.
1 Ph.D. dissertation under the direction of Stamatis Cambanis
Fall, 1973.
• This research was supported by the Air Force Office of Scientific
Research under Grant AFOSR 68-1415.
SOME RESULTS IN THE THEORY
OF STOCHASTIC PROCESSES1,·
Alan J. Lee
University of North Carolina, Chapel Hill
Institute of Statistics Mimeo Series No. 89'1
September 1973
~\Q
~~
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...
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~~~~.
~~~~
~
ALAN J. LEE
Some results in the theory of stochastic processes.
(Under the direction of STAJ4ATIS
C~tBANIS.)
•
ABSTRACT
Consider a stochastic process {x(t), t€T} of random elements of a
Hilbert space
space.
H, whose index set is a locally compact Hausdorff
The results obtained in this work fall into two broad categories,
first the study of weakly stationary processes and their representations,
and secondly the study of the sample path properties of not necessarily .
stationary processes.
Hilbert
spa~e
In each case, we choose the index set
T and the
H to be spaces appropriate to the investigation in hand.
We first deal with stationary complex valued stochastic processes
indexed by a locally compact topological group G, and consider their
spectral representations.
"inverted"
We show how the representation may be
to obtain an expression for the spectral measure; this re-
sult may be regarded as a weak law of large numbers.
We give a decom-
position of an arbitrary weakly stationary process into mean square
continuous (m.s.c.) and "almost" uncorrelated components, and also show
how the covariance function of a m.s.c. weakly stationary process may be
estimated from the sample values via a strong law of large numbers.
We then turn to a discussion of path properties, and show that any
measurable complex valued process indexed by an arbitrary locally compact Hausdorff space may be approximated as closely as we please in a
certain sense by a measurable process with continuous paths.
This result
is the analog of the well known measure theoretic fact that any measurable
function
i~r
"almost" 'continuous (see e. g. Royden (1963) p. 57).
Next
we return to the group-indexed stationary processes, and find conditions
on their spectral measures that are sufficient for path continuity with
probability one.
•
A special case of such processes are the so-called
band limited processes - we show that group indexed band
1imi~ed
processes
admit a sampling expansion, as do band limited processes over the real
line.
14/e a.lso discuss band limited processes that are not necessarily
stationary, extending an idea of .Zakai (1965).
We characterize such
non-stationary band limited processes and show they admit a modified
sampling expansion.
Finally we deal with the concept of group indexed second order processes having values in a separable Hilbert space.
We prove a general
representation theorem for such processes and from it derive a spectral
representation for stationary processes.
We prove a Hilbert space
analog of the decomposition theorem mentioned above and lastly extend
some of the results on path properties to the case of Hilbert space
valued processes.
.
TABLE OF CONTENTS
1.
INTRODUCTION
1.1
·1
Introduction and summary of
res~lts
1.2 Some notations used in the sequel
2. STATIONARY PROCESSES ON GROUPS
1
3
5
2.1
Stationary processes and their spectral representations
5
2.2
Decomposition of stationary covariances and stationary
processes
16
2.3 Estimation of the covariance function and the strong
law of large numbers.
3. SAMPLE PATH PROPERTIES OF STOCHASTIC PROCESSES
27
40
3.1
Continuous path approximations to measurable stochastic
processes
40
3.2
Sufficient conditions for path continuity of stationary
processes
52
3.3 General band limited processes on lR and the sampling
theorem
4. STOCHASTIC PROCESSES TAKING VALUES IN AHILBERT SPACE
62
83
4.1
Random elements taking values in a Hilbert space
84
4.2
H-va1ued stochastic processes
88
4.3
The sp~ctra1 representation of weakly continuous
stationary H-valued processes
96
4.4
Path properties and sampling of H-va1ued processes
112
APPENDIX
120
REFERENCES
123
iv
CHAPTER I
I NTRODUCTI ON
1.1
Introduction and a Summary of Results.
Traditionally, probabilists have treated stochastic processes as a
collection {x(t),t
a set T
~h~ch
or the integers
£
T} of real or complex random variables indexed by
has usually been assumed to be either the real line
Z, or a subset of these sets.
m
In this work we investi-
gate a more general situation; we allow the random variables x(t)
to
take values in vector spaces more general than the complex plane, and
the "parameter" set T to be a topological space more general than lR
or
Z.
More specifically, we will consider the case when the random variables
of the process take values in a Hilbert space, and the parameter set
is a locally compact abelian (LCA) group.
T
This choice of T is parti-
cularly natural when we are dealing with stationary processes, which
depend for their definition on some binary operation on the parameter
set.
Moreover, the existing theory of abstract harmonic analysis for
such groups enables us to utilize the methods of Fourier analysis so
useful in the study of stationary processes whose parameter sets T
are m. or
Z (briefly processes "over" m. or Z).
Finally, the fact that
m. and Z are themselves LCA groups indicates that the choice of T
as an LCA group is a reasonable generalization of the known cases.
In
a similar way, it is reasonable to suppose that the random variables of
1
2
our "generalized" process take values in a Hilbert space.
Then we have
at our disposal the theory of linear operators which will be used
frequently in Chapter VI.
This case includes the concept of a vector
valued stochastic process, so again is a natural generalization of
known cases.
The results obtained in this work fall into two broad categories:
the first, the study of covariance functions of weakly stationary
stochastic processes and their spectral representations; the second,
the study of the properties of the sample paths of not necessarily
stationary processes.
Chapter II is devoted to the first theme in the context of comp1exvalued weakly stationary processes over an LCA group G.
Section 2.1
is concerned with the spectral representation of mean square continuous
weakly stationary processes; the principal new result here is Theorem
2.1.1 which expresses the spectral measure of such processes in terms
of a mean square stochastic integral •. Section 2.2 gives a decomposition
of an arbitrary weakly stationary process into a mean square continuous
component and a component with "almost" uncorre1ated values, (Theorem
2.2.2), a result derived from an elegant theorem in Hewitt and Ross
(1970).
The last section, 2.3, contains some results on the estimation
of stationary covariances, together with a strong law of large numbers
(Theorem 2.3.1)
Our second theme of path properties is pursued in Chapter III.
In
Section 3.1 we prove-an analog for stochastic processes of the measure
theoretic result that every measurable function is "almost" a
ous function.
continu~
Our context here is measurable complex valued processes
3
over a Hausdorff space T, the main result is Theorem 3.1.3.
In section 3.2 we extend certain criteria sufficient for path continuity for stationary processes over R to the case where G is a
compactly generated LCA group (Theorem 3.2.3) in particular we show that
"band limitedll processes over such a grdup )h~ve continuous paths and
admit a sampling expansion (Theorem 3:2 .4j •. 'Section 3. 3 also concerns
band limited processes; we consider not necessarily stationary bandlimited processes over :R. and extend some results of 'Zaka:i (1965).
Our final chapter, Chapter IV, deals with stationary" Hilbert-space
. valued processes on an LCA group G.
Section 4.l:cOntains some basic
results on Hilbert space valued random elements (random variables), while
section 4.2 introduces the concept of covariance operators for Hilbert
space valued stationary processes.
2.2.2 is given in Theorem 4.2.4.
A Hilbert space analog of Theorem
Section 4.3 deals with spectral repre-
sentations of weakly continuous stationary processes (theorem 4.3.5)
while in our last section, 4.4, we give the Hilbert space analogs of
some of the results of Chapter III concerning path properties and
sampling.
More detailed descriptions of our results and bibliographic information are given at the beginning of each section.
10 2.'
Some Notations Used in the Sequel.
We will always denote the real line by
and the complexmimbers by C.
be written as
written as
-g.
