This researah was supported by the National, Science Foundation under
Grant GU-2059 and by the u.S. Air Forae Office of Saientifia Researah under
Contraat AFOSR-68-1415.
JOINT MEAsURES AND CRoss-COVARIANCE
OPERATORS
by
Charles R. Baker
Department of Statistias
University of North CaroZina at Chapel, HiZZ
Institute of Statistics Mimeo Series No. 728
Janwvr.y,
1971
it
Jon·IT I"lEASURES AND CROSS-COVARIANCE OPERATORS
Charles R. Baker
University of North CaroUna at Chapel, BiZZ
ABSTRACT
Let
field
f
HI
1
HZ) be a real and separable Hilbert space with Borel a-
(resp.,
(resp.,
f
2
),
and let
(HIXH ' f xf
Z 1
space generated by the measurable rectangles.
tween probability measures on
projections of such measures on
(HIXH Z '
f
1
(HI' f 1)
x f ),
2
and
2
)
be the product measurable
This paper develops relations bei.e., joint measures, and the
(HZ' f 2 ).
In particular, the
class of all joint Gaussian measures having two specified Gaussian measures as
projections is characterized, and conditions are obtained for two joint
Gaussian measures to be mutually absolutely continuous.
The cross-covariance
operator of a joint measure plays a major role in these results, and these operators are characterized.
JOINT l"lEAslJRES AND CROSS-COVARIANCE OPERATORS*
Charles R. Baker
University of North CarooZina at Chapel HiZZ
INTRODUCTION
Let
HI
(0,
duct
(resp., HZ) be a real and separable Hilbert space with inner pro-
')1
(0,
(resp.,
·)2) and Borel a-field
r1
(resp.,
denote the a-field generated by the measurable rectangles
Define
= {(~,~):
HlXHZ
~
2 in HI'
in HZ}'
Hl XH 2
k(2,~)
product
=
k~).
(k2,
[.,.]
HlXHZ
defiried by
AxB,
AEr 1 ,
(2,~)
[(2,~), (t,Z)]
= (2,t)1
+ (~,!)Z;
moreover, the
Let
11'//1
the inner product, and let
inner product.
~)
is a separable Hilbert space under the inner
sets under the norm obtained from this inner product generate
/1·ll z)
BEr 2 •
= (2+Z,
+ (!,X)
ope~
(resp.,
r 1xr 2
Let
is a real linear space,
with addition and scalar multiplication defined by
and
r 2 ).
denote the norm in
III ·111
HI
[1].
(resp., HZ) obtained from
denote the norm in
A probability measure on
r 1 xr 2
HIXH Z obtained from the
(HlxHZ,rlx~)
will be called a
joint measupe.
A probability measure
fHi II ~ /1/ dfl i (~)
defines an operator
I Hi (~'~)idfli(~)
and
Bi
in
on
<
Hi
(i
=1
or 2) that satisfies
(*)
00
and a mean element
~i
of
Hi
(Bi£'~)i = I H (~-~i'£)i (~-mi'~)idf.li(~);
i
by
Ri
(mi ,2)i =
is a aovaro-
ianae operator; i.e., it is linear, bounded, non-negative, self-adjoint, and
trace-class.
fl
i
is Gaussian if the probability distribution on the Borel sets
* This researoh was supported by the National Science Foundation under
Grant GU-2059 and by the U.S. Air Force Office of Scientific Research under
Contract AFOSR-68-1415.
2
of the real line induced from
is Gaussian.
If
iance operator
measure [2].
~i
~i
by every bounded linear functional on
Hi
is Gaussian, (*) is satisfied; moreover, to every covar-
E and element m in Hi there corresponds a unique Gaussian
All measures on
(Hi,r )
i
considered in this paper are probabil-
ity measures that satisfy (*).
We are interested in determining relations between joint measures and
their projections on
wered:
(1)
(Hi,r ).
i
In particular, the following questions are ans-
What is the relation between the covariance operator of a joint
measure and the covariance operators of its projections?
Gaussian measures
~i
on
(Hi,r ),
i
i
=
set of all joint Gaussian measures having
(3)
1 and 2,
~l
and
(2)
Given two
how can one characterize the
~2
as projections?
What are conditions for equivalence of two joint Gaussian measures, given
in terms of operators on
volve
given.
a~88-aovarianae
H1
and liZ?
The answers to all three questions in-
operators, and a characterization of such operators is
3
JOI NT
r'bsURES
A probability measure on
Suppose that
measure on
(HI,r 1)
induced from
].lxy
by the
Similarly, the projection
by the map
].lxy
].lxy
is the probability
].lX
r 1 xr 2 /r
measurable map
1
is the measure on
].ly
P2 , PZ(.!,X) = X·
eral, not be a unique joint measure having
notation
will be called a joint measure.
is a joint measure; the projection
].lxy
PI(.!,X) :: 35·
duced from
(HlxH2,flxr2)
].lX
(H ,r 2)
2
PI'
in-
Note that there will, in genand
].ly
as projections; the
is used to relate the joint measure to its (unique) projections.
A joint measure
].lxy
is Gaussian if the probability distribution on the
Borel sets of the real line defined by
=
is Gaussian for all
(~,!) in
H XH
I
2
(£,!)
HIXH .
in
Z
p(£,!)
is clearly Gaussian for all
if and only if the distribution
Po (£,!)
on
defined
by
is Gaussian for all
].l;cr
(u,v)
..........
will have a covariance operator
Rxy
mxy in Hl xH2
and a mean
if
(**)
this is always the case if
].lxy
is Gaussian [2].
We will assume that (**) is
satisfied for all joint measures considered in this paper.
since if
with the assumptions for the measures on
measure with projections
and
This is consistent
then
is a joint
4
= JR
=
so that
~XY
JHI
I
xH
11211/
satisfies (**) if and only if both
Given a j oint measure
~XY '
we will use
2
~X
and
~y
satisfy (*).
J?'x and mx (resp., By and my) to
denote the covariance operator and mean element of the projection
~).
~X
(resp.,
5
CROSS-CoVARIANCE OPERATORS
~XY
Suppose
on
is a joint measure satisfying
(**).
Define a functional
G
H1XH Z by
~
(resp.,
~),
variance operators of
~X
For fixed
G is a linear functional on
H
Z
Moreover,
and
Riesz ' theorem a unique element
every !
in
HZ.
such that
by
g~
in
Similarly, for fixed
G(~,~)
= ~. .ExY
.Exy~
Hence, for fixed
~Y.
