (GG*+V*G*)~~.Yu112
[-6Up R(t,t)]
t€T
EX(w)[Nt(w)+St(w)~]" 0
For
~
2
for all
-+ O.
tfT.
t
defined as above,
M
r
ak
M
RSN(tl,tk )
..
0,
Ito 1
since
RsN( t , t ) • 0
i k
for
k ~ i
(we are assumming that
~ ~
ti
for
k ~ i).
Hence
EN
=l>
X is contained in the subspace of
Suppose next that
~ .....
X in
L (S1,8,P)
L (S1,a,p)
2
2
with
t
(w}X(w)
•
0
1
spanned by
~
=
kL~~tk'
{N~},
where
k .. 2,3, •••
1 < m
~
M,
16
=>
EXJitm '"
{Ntk },
k
~
0,
m+l.
all
~(w)[S1:m(w)-ms
] D 0 for all M
tm
M => X is contained in the linear subspace spanned by
m fixed and independent of
Hence
M.
Then
EX (w}!ailrtk (w) ... 0,
=>
__2
Er(w)'" 0
=i>
Urn 1I~~.Yn112
n
0:_
=>
olAJII
...
0
all M
I ()
E \N
=i>
W
'.!lnl\2
11.1+11 ...
=0
0
17
Suppose that
L [O,bl,
2
H is
b <
GO,
and that
}.IN
is Gaussian with null
mean function and covariance function defined as follows:
(1)
~(t,s)
• min(t,s)
(2)
R (t,s)
2
R (t,s)
3
R (t,s)
4
=
(3)
(4)
= b - It-s I
•
e
-alt-sl
• f~GO
(5) R5 (t,s)
i
where
b - max.(t,s)
a > 0
eiA(t-s) i(A)dA
is a rational spectral density function with denominator of
degree exactly two greater than the degree of the numerator.
Suppose that
lJ
is induced by the stochastic process
S
(a)
Stew)
•
Jt Ys(w)ds,
(c)
St(~)
•
c(~)
.
(b)
0
or
+
f: Y9(~)ds,
where in (a) t (b) and (e)
(Y t )
functions almost surely in
Stew)
911 t
in
(S5)
•
ft
defined by
Y (w)ds
s
[O,b1.
is a measurable stochastic process with sample
L [O,bl.
2
c
is an a.s. finite random variable •.
Assume that one of the following two conditions is satisfied:
N
(B)
are independent;
and all
(i)
t
[O,b].
in
For
lJS+N
S
...,
S
is Gaussian with
St
independent of
(A)
N
v
Sand
for
<!
t
One then has the following results:
defined as in (a),
N
defined by (1), (3) , (4), or (5),
~.
(ii)
For
S
defined as in (b) ,
N
defined by (2) - (5),
lJS+N ...
~.
(iii)
For
S
defined as in (c) ,
N
defined by (3) - (5),
lJS+N ...
~.
These results are unchanged if any of the
by a positive real scalar.
variance function
v
~)
-
N covariance functions are multiplied
(The result given in (i) for the Wiener process (co-
has been previously obtained under weaker assmnptions [16].)
18
To obtain the preceding results we note the following [l1]:
(1')
The integral operator in
L [O,b]
2
with kemel
R (t,s)
1
h¥ square
root with range containing all absolutely continuous functions on
[O,b)
(2')
that vanish at
0
and have
L [O,b]
2
derivative.
The integral operator with keme1
R (t,s)
2
has square root with
range containing all absolutely cont~uous functions on
vanish at
(3') - (5')
b
and have
L [0,b ]
2
[O,b]
that
derivative.
The integral operators with
R (t,s),
3
R (t,s)
4
and
Rs(t,s)
as kernel have square roots with the same range; this range space contains all absolutely continuous functions on
belonging to
[Otb]
with derivatives
L [O,b].
2
The results stated above now follow directly from the corollaries to the
two theorems.
19
REFERENCES
[1]
U. Grenander,
Ark.
[2]
Mat.~
J. Feldman ,
"Stochastic Processes and Statistical Inference,"
Vol. 1, pp. 195-277 (1950).
''Equivalence and Perpendicularity of Gaussian Processes, II
Math.~ Vol. 8, pp. 699-708 (1958).
Pacific J.
[3]
J. Hajek,
[4]
J. Hajek,
[5]
"On a Property of Normal Distributions of any Stochastic
Process,"
Csech. Math. J.~ Vol. 8, pp. 610-618 (1958).
"On Linear Statistical Problems in Stochastic Processes,"
Czech. Math.
J.~
Vol. 12, pp. 404-444 (1962).
C. R. Rao and V. S. Varadarajan,
"Discrimination of Gaussian Processes,"
Sankhya, Series A, Vol 25, pp. 303-330 (1963).
[6]
M. Rosenblatt, editor, Proceedings of the Symposium on Time-Series
AnaZ.ysis.. Wiley, New York, 1963. See Chapter 11, (by E. Parzen),
Chapter 19 (by G. Ka11ianpur and H. Oodaira), Chapter 20 (by W.L.
Root), and Chapter 22 (by A. M. Yaglom).
[7]
L. A. Shepp,
"Radon-Nikodym Derivatives of Gaussian Measures,"
Stat.~ Vol. 37, No.2, pp. 321-354 (1966).
Ann. Math.
[8]
Yu. A. Rosanov, "On the Density of One Gaussian Measure with Respect to
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[9)
Yu. A. Rosanov, "On Probability Measures in Functional Spaces Corresponding to Stationary Gaussian Processes," Theo1'lJ of Prob. and
AppZ.ic... Vol. 9, pp. 404-420 (1964).
[ 10)
E. Mourier,
[ill
K. R. Parthasarathy, ProbabiUty Measures on Metria Spaces"
Academic, New York, 1967; Chapter lit Section 1.
[12]
c.
"Elements Aleatoires dan un espace de Banach,"
Ann. Inst. H. Poincare" Vol. 13, pp. 161-244 (1953).
R. Baker,
"Mutual Information for Gaussian Processes,"
pp. 451-458 (1970).
SIAM J. on AppUed Ma:thematies" Vol. 19, No.2,
[13]
C. R. Baker,
"On the Deflection of a Quadratic-Linear Test Statistic,"
IEEE Trans. on Information The01'lJ" Vol.
IT 15, No.1, pp 16-21
(1969).
[14]
C. R. Baker,
"Simultaneous Reduction of Covariance Operators,"
SIAM J. on AppUed Mathematics" Vol. 17, No.5, pp. 972-983 (1969).
[15)
E. J. Kelly, I. S. Reed and W. L. Root, "The Detection of Radar Echoes
in Noise, Part I, tI SIAM J. on Applied Math." Vol. 8. pp. 309-341
(1960) •
[16]
T. T. Kadota and L. A. Shepp, ItCondit1cms for Absolute Continuity between a Certain Pair of Probability Measures," to appear,
20
Zeitschzwf-ft j'fI:p Wahmchein1.ichkeitstheoztie und 1Je1fJJanate Gebiete.
[11]
C. R. Baker, nOn Covariance Operators," Mimeo Series report 11112 (to
appear, September 1910), Department of Statistic:s, University of
North Carolina at Chapel Hill.
-.p'