+,
:R., the"lntegers by
Z
The binary operation in a group will
and the inverse of an element
g of a group is
If Uis a subset of a group, -U denotes the set
4
{-g : g
€
U}
and
U + U the set {g+g' : g,g'
(n summands) is written nU.
of A by
U}.
The set U+U+ ••• +U
If A is any set we denote the complement
CA, and we use the sumbo!
proof or definition.
€
0 to signal the end of each
CHAPTER II
STATIONARY PROCESSES ON GROUPS
This chapter contains some results from the theory of stationary
processes indexed by locally compact abelian (LCAl groups.
In section
2.1 we describe a spectral representation of second order weakly stationary processes due to
Kamp~
de Feriet (1948), and show how this repre-
sentation can be inverted to yield the spectral measure of singleton
sets.
We also make a few remarks on harmonizable processes which will
be useful in Chapter III.
The spectral representation applies only to
mean square continuous (m.s.c.) processes.
The decomposition of arbit-
rary weakly stationary processes into two 1.D1correlated components, one
m.s.c. and the other with "almost" tmcorrelated values is the subject
of section 2.2.
Finally, in section 2.3 we give a few results on the
estimation of covariance ftmctions.
The results presented in this
chapter are generalizations of well known results valid for the case.
when the index groups is the real line IR or the integers
Z, such
results may be.found in Doob (1953) or Rozanov (1967).
2.1.
Stationary Processes and their Spectral Representations.
{x(g) : g
€
G}
for all g
€
G and Ex(g+h)x(g)
Let
be a weakly stationary second order stochastic process ,
i.e. a family of complex-valued random variables such that Elx(g)1 2 < 00
;
is independent of g.
We will initially
suppose that the index set G is an LCA group though later we will
6
assume that
G has additional properties.
will assume that the fUnction
that
g
+
Throughout section 2.1 we
Ex(g+gO)x(gO)
is continuous, i.e.
x is m.s. c.; and that Ex (g) = 0 for all g
€
indeed we
G;
assume throughout this work that all processes considered have zero
means.
The function
of the process.
R(g)
= Ex(g+gO)x(gO)
is called the covariance function
It is easy to see that
R is positive definite and is
continuous by our assumption of mean-square continuity.
It follows by
the well known Bochner theorem (see e.g. Rudin (1962) p. 19) that
R
has the representation
(2.1.1)
R(g)
= f (a,g) ~(da)
A
G
where
is a positive measure in the class M(G) of the Appendix.
~
measure
is known as the spectral measure of the process.
~
The
The corres-
ponding representation of the process is due to Kampe de Feriet (1948),
who showed that
(2.1.2)
x(g) has the representation
x(g) =
f (a,g)z'(da)
....
G
where
~(6)
Z is an orthogonal stochastic measure on
for all Borel subsets
60f~.
ed analogously to the real line case.
The
G with
EIZ(6)1
represe~tation
2
=
is estab1ish-
First we note that the characters
....
{(.,g) : g
€
G}
generate a dense linear manifold of
and Ross (1970) p. 211), so the correspondence
L2(G,~)
x(g)<--~
(o,g)
(Hewitt
can be
extended to an isometry between H(x) (the Hilbert space spanned by
the process) and L2(G,~).
The integral (2.1.2) is just the element of
7
H(x) correrPonding to (o,g) , in general a function
[~(a)Z(da)in H(x).
ponds to the random variable
In the case
measure
G=
~
G
-
.
or G = IR, it is possible to express the random
Z(6) in terms of the process
x(t) by means of an integral
representation which is an "inversion" of (2.1.2Y.
{x(t) , t e lit}
$ ~ L2(G,~) corres-
For' example, if
has spectral representation
00
x(t)
=f
e 2 '1TiAt Z(dA)
_00
'for a stochastic measure
Z, then if
Z({a})
= Z({b}) = 0
a.e. the Z-
measure of the interval [a,b] is given by
T
(2.1. 3)
Z( [a, b])
= lim
T-+oo
f
e
-2'11'ibt
-e
-21Tiat
x(t)dt •
-it
-T
If
Z({a}); 0
then
T
Z({a}) = lim
(2.1. 4)
T-+oo
2i
f
e-21Tiatx(t)dt
-T
Our aim in the sequel "is to give an expression simi! ar to (2.1. 4)
for the Z-measure of a singleton subset
{a}
of
G.
We have not been·
successful in obtaining an expression similar to (2.1.3) for the Zmeasure of more general sets.
We first make a few remarks as to the definition of a type of stochastic integral.
We wish to define an integral of the form [f(g)x(g)m(dg)
G
where
f ~ LI(G).
Let us·assume that
stationary measurable process.
Then
{x(g), g
~
G}
is a weakly
It
E J/f(g)1 Ix(g)lm(dg) = IE/X(g) I If(g)lm(d g).
G
Thus by Fubini's theorem the flUlction
[P]
f(g)x(g,w) is in Ll(G,m)a.e.
where we assume all the random variables
on the probability space (n,B,p).
{x(g), g € G} are defined
Hence the integral
~(w) = fX(g,W)f(g)m(dg)
G
defines a random variable except on a set of P-measure zero.
Since we have assumed
2.2.4 that
x(g) to be measurable, it follows from Theorem
x(g) has a spectral representation (2.1.2).
We will need
the following lemma.
Lemma 2.1.1.
Let
f
€
Ll(G)
and let
{x(g), g
(and hence m.s.c.) weakly stationary process on
representation (2.1.2).
(2.1.5)
Proof:
G} be a measurable
G, with
spectral
Then
f x(g)f(g)m(dg)
G
€
=
I f(-a)Z(da)
a.e. [Pl.
C
The integral on the right exists since
2
hence in L2 (G,lJ) when dlJ = EldZ1 . Now
f(a) is bounded and
9
)1
ElfX(g)f(g)m(d g
G
2
<
f
fIX(g)X(h)f(g)f(h)!m(dh)m(d g)
GG
E
J J(~IX(g)12EIX(h)12)~lf(g)1 If(h)lm(dh)m(dg)
<
GG
so the integral on the left side of (2.1.5) has a second absolute mpment.
Thus
(2.1.6)
ElfX(g)f(g)m(dg) -
f f(_a)Z(da)\2
.
G
G
= ElfX(g)f(g)m(dg) 1
2
G
+
- 2Re EfX(g)f(g)m(dg)ff(-a)Z(da)
G
G
Elff(-a)Z(da)/2 •
"
G
Let us now compute the second and third terms on the right hand side
of (2.1. 6).
Consider first the second term.
f
f(-a)Z(da).
Let us write
~(w)
for
Then since
"
G
we can apply Fubini' s theorem and obtain
EfX(g)f(g)m(dg)~(w)
=
fE(X(gJW)~(W)
G
G
=
f(g)m(dw)
f I(a.g)£'( -a)ll(da) f(g)m(dg)
A
GG
•
10
=J
G
=
J <atg>Jf(h)(a.h> m(dh)j.l(d'a)f(g)m(dg)
G
G
f JR(g-h)f(g)f(h)m(dh)m(dg)
GG
by another application of Fubini's theorem, which is justified since
f
~
Ll(G)
and
j.l
is a finite measure.
Consider the third term of the right side of (2.1.6).
Using Fubini's
theorem again, the third term is equal to
il
f (-a)/2j.l(da)
=J
f<a,g)f(g)m(dg)!<a,h)f(h)m(dh)j.l(da)
GG
G
G
= f ff(g)f(h)
GG
=f J
f<a,g-h)j.l(da)m(dh)m(d g)
A
G
R(g-h)f(g)f(u)m(dg)m(du)
GG
o
so (2.1.6) is equal to zero and the lemma is proved.
Remark 2 ~ 1.1.