= <Sv'~>l
...,
HZ
such that
~€HZ
for all
G(~,~)
HZ'
~€Hl.
Define a map
BxY*:
(~'.ExY*~)z
tor
BxY
Hl~HZ
for all
is defined by
u Hl
and
for
S
~
EJcr: HZ~Hl
is tmique •
is clearly linear, and is botmded since
=
Clearly
= <g~,~>Z
there exists a unique element
is single-valued, by the fact that
is defined everywhere in
there exists by
~,
~
BxY*~. g~.
in
HZ.
IQ(!,~)lz
II~II/
Thus
We define
will be called the cross-covariance operator of
~XY.
A partial
characterization of the cross-covariance operator was given in [3] for the case
where
Ex
and
.By are both strictly positive, HI = HZ' and
duced by a map from a probability space into
(HxH,fxf).
zation will be extended, and without these restrictions.
~XY
was in-
Here the characteri-
6
~X
Let
range(Sx)
of
(resp.,
range(B).
THEOREM
I.
~y)
(resp.,
HZ
be the projection operator mapping
onto
range(By»,
where
(A)
If
BxY
has a representation as
X is a unique bounded linear operator such that
and
X
= ~xX~Y'
is a bounded linear operator of trace class, then there
exists a joint Gaussian measure
J.I
Xy
such that
B is the cross-covariance op-
XY '
~ be any fixed element in range(ByJj), with !
PROOF:
(A)
of
satisfying
HZ
denotes the closure
is a joint measure with a covariance operator and mean
J.I}"'Y
BxY = Sx~Y-RyJj, where
X: HZ-+-H l , Ilxll ~ 1,
(B) If B: HZ-+-H
l
J.I
onto
We then have the following result.
element, then the cross-covariance operator
erator of
range (B)
HI
Let
By~! =~.
f~
Define a linear functional
any element
range(Sx~)
on
by
=
IH
l
xH
2
(~-mx'~)l
= <~!'~)l'
(X-my'!)2
all
f
Since
~
~
€
dJ.lxy(~'X)
HI'
k
range (By 2)
is bounded on
can be extended by continuity to a bounded linear functional on
(= ~X[Hl])'
Note that the extension has norm
there exists a unique element
~
in ~x[Hl]
h
in
~x[Hl] such that
Define a map
and
is defined for all
II Y' ~ III ~ II ~ 1/ •
the domain of
y
~
XI
2
uity to a bounded linear operator X
for
range(Bx)
By Riesz' theorem,
f~(E)
V'· H2-+-n 1
,,-'
= <h'~)l
y'~
by
for all
= h. y'
is clearly linear and single-valued,
and is bounded because
~ = ~xffy~
I I~I 12 ,
~
and thus
in ~y[H2]'
Ilyll
can thus be extended by contin-
defined on
~
1,
and
~y[H2];
note that
(w) = (Vs,w)l"
f S'"
~""""I
We extend
"-
to all of
HZ
by defining
~=
Q for
~
in
(~y[HZ])
l.
.
7
Thus, for any
fft, (Bxt~) '"
in
~
H2 ,
<!xY~'~>1
for
~
-= By~,
t
<':!!:tyt!:.;£?~>l'
•
and for any
in
~
H ,
l
one has
BxY = BxtYByt, Illll ~
so that
1,
and
Y. = F.xEy ·
and Q = fxQf y • Then
l~ =~,
all
BxY = B.!Q....'Ryt,
y. is unique, suppose that
To see that
~
(Y,-Q)By
in £y[H 2 ].
1:
1:
~ = Bx2(l-Q)By2~
l~
Since
~€H2'
= Q, all
= Q~ = Q
~ ~
for
with
IIQII ~
1
so that
f y [H 2 ], l = Q on
H2 •
(B)
!2
2
E = Q!,
By the polar decomposition theorem [4],
where !:
H2~H2'
= (B*E)t, and g: H2~Hl is partially isometric, isometric on fr[H 2 ] and
zero on
~
(fr[H 2 ]) ,
with
H with range equal to
2
Hilbert-Schmidt.
on
sure
= range(g)
range(!».
Further,
1]*'"
Since
range (gxlg*),
2
= (UT 2U*)tW*T.
R'" UT
I"ttJI"V
/"'oJ
I"Wt"'W
g
(fr
= the
is trace-class,
I"V
_
range(~) c
and with
on
(H ,r 2 )
2
with covariance operator!
~X(A)
(Hl,r 1),
= ~y{v:
X:
Q~€A},
mean element and covariance operator Bx
The map
range(!)
"""'"
are
=
is self-adjoint and Hilbert-Schmidt, there exists a Gaussian mea2
H2~Hl
and null mean element.
by
Xv
.....
measurable as a continuous map, and thus induces from
~X
and UT
/I'0o"I
as the identity map, and
sure
projection operator in
~(Q!2g*)t for a partially isometric
isometric on
range <1~*B) • Thus
Since !
range(Q)
(X,Y): H
2
thus induces from
~
H xH 2 ,
l
a measure
~X
=
(~,~),
2
is
(HlxH2,rlxr2),
I <~'~>l <~'~>2 d~xy(~'X)
= I <X,g*~>2 <X'~>2 d~y(X)
H
H xH
l 2
X is
a probability meais Gaussian, with null
= Q!2Q*.
(X,y)(~)
on
A€r 1 •
~y
= """'"
Uv.
De-
r 2 /r 1 xr 2 measurable,
defined by
8
= <~-u*v u) 2
NY~
Hence
~XY
BxY = £By • £X2 = B.
is Gaussian.
~,~
Finally, it is clear from the definitions that
This completes the proof.
9
U:>VARIANCE OPERATORS FOR JOINT flEAsuRES
Suppose that
is a joint measure satisfying (**).
~xy
We proceed to de-
termine the relations between the covariance operator and mean element of
and the covariance operators and mean elements of the projections
1:
Let
~
and
PROP.
Let
denote by
.Bx
PROOF:
=
and
fH xH2111 <'H,~) 11I2d~xy<'H'~) <
I
be the covariance operator and mean element of ~xy' and
and .t!!x
By and my)
(resp. ,
~x
(resp.,
(.BxB + BxY~' By~ + ByxB)
It is clear that
<.t!!x'B)1
mXY
=
fH
=
fH
Then,
for all (B,~)
(.t!!x,my);
mxy
~
in
HlxH2·
(.t!!x,my),
and
for example,
<~'B)l d~x(~)
I
I
=
~y).
the covariance operator and mean
xH
2
[(~,X),
d~xy(~,X)
(B,Q)]
=
To describe the covariance operator
null mean element.