If the process {x(g),g
~
G}
is weakly stationary and
H(X) of L2 (n,B,p) generated by the x(g) is separable by Theorem 2.2.4, and we may define
measurable, then the closed linear subspace
stochastic integrals of the form
fx(g)f(g)m(dg) for f
V
continuous
,
and V relatively compact as is done in Rozanov (1967) pp. 9-12.
•
These are mean-square integrals in contrast to the path integrals we
have used in Lemma 2.1.1.
The
~ozanovintegrals
satisfy a modified
form of Lemma 2.1.1 - if V is relatively compact then
11
Jx(g) f(g)m(dg)
V
We use Lemma 2.1.1 in the proof of Theorem 2.l.I J it is immaterial
for the purpose of the proof l'1hich definition of the integral we use.
With these facts at our disposal, we can proceed to the proof of the
inversion theorem, which can also be
regarded
Suppose that {x(g), g € G}
Theorem 2. 1. 1.
as
a law of large numbers.
is a wide sense stationary
measurable stochastic process over the LeA group G.
then there exist an increasing sequence
open sets in
(2.1. 7)
o-compac~,
of relatively compact
G such that
Z({a})
= lim m(~
)
n
n~
J x(g)(a,g)m(dg)
(limit in mean square)
E
n
and
~({a}) = lim tm(~n ) f
(2.1. 8)
n~
Proof:
{E}OO
n n=l
If G is
Let U(G)
R(g)<a,g)m(dg)
E
n
denote the class of almost periodic functions on
(See Hewitt and Ross (1963) p. 247).
G.
Then there exists a unique linear
functional M on U(G) with the properties
(i)
M(fg) = M(f)
Vf e: U(G), Vg
function f (h) = f(h-g) •
g
(ii)
(iii)
M(f)
>
€
G where
f
g
is the
"
0 if f
>
O.
M(l) = 1.
The functional
M is known as an invariant mean.
In their
boo~,
Hewitt
and Ross show that the unique functional H can be expressed in the
following way.
If f.€ U(G) is. fixed, then it can be shown that there
is a unique complex number
~
such that given
€ >
0 there exist group
12
elements
gl' •••
'~ €
G depending on
supl~ - ~
(2.1. 9)
g€G
The number
~
r
n k=l
E
such that
f(g+gk)I
< E •
is M(f).
Now if we consider the class Uc(G)
of continuous almost periodic
functions, an even more explicit expression
Hewitt and Ross.
sequence
{En}
If G is
for M(f) is obtained by
a-compact, there exists an increasing
of relatively compact open sets of G such that
= n-+<>o
lim (~)
m n
M(f)
(2.1.10)
J -f(g)m(dg)
E
n
for all continuous
En
f
in U(G).
is compact and hence m(En)
Note that the integral exists since
<
and
co
f
is bounded on En.
We now
apply these facts to prove a lemma essential to the proof of Theorem
2.1.1.
Lemma 2.1.2.
Let
{En} be the increasing sequence of sets introduced
Then if f n (g) = m(~ ) XE (g), where
n
n
function of the set En' the sequence {fn (C/.) }~=l
above.
for all
0. .;
c"
G
.
(the identity of G),
and
is the indicator
converges to zero
equals
1 if -
Proof:
=1
so
..
f (e )
n
G
+
1 as n
+
inCa) =
00.
Assume now that a:f. e..,
G
m(~
) J<a,g) m(dg)
n E
n
then
a
= e ..
G
13
A
is an almost periodic function. so lim fn(a) = MC{a,·)l.
Now {a,g)
Thus it
for any
r~mains
£ >
0
to prove that M({a ••
»)
= 0;
we can find group elements
we do this by showing thay
gl •••• '~ such that
Let N(K,o)
We adopt a result of Jajte (1967) to this end.
neighborhood of ea
A
in G with
N(K,o)
That
= {e
€
N(K,o),
G :I{e.g)
N(K,o) is a neighborhood of
Since a t N(K,o)
a ~
be a
- 11
0
<
Yg
€
ea is proved in Rudin
there exists an element
gl
€
compact
K} •
(1962) p. 10.
K with
Then
l
so
Sn ~ 0 as
n ~ ~.
gv = vg 1
1
Isnl
But
Let n be such that Isnl <
r
= In
E <
1, then if
n
r {a,gv)'
v=l
supl~
{a,g+g >1 = Is I
g n v=l
v
n
< E, 50
< E
by (2.1.9)
o
We may now proceed to the completion of the proof of Theorem 2.1.1.
In Lemma 2.1.1
set
f(g) = m(~ ) XE (g){a,g}.
n
n
Then
f
€
L1 (G)
•
14
and
where
f( -13) = m(~
f
n
J(13 ,g)(~,g)m(dg)
)
E
n
n
is the function of Lemma 2.1.2.
= fn (8-a)
,
Then by Lemma 2.1.1 we
have
. and so .
(2.1.1l)
f"n (l3-a)
~
X{a}(/3)
Ifn(a-a) - X{a}(B)I ~ 2 VI3
E
G
Now by Lemma 2.1.2
V/3
E
"
G
as
n
~
00
,
and
so by dominated convergence (2.1.11)
converges to zero, and (2.1. 7) is proved.
Also
~({a}) = f X{a}(I3)~(d~)
"
G
= lim J.£
n~" n
G
(B-a)
= lim (ro(E l)
n~
f
~(dS)
R(g)(o.,g)m(dg)
n E
n
which
prove~,
o
(2. ~. 8).
We conclude this section with a few remarks on harmonizable processes.
If a second order process
{x(t), t
R(t,s) which can be written as
E
IR} over the real line has covariance
00
= f J exp(2~i(tu-sv»~(du,dv)
R(t,s)
(2.1.12)
_00
where
1-1
15
00
_00
is a complex measure on the plane of bounded variation, then
x is said to be a harmonizable process.
Such processes have spectral
representations
00
= Je2~it~Z(dA)
x(t)
(2.1.13)
_00
where
Z is a random measure satisfying
E Z(A)Z(A')
= ~(AxA').
The integral (2.1.13) is defined by means of an isometry between 11(X)
and the Hilbert space
A2(~)
the integral
exists;
A2(~)
00
consisting of ail functions
$
for which
00
is a Hilbert space with inner product
00
(w,~) = f
00
00
J
w(u)~(v)~(du,dv)
•
00
The isometry in question is given by the correspondence x(t)<-+e2~itu.
These facts may be found in
Cram~r
(1951).
Similar definitions can be made in the case of a group parameter.
Specifically, we will say that
{x(g), g
€
G}
covariance R(g,h) can be written
R(g,h) =
J
J<a,g><8;fi)~(da,da)
A
"
G G
where
~ €
M(GxG) (see the appendix).
is harmonizable if its
16
The spectral representation of a harmonizable process is
x(g)
= I<a,g)Z(da)
'"
G
where
E
Z(~)Z(~I)
=
Jl(~X~I)
for all Borel subsets
~,~I
of G.
We
omit ptOofs;they are similar to the real lirie case.
Harmonizable processes over
IR will be used "in Chapter III, section 3.
2.2 Decomposition of Stationary Covariances and Stationary Processes.
Th~ ~pe}~ri'.1 representations described in tlieprevious 'secaon
to mean square continuous weakly stationary processes.
appl};'oniy
It is thus of
interest to determine how an arbitrary weakly stationary process can be
decomposed into m.s.c. and "otherlf components.
In this section we des-
cribe how an arbitrary process can be split up into two orthogonal parts,
one being m.s.c. and the other with "almost" tmcorrelated values.
As a
consequence of this, we will show that every measurable stochastic process is m.s.c., and thus has a spectral representation.