[~(B'~)'
~.
be a joint measure such that
element of the projection
Rxy(B'~)
~x
~xy'
(!,!)]
~,
we can asswne that
1lxY
Then
•
IH1XH
[(~,I), (B,~)][(!,I),
(!,!)
d~xy<!'I)
~ IH1XH2{(!'»)1 + (I.I)2} t~'!)l + (I'~)2~"xy(!.I) ~ (~»'!)l +
(BxY~ ,!) I + <.BxY! ,JA) 1 + <By! ,y.) 2
One thus sees that
.Ex'
•
[(.BxB +
BxYY., ByY. + Buy), (! ,!) ]•
By and .BxY completely characterize
~.
has
00.
10
CHARACTERIZATION OF joINT GAUSSIAN r·1EAsURES
One can now characterize the set of all Gaussian joint measures having
given Gaussian measures
THEOREM
2:
(H Z,r 2 ),
and
Suppose that
respectively.
as projections.
and
Let
are Gaussian measures on
Ex and Ex (resp., By
~X
covariance operator and mean element of
m as mean element, has
and
~,
A joint Gaussian measure
(A)
R(y,~)
= (Ex,!;! +
lJ
and
X
lJ
(resp.,
having
and~)
and
denote the
~y)'
R as covariance operator and
m""
as proj ections if and only if
y
Exy~, .By~ + ~,!;!)
for all
(mx ,~)
where
.BxY ""
Ex~~Ry~ for a bounded linear operator
Let
(B)
::;
II~II
1.
HZ
be any bounded linear operator mapping
~
by
Z 1 by
R..-__ : H -+H
Define an operator
N~Y
(Ex,!;! +
k
R..-__ "" R..-_7• VR..-_"2
N~
BxY~' By~ +
N~ ~y
Ex;,!;!)
H1 , with
and define
into
,
for all
(,!;!,~)
in
R is then a covariance operator, and the Gaussian joint measure having
H1XHZ'
R as covariance operator and
and
as mean element has
as
projections.
PROOF:
(A)
If
is a joint Gaussian measure with projections
lJ
then the assertion on the form of
Conversely, suppose that
lJ
R and
and
m follows from Prop. 1.
is a joint Gaussian measure with covariance
operator and mean element having the form given in the statement of (A).
pose also that
lJ
has projections
covariance operators
Bzw~' Bw~
Bz and 1:11 and mean elements !!1z and J!1w' The pro-
+ Bwz,!;!)
covariance operator of
{.!:!n}
and that these measures have
and
m=
jections must be Gaussian, and by Prop. 1,
(Bz,!;! +
Sup-
be a c.o.n. set in
for all
lJ.
(,!;!,~)
in
It is clear that
H ;
1
for any
(!!1z'~)
H1XHZ'
and also
where
Bzw
and
,!;! in H1 , one has that
R(£,~)
=
is the crossNow let
11
[R(~,Q), (~,Q)]
= <Bx~'~)l = <Bz~'~)l for each ~ in
Bz· Similarly, .By = Sw· Hence, llX = llZ and lly =
.Ex •
{~}, so that
llW'
by the unique
correspondence between a Gaussian measure and its covariance operator and
mean element.
It is sufficient to show that the operator
(B)
erator, with
R as defined in the theorem, and with y any bounded linear
operator mapping
HZ
into
Hl
IIY'I
with
R is defined everywhere in HlxH Z'
R is also bounded, by the closed-graph
[R(~,~), (~,~)}
R is non-negative,
To see that
<.By~,~)Z
s 1R is linear and self-adjoint; since
It is straightforward to verify that
theorem.
R is a covariance op-
= <.Ex~'~>l +
2<BxY~'~)1 = IIBx~~lllz + 11.By~~"/ + z<Y1.y~~'Bx~~)l
~ 11.Bx~~1I12 + 1I.By~~II/ - ZIIBx~~11111.By~~112 = (11~~lIl - 1I.By~~,,)2 ~ o.
It remains to show that R is trace-class. Let {u }, n = 1,2, ... be
""'11
+
c.o.n. in
{(~n,Q),
H ,
l
and let
n=l,Z, ••• } u
{v},
""'11
{(Q,~),
n
= 1,2, •••
n=l,Z, ••• } is c.o.n. in
HZ.
Then the set
H1XH '
Z
and
n
n
=
Thus,
having
lJ y
be c.o.n. in
Bx + Trace
Trace
.By.
R is a covariance operator, and by (A) the joint Gaussian measure
R as covariance operator and
(Elx,my) as mean element has
lJ
X
and
as projections.
CoROLLARY,
Let
R be a bounded linear operator in
bounded linear operators
.B3 : HZ-+H l ,
H1XH Z '
PROOF:
such that
H1XHZ.
If there exist
Bl ,.:8Z and .:83 ' wi th Bl : Hl-+H l , .BZ :
R(~,~) = (Bl~
+
B3~' .Bz~
+
B3*~)
for all
then these operators are unique.
Follows directly from the second part of the proof of (A).
HZ-+H '
Z
(~,~)
and
in
12
FURTHER PROPERTIES OF TI-lE CoVARIANCE OPERATOR
This section contains more details on the covariance operator
joint measure
~,
Rxy for a
Included is an explicit expression for the square root of
~xy.
and a description of spectral properties of operators related to the
cross-covariance operator.
The following result will be used in this and succeeding sections.
LeM"IA 2
[5].
Suppose that
H is a r.ea1 Hilbert space, and that
E1 :
are bounded linear operators,
the projection operator mapping
range(B *».
z
Then
range(B )
l
linear operator Q:
Hl~H2
C
range(B )
2
define
Q: Hl~H2
to show that
is closed.
CoROLLARY
(a)
B1
such that
The !Iif ll part is obvious.
Q~ = Qfl~ = ~Z~
~l
Let
(resp.,
range (B *)
I
and
B2
~2)
be
(resp.,
if and only if there exists a bounded
=
SZQ, Q =
Suppose that
when
~2Qfl
range(Sl)
B1~ = S2~.
= Qfl·
C
range (B 2 ) ,
and
It is straightforward
Q is single-valued; Q is obviously linear; and one can show Q
By the closed-graph theorem,
Q is bounded.