We begin with
a theorem of Hewitt and Ross (1970) p. 260; we give a slightly modified
version of their proof in some detail because it will be needed in
Chapter IV.
Let
Theorem 2.2.1.
G.
Then
<I>
tinuous and
...
4,q.
....
.,.
"
<I>
be a measurable, positive definite function on
has a unique decomposition
~
= <1>1+<1>2
<1>2
<1>1
and
locally a.e. zero, and
where
<1>2
<1>1
is con-
are positive
....
defiJiite. "
.0
Proof:
on
Let
IE denote the vector space of all complex valued functions
G that are zero except at finitely many points of G.
If
E:
h
17
funct~(>n
denotes the
on
G given by
. = {l
Ell (g)
then every element
U of
g = h
0
g ; h
IE has a canonical expression
n
=. LI C.E
(g)
1 g.
u(g)
1=
where
1
Now define the map
are complex numbers.
p : :IE >f'E + C
by
n
m
r
n m
r
L
L
p(
C..E
,
c!e: ,) =
c.c.!ep(g.-g!) •
i=l 1 gi j=l J gi
i=lj=l 1 J
1
J
Let
lE <p = {u
f
lE: p(u,u) =
IE, and that the map
space of
product on
lE / lE <p.
completing
lE llE cj>
map
Let
V
V
g
g
~'~)H
gl"."~
Hep denote the Hilbert space obtained by
by
Vg(u+IEcj»
Now define a
= ug+lEep , where
ugCh) =,
= p(u ,u ) = p(u,u) = l/u+lE",1/2 •
g g
'l'
is an isometry, and can be extended to an isometry
Moreover, V; = V_ g
g + (V
{u+1E cj>,v+lE ep)+ p(u,v) is an inner
is -linear and onto, and
g
Ilvg (u+E qt:'111 2
Thus
lE cj> is a SUb-
It can be shown that
with respect to this inner product.
Vg : lE/lEt/> + lE/lEt/>
u(g+h).
oJ.
f
and
Vg +h = VgV ' and for every
h
~
€
Hcj>+ Hep.
Hcj>
is positive definite since for scalars cl, •.• ,c
n
</l
G
n
n
n
n
r r c.c:(V
~,~) = L I c.~(V ~,v ~)
i=l j=l
J gi-gj
i=lj=1
J gi
gj
1
1
n
=
1/ . L1 c.1 Vg. ~"
1=
1
2
~ 0 •
the map
and
18
~,n ~ H$ ,
Also, if
that
$
by an approximation argument and using the fact
+ CVg~,n)
is measurable, it can be shown that the map g
measurable.
Now if
€
"is
L1 (G)
J1fCg)(Vg~,n)
/med g)
G
I
~ IIf(g) "Vg~III,lnllmCdg)
G
~ I I~I I
Ilnll flfeg)lm(dg)
<
=
G
so we may define a bounded linear operator Tf
on H$
(Tf~,n) = JfCg)eVg~,n)m(dg)
by
•
G
Consider the map
g
+
eVgTf~,n).
Then
IVgTf ~,n) - eVhTf~,n)1
= Ifevg f~'V -gn
= leTf~,V_gn
- V_hn) I
- V- hn)fegf)m(dg)I
G
= IJ(Vgf~,n)ef(gf-g)
-
fegf~h))m(dg)1
G
By a theorem in Rudin (1963) p. 3, given
borhood V of e
Hence the map
g
+
;,
so Vg C5i
)
§ 5
there exists a neigh-
{Tf~:
f
€
Ll(G),
~
€
H$}'
<
It can
so Vg (5) £ 5 Yg € G. Also Vg* (5)" g 5
g
i.e. Vg reduces 5 Yg € G. If ~ € 5, then the map
VgTf
i
> 0
in G such that if g-h ~ V then II f g-fh II L (G)
I
(VgTf~;n)
is uniformly continuous. Now let 5
denote the closed linear span of
be shown that
e
= Tf
e.
19
g
(Vg ~,n) is continuous since it is the uniform
~
fum
g~
n .
I
c~(V
. 1
1=
1
T
g f 1.
~.,n)·
which are continuous as proved above.
G.
Tf~ E:
n
E:
~ E:
J.
5 and
f(g)m(dg)
n
E:
5, then
=0
G
J.
Tf~
5 • Thus
H4J ' Yf
1
If
'(Tf~,n) = f(Vg~,n)f(g)m(dg) = fo
so
of maps of the
l~mit
E: Ll (G)
=0
since
Tf~ E:
5.
Thus for all
~ E:
J.
5 ,
f(Vg~,n)f(g)m(dg) = O.
A
G
so (Vg ~Jn)
Now let
~1 E:
=0
~
5 and
locally a.e. [m].
= Ee
+
~2 E:
5.
IE 4J'
J.
Then
Then
(Vgl;,~) =
ep (g).
Let
~
= ~l +~2
with
.
From the above remarks it is clear that <1>1 and $2 are positive definite,
$1
is continuous and <1>2 is zero. locally a.e.
If <I> = <l>i+$2
is
another representation with the desired properties, then
$l- 4J i =<1>2-<1>2'
The left side of this equation is continuous, the
right side is zero locally a.e. so that both are zero i.e. the repre-
o
Corollary 2.2.1.
If the Hilbert space
H<I>
has countable dimension,
then <1>2 = O.
;;
20
If H$ has countable dimension, then S~ has a countable
Proof:
dense set {~n}:=l. For each pair (m,n) there is a locally null set
Am,n. with (Vg ~m,~n ) = 0 for g fA.
Let A = m,n
u A.
Then A
mn
m,n
is locally null and (Vg ~m,~n ) = 0 for all m,n, all g f A. Thus for
all
~,n
Thus
I I~/I
€
S~, (Vg
~,n)
.
=
$2 = 0, and
Ilvg;//
lJ>
=0
Vg ~ A, so Vg~
= 0 V~
€
€
S~ n S = {OJ Vg
S~ and so S~ = {OJ i.e.
S=
A.
H$
and
0
is continuous.
We can now state and prove the corresponding decomposition for" a
weakly stationary process over an LCA group G.
As
before H(x)
the closed subspace of L2 (O,B,P) spanned by {x(g) : g
Theorem 2.2.2. ,Let {x(g), g
€
G}.
be a weakly stationary stochastic
G}
process over the LCA group G.
€
denotes
If
~,
the covariance function of x(g),
is measul'{i.ole, .then x(g) can be expressed uniquely as the sum of"!wo"proCesses xl and x2 such that
= 0 Vg,hE
(a)
E xl (g)x (h)
2
(b)
xl
is stationary with continuous covariance
(c)
x2
is stationary with covariance $2 that is zero locally,
G•
$1.
a.e. [m].
(d)
If H(x)
. Proof:
H$
H(xl ) 19 H(x2 ) = H(x).
is separable then x2 =0, i.e.
Consider a map
~
X
is m.s.c •
from the linear span of the r.v's
defined by
~
n
n
(. La.
X (g. »)
III
1=
=
L a.1 € gi' +
i=l
JE
$'
x(g) to
21
~
is well defined and onto
lE
/lE</>, moreover
n
/2 n n __
E . I -I a . x Cg. )
=. l 1 . l 1 a.a.</>(g.-g.)
J. J
J. J
1= 1
1
1= J=
I
so
~
preserves inner products.
Thus
~
= p(la.E
J
J. g. ,Ia.E
1 g.
J.
1
may be extended to an isometry
between H (x) and H</>.
Define xi (g)
~2 e: SJ..
(i)
= ~-lVg~i
where
Ee + :IE </>= ~ = ~1+~2
~l
with
e: S,
Then
E
xl (g)x2 (h)
= (Vg~1,Vh~2) = 0
since
V
g
reduces
S,
which proves (a).