[5] •
Range(B )
I
I IB1*B I12
C
range(B )
2
there exists a scalar k
<=>
~ klIB2 *BI1 2 for all ~ in H
bounded linear operator Q:
(b)
H2~H.
(resp., HZ) onto
HI
PROOF:
by
E2 :
Hl~H,
B1
Range(!l)
= range(Bz)
such that
00
such that
there exists a
BIB I
* = B2QB2*.
there exists a bounded linear operator
<=>
having bounded inverse with
bounded linear operator
H2~H2
<=>
<
(R R
""1""1
*)~ = (R""2""2.::s
R *)~n
<=>
there exists a
!
having bounded inverse with !lBI*
A
is partially isometric, isometric on
=
(B2B2*)~!(B2!2*)~·
(c)
(R R *)~
""1""1
= .-RIA*,
'-
where
""
g
13
In many applications, such as determining equivalence of two joint
Gaussian measures, one must verify conditions involving the square root of a
covariance operator.
In this
section~
we give an explicit representation for
the square root of the covariance operator of a joint measure
Let
RX@y
sure for
~X
denote the covariance operator of
on
and
@ ~y'
~xy'
the product mea-
the cross-covari-
For
ance operator is the null operator, so that
=
Hence
k
~@y2(2'~)
=
(Bx.l:!,By~)
+
~
Ex~y.By,
y': H2~Hl'
(~, Y.*.l:!)'
~
2
~
(Bx 2,By ~),
(BxY~' Byx~)
=
II y. II
and one can write
RX@y(2'~)
~ 1)
~
as
+ (E.!Y.J1y'~~, By~y.*Bx.~.l:!)
~@y~ VRX@y~ (.l:!,~)'
= RX@y(.l:!,~) +
V is a self-adjoint bounded linear operator with
Rxy(2'~)
=
(where
BxY =
with
V
)
(.l:!,~
=
III ViII = 11y.11,
as can be easily verified.
We have established the following result.
partially isometric operator, isometric on
space of
~,
and
1
range (R xy ) ,
is the identity operator in
and zero on the null
HlXH2.
The second part of this result follows from the corollary to Lemma 1, and
the fact that
range(Rxy )
= range(Rxy~).
Note that Prop. 2 and Lemma 1 imply that
with equality if and only if
The operators
1 + V has bounded inverse.
I+V and V,
defined above, play an important role in the
study of equivalence of joint Gaussian measures.
spectral properties of these operators.
We proceed to examine the
Recall that the set of limit points of
the spectrum of a self-adjoint bounded linear operator consists of all points
of the continuous spectrum, all limit points of the point spectrum, and all
14
eigenvalues of infinite multiplicity.
The identity operator in
H
l
and the
identity operator in H will both be denoted by the symbol !; the appropri2
ate space will be clear from the context.
THEOREM
~*~
3.
~* has a C.o.n. (in H ) set of eigenvectors if and only if
(a)
l
has a c.o.n. (in H ) set of eigenvectors.
2
V(~,~)
(b)
V(~,-~)
= A(~,~)
= -A(U,-V). V
<=> ~*£
= A2£~
Y*~
= A2~,
-V*u- = AV-
and
<=>
has a c.o.n. set of eigenvectors if and only if
~*
has a c.o.n. set of eigenvectors.
~
(c)
V is compact if and only if
is compact.
(d)
1 + V is compact if and only if both
H
and
l
Il
2
are finite-
dimensional spaces.
(e)
all
If there exists a scalar
in
£
range, and
(f)
a
- --
range (I-VV*),
then
I+V
on
[l-j2]1
~
a < 1
I+V,
Av
(a)
=
VV*u = A2 u
n-n'
-- -n
for all n such that
so that
contains a set that is c.o.n. in
Then
-V*Vv-- = -
each has closed
I-VV*
' 2
I+V if and only if l-(l-a)
c.o.n. in
Hl • Define
A2n-n
v • Now suppose
V*Vv =
- --n
for all
v •
-n
Then
U
and
-A(-~)
= ~*~
~
Hence
and
and
V*u
--VV*u- = A2--
for
{v} u {null space of
-n
H • The converse follows by symmetry.
2
O.
(b)
<=>
- --
and
{u }
-n
with
O.
An '"
such that
y*,
I IY*.YI 1/ : :; all~lll2
-I-VV*.
--
Suppose
is in the null space of
y*y}
/'OW
t"<IJ
is a limit point of the spectrum of
V*u
- -n
that there exists
n-n
I-V*V t
f'toJ
range (I+V).
is a limit point of the spectrum of
PROOF:
such that
<=> V(y,-~)
= -A(£,-~).
If
-=
AV
{(u ,v)}
-n -n
H xIl , then {u } must be complete in H ,
-n
l
l 2
has a complete set of eigenvectors if V has a complete set.
is a complete set in
so that
--VV*
Conversely,
define
~
by
V*u
- = --
AV
if
A '" 0,
and
-v· -0
if
A = O.
15
part of (b», so that
m
,.,
= ,.,0
if the eigenvectors of ,.,
V*V
,.,
~*~
In this case, the eigenvectors of
are complete in
proof of (a) shows that the non-zero point spectrum of
the non-zero point spectrum of
~*y.
to the null space of
{(Q,~)}
set
If
is identical to
V*V,.,
,.,
{x}
is c.o.n. in the null space of
"'D
above must belong
the
{(~,~)}
derived as above
w* constitutes a complete set in
..."...
Follows directly from (b).
(d)
From (b)J
I+V cannot have zero as the only limit point of its non-
zero eigenvalues, since
A is an eigenvalue of
I+V if and only if 2-A is
I+V is thus compact if and only if range (I+V)
an eigenvalue.
~*~,
V corresponding to the eigenvalue zero.
(c)
is finite-
The above eigenvalue relation also shows that the dimension of
I+V is equal to the multiplicity of the eigenvalue
the null space of
only if
H1xH2
and
H
2
are finite-dimensional.
(e)
Suppose there exists
range(!-yy*).
is finite-dimensional.
Then,
2
V s a1
<
1 such that
w* s a1,., on
on
range(1-V2).
2:
H1
This occurs if and only if both
I IY*~I 122
V*V
,.,
,.,
..."...
[(1+V)u,u]
equalities yield
a
A = 2.
I+V will be finite-dimensional if and
Hence, the dimension of the range of
and hence
by (a), and the
Thus, the element !
Hence, the union of this set with the eigenvectors
dimensional.
H ,
2
~*.
are eigenvectors of
from the eigenvectors of
are complete in
If
U
is in
::; a1
,.,
s al I~I 11
on
on
range(1-V2),
(l-a~) Illul1l 2 • Suppose that
2
U.L
the above inrange(I-V2);
(1+V)u = 2u. Noting that the null space
of I+V is contained in the null space of I_V 2 , one sees that 1+V ~ (l-a~)1
then either
on
range(I+V).
!-yy*
and
range(1+V)
U.L
or else
This implies that
!-~*~
I+V has closed range.
The fact that
each has closed range follows from the preceding inequalities.
16
(f)
a
exists a normalized sequence
zero, and such that
and only if
If
I+V if and only if there
is a limit point of the spectrum of
(u
v)
~'~n
in
H1 xH 2 which is weakly convergent to
111(I+V-al)(2n'~n)III-+0 [6].
(l-a)u
+ Vv -+ 0,
""n
~n
(I+V-al)(u ,v ) -+ (0,0)
~
(I-a) v + V*u -+ 0,
~
/"OJ
~~
~
and
is bounded away from zero, it is clear that
limi t point for
since
2n
suppose that
vnk -+ 0;
{2n}.
-+ Q for some subsequence
Hence,
Then
k
this contradicts the assumption that
~
(I-a) 2 u
-~n
1 - (l-a)2
{(u ,v)}
~~
V*u
-+
~k
~
if
~
- VV*u -+ Q.
~n
°
~
~
is a
Thus,
and hence
is normalized in
112n I 11 must be bounded away from zero, and thus
is a limit point for
~
must be weakly convergent to zero.
k
~
~
1 - (I-a)
2
!-yy*.
Conversely, suppose that
convergent to zero, with
by
{2 } is a normalized sequence in HI' weakly
n
(l-a)2 u - VV*u -+ 0. Define positive scalars {k}
~
2•
kn 2 = 1 + (1-a)-2 Ilv*u
11
~~
2
quence in
HlXH2,
when
1 - (1-a)2
theorem.
n
~
{k -l(u ,(a-l)-lV*u )} is anormalized sen
~
~
~
weakly convergent to zero, with
[1+V-al](kn -lu , kn -l(a-l)-~*u )
~
"""" "'n
~~n
-+
(0,0).
~~
is a limit point of
a
is thus a limit point of
I+V
I-VV* This completes the proof of the
17
EQUIVALENCE OF JOINT GAUSSIAN f'1EASURES
For two Gaussian measures on a real and separable Hilbert space, necessary
and sufficient conditions for equivalence (mutual absolute continuity) have
been the subject of extensive investigations.
sures are either equivalent (denoted by
~)
It is known that the two mea-
or orthogonal [7][8]; the neces-
sary and sufficient conditions for equivalence can be expressed in terms of the
covariance operators and mean elements of the two measures (see, e.g. [9],
Theorem S.l,or [10]).
Since these results apply to any two Gaussian measures
on any real and separable Hilbert space, they can be used to determine equivalence or orthogonality of two joint Gaussian measures on
(HIXH ' r l,xr 2).
Z
Our objective here is to formulate conditions for equivalence in terms of operators and elements in the individual spaces
HI
and
HZ.
Throughout this section, we will consider two Gaussian joint measures,
llxy
llZW.
and
B/.2~~~
IIIII
and
~ 1.
The cross-covariance operators will have the form
R
~Zw
llXY
= ""L;
R_ tT~~
~lV '
where
Y
has mean
II YII ~ I
are linear,
I
and
has mean
and
the covariance operator of
llX 0 lly'
(Z)
conditions for equivalence of
by Lemma Z below;
?) :f:
range (R X0
range (RZfi!}}> ,
llX 0 lly
~
(3)
llX 0 lly
~
is
'R
"lC0Y
This is
~
range(R
),
X0Y
(1)
pend only on elements in, and operators defined on,
range (Rzll> ,
zw
o llU.
similarly
no restriction, for the following reasons:
as shown previously;
= (mz ,mw) •
m
where
We will also assume that
and
llZW
and
range (Rxy¥)
llZ 0 llW
.t...
llxy
de-
and
if
as one can easily show, and we will see that
llZ 0 llW'
independently of the assumption that
= range(RZ0W~) = H1xH Z•
The two lemmas below are fundamental to our results.
18
LEMv~ 2. [9].
~l
Suppose
~2
and
are two Gaussian measures on the Borel
cr-field of a real and separable Hilbert space,
covariance operator and mean element of
H.
Bi
Let
~i
and
Then
be the
if and only if
m2
(a)
ml -
(b)
Bl =
is in the range of
J; J;
B2 + B2 HB2 ' where
does not have
-1
W is a Hilbert-Schmidt operator that
as a eigenvalue, and
the null space of
is identically zero on
~
B •
2
This is simply a restatement of Rao-Varadarajan Theorem 5.1 [9], slightly
range(B2~); H.
modified to allow for
The extension follows easily from
Theorem 4.1 of [9], and the fact that the support of a Gaussian measure is the
closure of the range of its cOVari&lCe operator [11] (note that
=
range(.B 2'i)
Since the values of ~' on the nullspace of B do not affect
2
J;,
J;
the values of B = B2 +B2~B2 on H, it is clear that if one defines ~
l
.1
J; J;
by !:! = li' on range(B ), and W - Q on range (B ) , then B = B +B !iB •
2
2
2
l
2
2
'"
range (B ) ) •
2
l.E~tJ1A
3,
PROOF:
~x @
Suppose
IH ].lx{15:
~x{x:
€
~X
...,
A}
and
~Z
=0
follows easily that
if and only if
].lW
@
€ A}d~y(l)
(15,l)
(15 ,l)
].ly '" ].lz
,
~y
so that
a.e. d].ly(l)
].lX
'" ].lW-
@ ~y
...,
lJ X
and
~Z @ ~W.
Then
@
and
].lX'" ].lZ
].lX
lJy[A]
@
].ly{y: (15,l)
=
J.ly [A]
=0
].ly '" ].lW.
if and only if
€
=0
A}
It
a.e. d].lX(15).