(ii)
(iii)
x.(g)x..
J.
J. (h)
and (c).
E
= (Vg~"Vh~')
= </>J..(g-h)
1
1
xl (g) + x2 (g)
= ~l(Vg~l)
which proves (b)
+ ep -1 (V g~1)
= ~-1 (Vg~)
=
~
-1 (8
g + :IE
</»
= x(g)
,which proves the decomposition.
(iv)
(d) follows from (i) and (iii).
It is also true that'S
= sP{Vg~l
: g e: Gl~ SJ.
= sP{Vg~2
: g e: G}.
By virtue of the isometry, if H(x) has countable dimension so does
J.
H<t> and so S = {a} whence x2 = O. Lastly, we show that the
decomposition is unique. Suppose that xi Cg) + xi(g) is another'
.
22
From the unique-
decomposition of x with the properties (a) - (d).
ness of the covariance decomposition, the covariances of xi
~i'
Let UI be the mapvg;l ~ xi(g), U2 the map Vg~2 ~ x2(g)
and U2 can be extended to isometries between S and H(xi) and
UI
S~
and H(x
U(n)
Z)'
= UI(n l )
uls~
and
For n
€
H~ , let
U be the map given by
U2 (n 2) where nl and n2 are the projections of n
S~. U is an isometry between H$ and H(x), uls = UI
+
onto S and
= U2 •
Now
U(Vg;) =. UI Vg~l + U2Vg~2 = xi(g) + x2(g)
so
X
are just
q>
z(g)
= U-1 •
Thus
xi (g) = U(Vgt;l) = ~-l(Vg~l)
= x(g)
= Xl (g).
Similarly
o
= x (g) •
2
We now apply these results to prove that every measurable stationary
process over G is mean square continuous.
Let {x(g), g
€
G} be any
second order process over G, not necessarily weakly stationary, with
each random variable x(g) defined on the sarne probability space
(n,B,p).
Let M denote the vector space of all complex random
variables on
n.
M is a metric space with metric
d metrizes the topology of convergence in probability.
We will need an unpublished result of Hoffmann-Jorgensen (see also
Cohn (1972)).
Theorem 2.2.3.
We 'give Hoffman-Jorgensen' s proof.
If the stochastic process {x(g),g
then the set M = {x(g):g
;;
€
G}
€
G}
is measurable,
is a d-separable subset of M.
23
Proof:
Let S
such that
denote the class of all stochastic processes over G.:
M above is separable.
process is in the class S.
A € B(G) x B where B(G)
We will show that every measurable
To this end, let xl
and' x2
€
S, and
denotes the Borel a-field of G and
B is
the a-field of the probability space (n,B,p) on which allr.v. 'sare
assumed to be defined.
Set
YA(g,w)
t = {A
Let
limits,
C.
J
£
€
B(G) x B :
Y
= xl(g,w)
(g,w)
€
A
xz(g,w)
(g,w)
t
A •
€
A
S}.
Since S
is closed under pointwise
n
Now suppose A = u B.xC. B. € B(G),
j=l J J
J
B. x C. are disjoint. Choose a measur-
is a monotone class.
L
B where the rectangles
J
J
able partition S1'''' ,Sk of G and subsets TI , ... , Tk
€
B such that
Then
Suppose
g
€
Sk and w € Tk k=l, ••• ,m
g
€
Sk and w ~ Tk k=l, ••• ,m
nl and nZ belonging
to the countable sets Nl and N2 which are dense in {Xl (g) : g € G}
and {x2 (g) : g € G} such that
€
Sk'
g
€ >
0, then there are r.v.'s
i=l,2
Define
new) =
flew)
n2 (w)
In
w E: u Tk
k=l
m
w t u Tk
k=l
;;
24
(2.2.1)
Since M1 and N2 are countable, the family of r.v.'s n of the type
defined above is countable; (2.2.1) shows that the set of such r.v.'s
n is dense, and so YA E
with the rectangles
whenever A is of the form
S
B.xC. disjoint.
J
J
taining the ring which generates
By an induction
Al""'~
the process
argum~nt
it
Thus
E is a monotone class con-
B(G) x B, it follows thatB(G)xB £ E.
follow~
that" if xl"",xn are in Sand
,
is a partition of G x n with each Ai E B(G) x B , then
y given by
for (g,w)
is a process in the class S
in S.
n
u B.xC.
j=l J J
€
A.1
i.e. every B(G) x B simple function is
Since S is closed under pointwise limits and every measurable
process is the pointwise
~~mit
of B(G) x B simple functions, it
follows that every measurable process in in the class S.
0
We now apply this result to prove that every measurable weakly
stationary process is m.s.c.
Theorem 2.2.4.
Every measurable weakly stationary process {x(g),g
over an LCA group
€
G}
G is mean square continuous, and thus has a
spectral representation.
Proof:
By Theorem 2."2.2 it is enough to prove that H(x) is separable,
i.e. has countable dimension.
subset
By Theorem 2.2.3 there exists a countable
GO of G such that the set {x(g) : g
€
GO}
is d-dense in
set {x(g} : g e G}.
Thus if
25
L denotes the linear manifold.generated
by {x(g) : g eGO};T is separable.
It is clear that
We will show that
r
= H(X).
T £ H(X); to show the reverse inequality let
Then there exists a sequence
{~}:=1 in
such that
GO
g e G.
x(g) is the
limit in probability of the sequence
{x(gn} }:=l " and '50 a subsaquence
[\toreover; 'E"lx(~) 12= R(e) so' t-he 1 ·seq-.
x(g~) converges to x (g). a.e. [P].
uenCe x(g~) is uniformly norm~bounded;Jn n.
x(~)
and Stromberg (1965) p. 207,
By Theorem 13.44 of Hewitt
converges weakly to
By II 3.27 of Dunford and Schwarz (1958) p. 68,
linear manifold generated by the
x(~} i. e.
s T.
sp (x (g) : g e G) S T i. e. H(x)
that
and
{x(g) : g e G}
x(g) in H(x).
x(g} is in the' closed
x(g} e T.
Hence
It follows
H(x) is 'separabIe
0
is m.s.c.
An alternative proof to Theorem 2.2.4:
We present below a direct
proof of Theorem 2.2.4 which is an adaptation of a result of M.M. Crum
(l956).
Let
cfl
be the covariance function ofx(g) and
decomposition.
Let
f e L2 (G)
be a positive function.
it follows by Fubini's theorem that
Define random variables
cfl = cfl +cfl
l 2
n(h,w)
its
Then since
f(g)x(g,w) e L (G) a.e. [Pl.
2
by
n(h,w) = f1f(g)x(g,W) - f(g+h)x(g+h,w)12m(dg)
G
If
a
denotes the Haar measure on
G,
then by Plancherel's theorem
;;
n(h,w) = .fll-<a,g)12IF(a,w)12e(da)
A
G
26
where F(a,w) is the L2 Fourier transform of the function f(g)x(g,w).
Moreover,
2
2
!EIF(a,W) 1 e(da) = EJIF(a,W) 1 e(da) = Ej1f(g)X(g,W) I2m(dg)
A
A
G
G
Ilf/l~
=
(G) <He) <
co
2
so
EIF(a,w)1
is a function in LI(G).
Eln(h,w)1
2
= H(a)
say,
Thus
2
= J 1I-(a,h)1 2H(a)6(da).
A
G
Another expression for
Eln(h,wlI
Eln(h,wlI 2 is
2
= E 2J{lf(g) 12j x(g,w)/2
A
- Re f(g+h)f(g)x(g+h,w)x(g,w)}m(dg}
= CI
~(hl
¢(e) - C2 Re
= C1~I(e)
- c2Re~ICh) + CI~2(e) - C2Re~2(h)
A
for positive constants C1 and C2 , Now since H(a) € LI(G) it
2
follows that £II-(a,h),1 H(a)6Cda) is a continuous function of h,
and since
~1 Gis continuous we deduce that Re ~2Ch) is continuous.
But by a result Of, Rudin (1962) p. 18
1~2(h)-<I>2(hr)
so
that
~2
is continuous,
~2
I
~2
Since
is zero and hence
~
2 (e) - Re ~2(h-h'))
is zero locally a.e., it follows
2 2(e)(Re
<
<P
is continuous.
o
27
2.3 Estimation of the Covariance Function and the Strong Law of
Large Numbers. Our goal in this section is to discuss an estimator of the covariance fWlction of a stationary process over an LCA
group G of special type, and establish some of the properties of such
an estimator.
This work generalizes results known for the cases
G = m and G = Z, which may be fOWld, for example, in Doob (1953)
Chapter X §7, Chapter XI §7.
A key result is Theorem 2.3.1, which asserts
that Wlder certain conditions integral averages of the form (2.1.7)
converge almost surely.
Theorem 2.lll
can be regarded as an ergodic
theorem in mean square or a weak law of large numbers, while Theorem
2.3.1 i$ the corresponding strong law of large numbers.
Theorem
2.3.1 is needed to establish the properties of our covariance estimator.
Throughout this section, we will assume that our indexing group
G
has the following property:
Property 2.3.1.
hood
U of
0,
There exists in G a relatively compact neighborand an element
if Un = ng + U , then
o
For example
get
go = 1.
:
....
~tationary
G such that
m=
cf>
m rf n
Also
n
n U
=1
and
and
u U = G.
n=-oo n
we may take U = [ -~,-t}
U = {OJ and
Z has the prcperty:
take
A stochastic process
{x(g) : g
if for any finite subset
E
m(U)
00
U
distribution of (x(gl)' •••
for any h
E:
R has this property:
Definition 2.3.1.
.
go
,x(~))
{gl' •••
'~}
E:
G}
~o
and
= 1.
is strictly
of G. the joint
is the same as (x(gl+h)J ••• ,x(~+h))
G.
.
Before we can formulate our strong law of large numbers for strictly
;;
28
stationary processes, we must first discuss the canonical form of a
stochastic process and introduce a certain transformation.
all the random variables of our process
{x(g), g
£
Suppose
G} are defined
on a pr.obability space (n,B,p). Consider the cartesian product space
CG of all flDlctions G + C. Let B(GC) denote the a-algebra of
subsets of CG generated by the "cylinder sets" of the form
G
{¢J EO C :
where
(4)(gl),·'' ,4>(ltn»
en.
Bn }
It is well known (see e.g.
£
n is a Borel subset of
Gikhman and Skorokhod (1969), p. 108) that there exists a probability
measure Q on (CG,B(GC)) and a stochastic process {i(g),g EOG}
B
on the probability space {CG,B(GC),Q} given by i(g)(4»
the
= 4>(g)
such that .
finite dimensional distributions of x coincide with those of i.
x is called the canonical representation of x. Moreoever, if
'l' : n + CG is the map w + x (. ,w) then 'l' is a measurable transformation and Q = p'l'-l.
Lemma 2.3.1.
The following lemma will be useful.
With the notation of the previous remarks, let
,..,
f n be r.v.'s on.
....
Then f n +0 a.e. [Q] iff
be a sequence of r.v.'s on (n,B,p) and let
{cG,B(CG),Q}
f
n
such· that
,..,
f n CW(w)) = f n (w).
+ 0 a. e. [Pl·
Proof:
Let
A =
.
G
{4> £ c
lim f (~)
n+oo n
00
so
A EO BCC G)
since
= O}.
Then
r ..:
co
n
u n
A=
r>O
n=lk=n
r rational
;;
fn
{~
f k (4)) < r}
{4> : -r < '"
Ifk (4)) I
<r}
is a cylinder set.
Hence
29
Q(A) = p\f-l (A)
= P{w
E:
n
=
p{w
€
n
=
P{w
E:
n
'1'(w)
= ~g
where
A}
...,
Now let Tg be the transformation CG +
Tg(~)
E:
f \few) + o}
n
f (w) + o}
n
CG given by
~g(h)
0
= ~(h+g).
Tg is a measurable transfonnation, and if x(g) is a strictly stationary process, then Q(A) = Q(Tg(A)) for every A E: B(CG). Note
also that T
-g
= T-gl
A set A € S(C G)
Definition 2.3.2.
is said to be T -invariant if
go
go is the element of G in property 2.3.1.
Q(T A~A) = 0 where
go
It is easy to see that the class of all Tg invariant. sets is a suba-algebra of B(CG).
also denote by A*
This
the class of sets
sub-a-algebra of B.
invariant.
o
a-algebra will be denoted by A.
Note that
{ljJ-l(A): A € A} ;. A*
A is T
go
invariant
<~
We shall
is a
A is T
-go
We can now state our strong law of large numbers.
Theorem 2.3.1.
Let
{x(g), g
€
G}
be a strictly stationary measurable
process over the LCA group G which has property 2.3.1.
Let
n
En .= U Un;
k=-n
(2.3.1)
Proof:
lim
n+oo
then if Elx(e)1 <
m(~
n
f
)
x(g,w)m(dg)
00
= B(x(e) IA* )
a.e. [P].
E
n
The proof will consist of two lemmas.
Let
Lemma 2.3.2.
function
[Q], and
f*
€
f
E:
LI(C G,B(C G),Q).
Then there exists an A-measurable
Ll~C G,B(CG),Q) such that f * (TgOIj»
= f*(T_geJl) = f * (~)
a.e.
J
l
lim meE )
f(Th($))m(dh) =f*($)
n~
n E
30
a.e. [Q]
n
and
Proof:
Let
F(~)
jf($)Q(d$) = J~*($)Q(d$) VA
A
A
= Jf(Th(~))m(dh)
Then
JIF(~)IQ(d~)
CG
<
~
A
Jlf(Th(~))lm(dh)Q(d~)
f
CG U
= J Ilf(~) IQT~l(d~)m(dh)
.. .,._ 0_'_.'.
U CG
=
=
variable and Fubini's
theorem
J
J1f($)IQ(d$)m(dh)
U
G
C
"f
II L
by change of
m(U) <
00
1
so F($) exists and is in L 1 (C G,BlcG) ,Q).
By a trivial corollary of the Birkhoff ergodic theorem (see e.g.
H~lmos
(1956))
Ll (G
C , (C G),Q ) , such that VA ~ A,f F($)Q(d$) =
*
*
*
A
f (Tg$) = f (T_g~~ = f ($) a.e. [Q].
This last
statement implies that f* is A measurable. But
for some f * in
*
.
. !f($)Q(d$) ·and
n
n
m(E ) .:;, I m(Uk)· = I m(kgo+U) = 2n+l
n k=-n
k=-n
31
and
I
=
f
fCThl/»mCdh)
k=-n kgo+U
=
. m(E)
1
so 11m
n-+oo
n
.,
!
f fCTh$)mCdh)
,
En
fCTh$)mCdh) = f * ($)
E
n
a.e. [Q].
..
Also,
fFC$)QCd$)
A
=f
!fCTh$)mCdh)QCd$)
AU
=
f ffCTh<l»QCd 4»mCdh)
UA
f
= ff(l/»QTh1Cd4»m Cdh)
UA
=f
!fC</»QCd4»m Cdh)
UA
= mCU) ffC</»QCd.)
A
=
ffC</»QCdl/»
A
so
!fCl/»QCd</»= ff*C$)QCdl/»
VA
€
A.