For the converse, one notes that
We now examine the equivalence or singularity of the joint Gaussian measures
].lxy
and
RZW = v;'Z@v]~(I+T)V·'Z@W~ ' where
~
range(Ex 2)
:k
= range(Bz2)
and
exist bounded linear operators
and
~2
and
He have seen that
].lZtv"
V(~,X)
=
range(ByJ;)
(~,Y*~)
=
~l: Hl~Hl
have bounded inverse, and
and
range(~J~)'
and
J;
EzJ; = EX
~l'
T(~,X)
(Ix,!*~).
If
then by Lemma 1 there
~2: H2~H2
J;
=
such that both
J;
.BvJ = By
~2·
vIe
~l
can then
19
define an operator
H1xH 2 by
in
B
B(g,~)
= (1!1.!;!,1!2~).
B
is linear,
bounded, and has bounded inverse.
THEOR8~
4,
llxy"'" llZW
if and only if
and
(a)
There exists a Hilbert-Schmidt operator W in
(b)
does not have
-1
HlxH2
such that
W
as an eigenvalue, and
= B[l+T]B* + B[l+T]~ W[l+T]~B*
1 + V
[l+V ]J;.
mxy - mZW belongs to the range of Rv,,",yJ.?
A~
(c)
REP~K: Note that condition (a) implies (Lemma 2) that the operator B (defined above) exists and is linear, bounded, and has bounded inverse.
PROOF:
(A)
Suppose
for example,
llX(A)
llX 0 lly ,.... llZ 0 llW;
llxy"'" llZW.
Condition (a) is obviously satisfied, since,
= llxy(Axll).
n
"Z0W
since
This implies (Lemma 3) that
k
= RX0YJ.?BB *~0Y~'
Hilbert-Schmidt and does not have
llXY ,.... llZW
Schmidt,
also implies that
Rxy
-1
as an eigenvalue, by Lemma 2.
~
=
Rz~(l+K)Rz~'
K can be taken as zero on the null space of
Rzw '
Rxy
= ~0:BA(1+K)A*B*RX0YJ,;,
also can be written as
Rxy
= RX0YJ.? (I+V)R X0YJ;
K.
I + V
K is Hi1bert-
where
Hence
an eigenvalue of
1-8B* is
the operator
and
where
-1
M*
is not
= 1+T.
Hence
=
=
where
R J;
ZW
W
= V*KV, 0
the partially isometric operator satisfying
= RZ0W~[l+T]J;D* ' A = (I+T)~V*.
Finally,
llxy"'" llZW
implies that
Condition (b) is satisfied.
mx~ - mZW
belongs to
range(RxyJ;).
20
R
.lIT~
Since
= RX@Y~[1+V]~G* , G
range (Rxy ),
(B)
partially isometric and isometric on
~
}§
= Rz~v
zw
)
KR ZW '
----,.,,::---:- .J.
range(~W).
and zero on
Rzw '
the null space of
(c) implies
thus imply
K = VW1J*,
where
mxy-mZW
range (R xy~ ),
H1xH2.
1-BB* Hi1bert-
Assumption (b) then implies that
V partially isometric, isometric on
K is thus Hilbert-Schmidt, vanishes on
and does not have
in
m in
Condition (a) im-
B, Rz@w = RX@y~BB*RX@Y~'
BB* having bounded inverse.
Schmidt, and
range (R
for some
Suppose conditions (a), (b) and (c) are satisfied.
plies the existence of the operator
~-Rzw
= RX@y~[I+V]~m
mxy - mZW
one sees that
-1
as an eigenvalue.
by Prop. 2.
Assumption
Conditions (a), (b) and (c)
by Lemma 2, and the theorem is proved.
~xy ~ ~zW'
The necessary and sufficient conditions given in Theorem 4 are completely
general.
However, they are stated in terms of operators in
one might prefer conditions given in terms of operators in
H1XH2,
H
1
whereas
and in
H .
2
The next result gives such conditions for equivalence for a wide class of joint
Gaussian measures.
THEOREM 5.
some
a < 1.
(a)
(b)
(c)
~Xy ~ ~zW
Then
and
~x ~ ~z
if and only if
~y ~ ~W
is Hilbert-Schmidt
V-B
TB*
~1~ 2
~
There exists a finite and strictly positive scalar
1B (I-TT*)B*
I-VV* ~ A- ~l
~ ~
~ I
ABt'toJl (I-TT*)B* 1
~
I'"t.I
XB
f"t>J2 (I-T*T)B* 2
~
"*J
""'J
""'J
(d)
I ly*~1 12 ~ al I~I 11 for all ~ in range (!-yy*),
Suppose that
"If'ttJ
"...,
f"IW
I"tJ
f"Iit.J
There exist
f't<J
I"'oJ
elements
~ X-l~2(!-!*I)~*2·
~
in
IDy-~
PROOF:
(A)
Suppose
and implies that
and
f'<tJt'OtJ
I-V*V
~xy ~ ~zW.
A such that
HI
and
k
~
= &/ (~y*~)
in
H2 such that
•
Condition (a) is necessary, from Theorem 4,
1-B8* in Hilbert-Schmidt.
Condition (b) is then implied by
21
condition (b) of Theorem 4.
= range (RZW~ )
S2
To see that (c) holds, one notes that
implies the existence of strictly positive and finite scalars
SI
and
to
SIB(1+T)S*
SI RZW ~ Rxy ~ S2 RZW'
such that
~
for
I-VV*
,..,
,.."..,
and that the null space of .J.-Y*y'
.- .-
~2(!-!*!)~*2. The assumption that
to the null space of
range(I+V)
in
each has closed range.
= range(B[I+T]B*),
range~z[!-!*!]~*z.
range(B[I+T]B*)
and
~1[!-1!*]~*1
N
""JI"oJ
-
I
1
(I+V)~ has closed range,
Hence,
so that
= rangeB"""1 [I~TT*]B*'1
range(I-VV*)
,...., f"ItJ,....,
I~*~I
is identical
2 is bounded
implies, by Theorem 3, that I+V,
range(!-~*)
implies that the range spaces of
so that
This is equivalent
!-~* is identical to
This implies that the null space of
the null space of ""1
B (I-TT*)B*
,..,,..,,.., ,.., 1
and
from Lemma 1.
~ I+V ~ B2 B(1+T)B*, which is equivalent to range([B(I+T)B*]~)
= range[(I+V)~].
and
k
range (Rxy 2)
is closed.
B [I-T*T]B*
-2
,. . , ,. . ,
2
N
N
This
are closed,
range(I-V*V)
,.., ,.., ,.., =.
and
Since these range spaces are all closed, condition (c)
follows.