A
Lemma 2.3.3.
With the notation of the preceeding page,
o
,
32
lim
n-+-oo
provided that
Proof:
m(~
J ~(h,~)m(dh) = f*(~)
)
n
Elx(e)1
a.e. [Q]
E
n
< ~.
In Lemma 2.3.2, set
f(~)
= ~(e).
Then
f If(~)/Q(d~) = J/f(~(W»/P(dW)
n
CG
= J1x(e,W)/p(dW)
n
= Elx(e) /
.
Thus if Elx(e)1 < ~ we can apply Lemma 2.3.2 with f(~)
= ~(e)
and
obtain the result, since
= Tg~(e) = ~(e+g) = ~(g) = i(g,~).
f(T g~)
0
The prqof of Theorem 2.3.1 is completed by setting
f n (w)
=
m(~n)
fx(h,w)m(dh)
- £: (x(e}~A*) ,
En
and
fn(~) = m(~n) Ii(h,~)m(dh)
-
f*(~),
En
and noting that since the process
x(g,w) is measurable, f (w) is
n
measurable, also
i(h,~(w»
•
f * (~(w»
= ~(w)(h) = x(h,w),
= €(x(e)/A*)
and
since for all A € A
I
33
f
f*('1'(w»dP
= fl(~)P'1'-I(d~)"·
'1'-I.(A)
A
=
fl (~)Q(d~)
A
r
= Jep(e)Q(d~)
A
= J'1'(w)(e)p(dW)
'1'-1 (A)
=
f x(e,w)P(dw)
•
'1'-1 (A)
Hence fn('1'(w»
~ fn(w).
thus by Lemma 2.3.1
fn(w)
Definition 2.3.3.
the LCA group G.
By Lemma 2~3.3
~
0
fn(~) ~ 0 a.e. [Q], and
a.e. [Pl, which proves (2.3.1).
Let {x(g), g
€
G} be a stochastic process over
With the notation already established, the process
x is said to be metrically transitive if for all
Q(A)
=1
or Q(A)
0
A € A, either
= o.
With this definition we can state the following corollary to Theorem
2.3.1.
Corollary 2.3.1.
Let {x(g), g
process with Elx(e)1 <
00.
€
G}
be a strictly stationary measurable
If in addition {x(g), g
€
G} is metrically
transitive, then
(2.3.2)
lim
n~
m(~
)
f
n E
n
x(g,w)m(dg)
= Ex(e)
a.e. [Pl·
•
;;
34
Since f * is A-measurable, if x(g) is metrically transi-
Proof:
tive
f * is a constant a.e. [Q], and thus
e(x(e)IA*) is a constant
* = E(x(e))a.e. [Pl.
e:(x(e)IA)
a.e. [P]' Le.
0
We now turn to the problem of covariance estimation.
that {x(g), g
the
€
G}
Suppose
is a weakly stationary stochastic process over
LCA group G which has Property 3.2.1.
By generalizing £rom the
known results in the cases G = Z or lR, a natural estimate for the
covariance R(g) of the process
(2.3.3)
RnCg')
= m(~}
n
x(g) is
I
x(g+g' )x(g)m(dg)
E
n
We conclude this section by establishing a few properties of the
estimator (2.3.3), generalizing results of Doob (1953) Ch. X §§7.
OU7
first result in this direction concerns the mean square limit of
the sequence of estimators (2.3.3).
Theorem 2.3.2.
Let {x(g), g
€
G}
be a weakly stationary measurable
process over the LCA group having Property 2.3.1.
that the sets
En satisfy
(2.3.4)
lim
n+oo
= X(g)
process.
(2.3.5)
-
Then
.
1
1 .1.m· (E )
m
n+oo
m(En n (g+En ))
m(En )
= 1
= x(g+g'.)x(g) for
E(X(g)) = X(g) - R(g')
Also suppose that if X(g)
process Y(g)
Suppose in addition
n
Vg
€
fixed
G•
g'
then the
is a weakly stationary
35
exists in .mean square and the limit.(2.3.5) is equal to R(g') with
probability 1 iff
limm(~ )
(2.3.6)
JE(X(g)X(e»m(d g)
n E
n+»
= IR(g')1 2
n
Proof:
The sets En
union i.s G.
are relatively compact, increasing and their
If they satisfy (2.3.4) then they form a sequence such
that equation (2.1.10) is satisfied.
(Hewitt and Ross (1963) p. 255)
Hence, if Zy is the orthogonal random measure in tne spectral representation of the process Y(g), (Y(g) has a spectr41 representation
since it is measurable and hence
LLrn.
n-+co
m(~
by Theorem 2.1.1 we have
m.s.c.)~
) Iy(g)m(d g) = Zy({e})
. n
E·
n
Hence
l.~~. m(~n) IX (g)m(dgl=
Zy({el) +R(g')
E
n
which proves (2.3.5).
EIZy({e})/2
Also by Theorem 2.1.1
= ~y({e}) = lim m(~
) JRy(g)m(d g)
n E
n+»
n
where
Ry(g)
Hence
EIZy({e}) /2
= E(Y(g+e)Y(e)) = E(X(g)X(e»
= lim m(~
n-+cc
_ IR(g') 12 •
2
) JE(X(g)X(e»m(d g) _ IR(g') 1 •
n
E
.
n
Thus if (2.3.6) is satisfied,
is R(g') with probability 1.
Elz y ({el)1
2
=0
so the limit (2.3.5)
Conversely, if the limit is R(g') with
probability 1, then (2.3.6) is true.
o
•
;;
36
If the process x(g) is strictly stationary, we can say more •
. Theorem 2.3.3.
Let {x(g), ge G} satisfy the hypotheses of Theorem
2.3.2; then if . x(g) is strictly stationary and metrically transitive,
(2.3.7)
lim
m(~
= R(g')
) IX(g+g' ,w)x(g,w)m(dg)
n
E .
n
Proof:
In Lemma 2.3.2, set
X(g,w).
Then
a.e. [Pl.
.
f(~)
= ~(g')~(e)
=
so that f(Tg'(w»
flf(~)IQ(d~) = flf(~(w»ldP
CG
Q
= f1x(e)ldP
Q
=
fX(g' ,w)x(e,w)dP
Q
~'Elx(e) 12
since x· is weakly stationary so that
f e
<
ClO
G
G
Ll(C ,B(C ),Q).
Thus since
x(g) is metrically transitive
a. e. [Q]
where
f * (~) is constant.
. m(E)
1
11m
n-+oo
But f * (~(w»
n
It follows from Lemma 2.3.1 that
fX(g,w)m(dg) = f * (~(w»
= E(X(e)IA*);
that f * (~(w»= EX(e)
=
a.e. [Pl.
E
n
since f *~
is constant a.e. [P] it follows
R(g') which proves (2.3.7).
o
37
Corollary 2.3.2.
Let {x(g) : g
measurable Gaussian process.
spectral measure
~
E
G} be a real-valued, zero mean
If x(g) is weakly stationary, and
its
is absolutely continuous with respect to Haar
measure and G is not compact, then the' sequence
m(~
(2.3.8)
f x(g+g~)x(g)m(dg)
)
n m(E )
n
converges to R(g') with probability I
Proof:
and also in mean square as n
By a result of Blum and Eisenberg (1972), if
~
-l- co.
is absolutely
continuous with tespect to Haar measure on G then x(g) is metrically
transitive, hence the sequence (2.3.8) converges to R(g') with probability 1 by Theorem 2.3.3.