Finally,
J.?
~,
so that
]..Ixy ,.., ]..IZ\-1
mxy-mZW
belongs to the range of
= RX@y~ [ 1+V] ~m for some
range([I+V]~) = range(I+V).
mxy-mZW
condition (d), since
(B)
implies that
m in
Suppose conditions (a)-(d) are satisfied.
H1XHZ·
This implies
(a) implies that
RX@Y
=
~
Rz@w~ BB Rz@w'
where 1-88* has bounded inverse. The assumption that
II ~*~ liz
!-~*,
~ ex II ~ 111
!-y*y,
for all
and
~
in
I+V have closed range.
range[~l(!-!!*)~*l] = range(!-~*)
so that
range(B[I+T]B*)
implies that
range (!-yy*) ,
and
= range(1+V).
some
ex < 1,
implies that
Condition (c) then implies
range(!-y*y) =
Since
range(~Z[!+!*!]~*Z)'
J.>
range([I+V]~)
is closed, this
range([I+V]~) = range([B(I+T)B*]~), so that by Lemma 1 there
exist finite and strictly positive scalars
This is equivalent to
SI Rzw
~
RxY
s: S2Rzw
SI' S2
such that
which implies
range(Rxy~) =
22
range(Rzw~).
Thus, there exists a bounded linear operator
on the null space of
away from zero on
RzW'
~
with
= Rzw[1+K]RzW'
K such that K = 0
1+K is bounded
and
range (R ) •
zw
One now has
I + V = B(I+T)B* + BAKA*B*
AA* = 1+T,
where
and
1-BB*+V-BTB*
or
= 3AKA*B*.
B has bounded inverse,
Since
1-88* is Hilbert-Schmidt, condition (b) implies that AKA* is Hilbert-
Schmidt.
K contains the null space of RZW '
The null space of
range(K)
c
range(Rzw ).
Rz~ = A*Rz~~' so that range(K)
But
II ~ 11 2
The assumption that
in
range (A*A)
= range(A*).
Hence if
and some
k
{(,Hn'~n)}
L III AKA*(,Hn '~n) 111 2
;;:
k
n
so that
>
O.
Moreover,
range(A*A)
is a c.o.n. set in
~K:
for all
= range[(A*A)~]
range(AA*),
then
L IIIKA*(.Hn'~n) 111 2
n
-1
as an eigenvalue (the
1+K is invertible).
mxy-mZW
Finally, it is easy to show that condition (d) implies
to the range of
range(A*).
II ~ 111 for ~ in
IIIA(~,~) 111 2 ;;: !sill (,H,~) III
K is Hi.lbert-Schmidt and does not have
latter because
c
is bounded away from
range(!-~*y), and condition (c), imply that
(,H,~)
so that
Jj
~y •
Note that if
This concludes the proof.
Rxy
= RZW'
and either
J!!x = lDz
or
dition (a) of Theorem 5 is necessary and sufficient for
In the case where
belongs
~ZW
= ~X~Y'
my = Jaw'
then con-
~xy ~ ~ZW.
simple and general condition for equiva-
lence can be given.
THEOREM
Schmidt.
6.
~xy ~ ~x ~ ~y
if and only if
IIYII
<
1
and
Y
is Hi1bert-
23
PROOF: ~y = RX~y~[I+V]RX~y~.
~
only i f
~*
if
From Theorem 3,
is Hilbert-Schmidt, and
has
+1
as an eigenvalue.
and only if
has
V has
-1
y
If
V is Hilbert-Schmidt if and
as an eigenvalue if and only
IIyl I = 1
is Hilbert-Schmidt,
if
as an eigenvalue, since the set of limit points of
1
the spectrum of a self-adjoint compact operator contains only zero.
The result
follows from Lemma 2.
CoROL.LARY:
lJ xy "" lJ ZW
(a)
lJ
(b)
Ilyl\
II!II
(c)
PROOF:
X
,.., lJ
and
Z
lJ
X
~
lJy ,.., lJl-J
< 1
and y
is Hilbert-Schmidt.
< 1
and
!
is Hilbert-Schnudt.
lJ xy .... lJ
(b) implies
implies
if
lJy ,.., lJZ
@
X
~
lJy '
(c) implies
lJ ZW .... lJZ
@
lJ W'
and (a)
lJW-
We give an example of two equivalent joint Gaussian measures which satisfy neither condition (b) or condition (c) of the corollary.
lJ
X
....
lJy
~Z'
by
J.ly[B]
J.l Z{x: EZEB}.
Then
H be any bounded linear operator mapping
and let
= J.lX{~:
for
ReEB}
Also define
llXY
and
=
llX {,e:
(x,\Vx)
,.., ,..,,.., e: C},
J.lZW[C]
=
lJ Z{,e:
(x,Wx)
,.., ,..,.... E C}.
J.l
XY
.... lJ ZW '
and both
However,
!!xy
lJ xy
~
neither
is a partial isometry of
Q nor
~
J.l W by
=
are
are Gaussian if
=
into
H2
EExE*,
HI'
into
is Hilbert-Schmidt if
and
ExY = BxH*) ,
similarly,
HI.
Thus
range(B~)
From the proof of the corollary, it is clear that
(c) (but not both (b) and (c»
lJ\,,[B]
Define
C e: f 1 x r 2
and
H
2
Hl~H2.
J.l ZH by
= B-!JL...Ry~ (By
is a partially isometric map of
and define
2
llXY [C]
Gaussian.
where
BEf ,
Suppose that
where
Q
Bzw = Bz~~Bw~'
IIQII=II~II= 1,
and
is infinite-dimensional.
J.l XY
L
of the corollary is satisfied.
lJ ZW
if (b) or
Z4
APPLICATIONS TO INFORMATION THEORY
The average mutual information
(~~I)
of a joint measure
is defined
as
if
lJXY « lJ X
~
lJy '
and equal to
is clearly finite if and only if
+00
otherwise.
lJ xy ~ lJ
0
AJII(lJXY )
operators, were obtained for
provided
lJ
In [3], necessary and suf-
to be finite, stated in terms of covariance
lJ xy
Gaussian, and
an extension of the work of Gel'fand and Yaglom.
sufficient conditions for
lJXY ' AMI(lJ XY )
Vy. The most comprehensive re-
X
sults on AMI are due to Gel'fand and Yaglom [lZ].
ficient conditions for
For Gaussian
AMI (lJ xy )
HI
= HZ.
That result was
Theorem 6 gives necessary and
to be finite without requiring HI
= HZ'
is Gaussian, and is independent of the results of Gel'fand and
XY
Yaglom.
Our final result is of interest in information theory applications.
HI
= HZ = H,
lJY_X(A)
f(~,~)
f1
= f 2 = f,
= lJA~{(~'X):
= ~-~
and consider the measure
X-~€A}
is a continuous linear map from (HxH,fxf)
"signal-pIus-noise".
u..
"I-X
T.
to
represents "noise",
(R,f».
11
"'x
In in-
"signal" and
Typically, these measures are induced by measurable
mean-square continuous stochastic processes, with
some finite
lJy _X defined by
(this is a well-defined Gaussian measure, since
formation theory applications,
lJy
Let
Of interest is the
AMI (lJ xy )
HI
= HZ = H = LZ[O,T]
for
(the "average information about
the signal obtained by observing signal-pIus-noise") and the equivalence or
orthogonality of
and
one might intuitively
allows one to
and conversely, since
expect that
discriminate perfectly between noise and signal-pIus-noise.
lJy _
X
~
lJy
does not imply
AMI(lJ
XY
) < co,
[13], [3].
It is known that
In the case where
25
= lly-x
lJx-y,x
Hajek [9] has shown that
lJ X'
lJy _
are strongly equivalent; i.e.,
X
and
lJy
Q:'O
is tr>ace-alass and does not have
AMI( lJ
xy
.By-x
= .By~ U+JD.By~,
) <
where
!i
The significance of
as an eigenvalue.
-1
if and only if
co
strong equivalence is partly because one can then explicitly express the
Radon-Nikodym derivative in series form [13], [9].
that not only is strong equivalence of
lJ xy ~ lJ X Q:'O lJ y when
not even imply that
THEOREM 7,
If
and only i f
PROOF:
lJ xy
~
lJ xy
lJy-X,X f lJ y _X
~
lJ
X
0
lJyex
by
Ilmxll = II my II = 0,
and
for
g Hilbert-Schmidt.
= lJX 0
lJyex(A)
0 lJy -> lJy _ ~ lJ
yex •
X
X
Suppose first that
Ex;
hence
g
since
As
lJ y _ ~ lJy
X
then
lJy{(~,X): X-~€A},
then
.Ex = .By~~Ry~,
suppose
is non-negative.
an example where
variance operator
if
Note that
AEr.
.&z
~_3/2
....
-y
and
lJy
Ex = By~g.By~,
Hilbert-Schmidt.
Then
~
lJX
0
lly
IIByII
such that
lJ xy
but
lly_X.l lly,
lJX'
since
i t is true that if
strongly equivalent to
lJy
=
lJyex
N
lly,
< 1,
having
one can take any co-
and define
lJ x
Bx
=~, .BxY
= Ey3/2.
as projections and
and
However,
.Ex = By~LRy~.
Although the result of Hajek quoted above does not hold when
0
R.._
-yex
....
Hence
lJ xy
by Theorem 7,
But
n.
;$
Q is Hilbert-Schmidt.
i i
as cross-covari
ance operator
s equ va 1ent to
.1
lly ~ lJyex => Ryex •
not an eigenvalue of
-1
where
Then the Gaussian joint measure
lly-x
does
lJ
Conversely,
lly-x
lJ xy ~ lJ X 0 lJy
but also that
lJy '
.By + .By~~Ry~, g Hilbert-Schmidt,
By +
not equivalent to
lJy
lJ y _ ~ lJ y •
X
Ex = .By~9.By~
Define
lJ y _
and
X
lJ X'
0
In the following, we show
By
~
By-x'
then
lJ xy
~
lJy-X,X f
lJ X 0 lJy => lJ y _X is
[3].
These results indicate that the concept of mutual information is not
co~
pletely satisfactory, since one may be able to discriminate perfectly between
26
~x
and
~Y-X
while having only a finite value of
AMI(~xy),
or,
~~I(~xy)
can be infinite while it is impossible to discriminate perfectly between
and
~Y-X.
~X
27
REFERENCES
[1]
K. R. Parthasarathy, P'l'obability Measu'l'es on Metnc Spaces,
New York, 1967; Chapter I.
[2]
E. Mourier,
"E1~ments A1~atoires
H. Poinca'l'e,
[3]
C. R. Baker,
Academic,
dans un espace de Banach,"
vol. 13, pp. 161-244 (1953).
"Mutual Information for Gaussian Processes,"
vol. 19, no. 2, pp. 451-458 (1970).
Ann. Inst.
SIAM J. on
Applied Math.,
Gene'l'alized Functions, vol. 4;
Applications of Ha'1'l7lonic Analysis, Academic, New York, 1964.
[4]
I. M. Gel'fand and N. Ya. Vi1enkin,
[5]
C. R. Baker,
"On Covariance Operators," Institute of Statistics Mimeo
University of North Carolina at Chapel Hill,
September, 1970.
Senes No. 712,
[6]
F. Riesz and B. Sz-Nagy,
[7]
J. Feldman,
Functional Analysis,
Ungar, New York, 1955.
"Equivalence and Perpendicularity of Gaussian Processes,"
Pacif~c
J. of Math., vol. 8, pp. 699-708 (1958).
[8]
J. Hajek,
"On a Property of Normal Distributions of any Stochastic
Process," Czech. Math. J., vol. 8, pp. 610-618 (1958).
[9]
C. R. Rao and V. S. Varadarajan, "Discrimination of Gaussian Processes,"
Sankhya, Series A. vol. 25, pp. 303-330 (1963).
[10]
M. Rosenblatt, editor, P'l'oceedings of the Symposium on Time Series
Analysis, Wiley, New York, 1963. See Chapter 11 (by E. Parzen),
Chapter 19 (by G. Kallianpur and H. Oodaira), Chapter 20 (by W. L.
Root), and Chapter 22 (by A. M. Yaglom).
[11]
K. Ito,
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Nagoya Math. JOU'l'nal, vol. 38, pp. 181-183 (1970).
[12]
1. M. Gel'fand and A. M. Yaglom,
"Calculation of the Amount of Information
about a Random Function, contained in another such Function,"
Ame'l'. Math. Soc. T'l'ans. (2), no. 12, pp. 199-246 (1959).
[13]
J. Hajek,
"On Linear Statistical Problems in Stochastic Processes,"
vol. 12, pp. 404-444 (1962).
Czech. Math. J.,