Since x(g) is Gaussian and real, with zero mean
EX(g)X(h) = E(x(g+g' )x(g)x(h+g' )x(h»
= Ex(g+g')x(g)Ex(h+g')x(h)
x
Ex(g)x(h) + Ex(g+g')x(h)Ex(g)x(h+g')
= R(g)2
+ R(g_h)2 + R(g-h+g')R(g-h-g') •
Hence X(g) is stationary.
m(~n)
fEX(g)x(e)m(d g)
En
+ Ex(g+g')x(h+g')
Moreover,
= m(~n)
fCR(g,)2+R(g)2+R(g+g')R(g-g'»m(dg)
En
= R(g,)2+m(~n)
f(R(g)2+ R(g+g')R(g-g'»m(d g)
En
Thus (2.3.6) is satisfied and the result proved by Theorem 2.3.2 if
we can show that
38
lim
n-+oo
Now
m(~
I
) J(R(g)20'11- R(g+g')R(g-g'))m(dg)
n
E
n
=0
.
f(R(g)2 + R(g+g')R(g-g')m(dg)\
En
~ f R(g)2m(dg)
(2.3.9)
+ fIR(g+g')R(g-g')lm(d g)
En
<
f
R(g)2m(dg) +
En
Consider
fE R(g+g,)2m(dg).
Set
F
n
= En
+ g'.
Then
n
m(Fn n Fn + g) = meg' + Eng'
+ g + En )
n
= m(En
Hence
m(F
lim
n-+oo
n F + g)
m(Fn)
_..;:.;n~~n..--_
n (En+g)) + g')
=m(~n
n (En+g))
=1
Thus by Hewitt and Ross (1963) p. 255,
(2.3.10)
limm(~)
m
2
IIR(gJ/ m(d gJ
n.F
n
n+
Now let
f n (g)
=
limm(~)
n-+oo
IIR(g)12 m(d g).
n E
n
be the sequence of functions defined in the state-
ment of Lemma 2.1.2.
Then
39
(2.3.11)
=
IJ
4l(a,B)u(da)u(dB)
" "
GG
by the dominated convergence theorem where
Ha,B)
={:
ll=
13
~
B
a
"
, since
f n (a)
+ x{
,,}(a)
eG
Hence (2.3. 11) is equal to
JU{B}u (dB) = I"
8EG
II ({S})
2
"
G
where at most countably many terms in the sum are non zero.
hypothesis
u
But by
is absolutely continous with respect to Haar measure,
so ~{a} =0 V8
E
G.
Thus
lim m(~ ) JIR(g)1 m(dg) = o.
n+oo
nE
2
Thus by
n
(2.3.0) and
(2~
3.10) .
lim
n+oo
m(~.)
n
I(R(g)2 + R(g+g')R(g-g'))m(dg)
=0
En
which completes the proof of Corollary 2.3.2.
o
·CHAPTER II I .
SAMPLE PATH PROPERTIES
OF STOCHASTIC PROCESSES
In this chapter we investigate questions relating to the sample paths
of stochastic processes.
In section 3.1 we discuss a stochastic analog
of a well known result in measure theory, relating to the approximation of
measurable functions by continuous functions,
In section 3.2, we study
the paths of a stationary second order process over a compactly generated
group~
and give conditions sUffi<;ient for the sample paths to be continu-
ous with probability one, and also prove a sampling theorem.for such
processes.
Finally, in section 3.3, we discuss a generalized notion of
a "band limited process. tI
3.1
Continuous-Path Approximations to Measurable Stochastic Processes.
It is a well known fact in measure theory that any measurable
function is "almost" continuous in the following sense:
if f
urable function on a finite interval [a,b] then given any
exists a continuous function
__ less than
€
such that
g and a measurable set
Sup If(x)-g(x)I <
aSxsb
x~E
€
€
>
is a meas0 there
E with measure
(See e.g. Royden
(l~63) p. 57)
The aim of this section is to formulate similar results for measurable
stochastic processes.
We treat a class of processes that includes meas-
urable processes and also processes having a finite p-th absolute moment.
40
41
The case p=2 is treated separately.
Let
(~,B,P)
be a probability space, and let
space of all complex valued random variables on
~.
function defined on the positive reals such that
creasing and
= O.
N(O)
Also, let
(3.1.1)
Let
M denote the vector
Let
N be a real
N is continuous, in-
N satisfy, for some fixed constant k
Vu ;::; 0 •
N(2u) < kN(n)
PN be a function defined on M by
PN(x)
= !N{lx(w)l)dP
•
~
Now let
that
N(L(~,B,P))
PN(kx)
denote the space of all random variables x such
is finite for some constant
k.
If we identify r.v.'s
that are equal a. e. [P]' then N(L(Q,.B ,P)) is a modular space with modular
PN (see Rolewicz (1972) p. 18).
metric d N on N(L(Q,B,P))
The spaces
The modular PN induces a complete
by
N(L(Q,B,P)) are thus complete metric linear spaces, with
invariant metrics:
they are thus
F-spaces, in the sense of Rolewicz
(1972) •
Remark 3.3.1
If N(u)
= u/l+u,
then
N satisfies (3.1.1)
and
N(L(Q,B,P)) is just the space of all random variables, with the topology
of convergence in probability, which is induced by the metric
d(x,y)
= E~
of all r.v.' s
•
= uP then N(L(Q,B,P)) is just the
EIx IP < 00, which is an F-space for
If N(u)
x satisfying
o <p s 1 and a Banach space for p.~ 1 with F-norm Ilxll
=
space
Elxl P in
the first case and
a-field B if there
P is separable with respect to the
exists a countable subclass A of B such that for all
B € B there exists A € A with
N(L(n,B,p)
Theorem 3.. 1.1.
42
o
= {ElxIP}l/p in the second.
If (n,B ,P) is a probability space',"" we will say that
Definition 3;1.1.
the measure
I Ixl I
E
> 0
and
o
P(A~B) < E.
is separable as a metric space iff p" is
separable with respect to B.
Proof:
See
Let
Definition 3.1.2.
{ek};=l
(i)
of vectors in
V be a metric linear space.
V is called a quasibasis
The subspace spanned by the
00
"Cii)
o
Rolewicz (1972) p. 30.
If
L Ake k
= 0 for scalars
k=l
A sequence
if
is dense in V,
k
Ak then each A = O.
k
e
According to Peck (1968) every separable
F-~pace
has a quasibasis.
o
We now formulate our first approximation theorem.
sider measurable stochastic processes
ability space
(n,B,p)
where
T isa locally compact
{x(t) : t€T}
defined on a prob-
P is separable with respect to B, and
normal space.
measurable with respect to the
We shall con-,
The processes x will be
a-field B(T)
x
B where B(T)
is the
a-field of Borel sets of T. "
We need a result similar to that mentioned in the introduction.
Theorem 3. 1. 2.
Let
T be a locally compact
finite positive Borel measure on B(T).
·normal- space,
Then given
E >
~
a
0 and any
43
measurable function
~(ce) < €
with
f
on T, there exists a compact set eST
such that the restriction of f
uous extension to
to e
T.
By Lusids theorem (Halmos (1950) p. 243)
Proof:
set e
~(Ce) < €
with
continuous.
normal,
has a contin-
Since e
there exists a compact
such that the restriction
is compact-3nd
g of f
T is locally
con~act,
to e
is
and
g can be extended to a continuous function of T.
o
We can now formulate our theorem.
Theorem 3.1.3.
Let
{x(t) : tE:T}
be a
B(T) x B
defined on the probability space (n,B,p)
respect to B, and
also that
x(t)
€
N(L(n,B,p))
for each tE:T.
Then given any
~(Ce) < £
with
yet) E: N(L(n,B,p))
sup PN(x(t)-y(t))
tE:e
<€
3.1.1.
z(t)
Since
and
€ >
Let
~
space.
Suppose
be a finite positive
there exists a compact
0
and a measurable process
paths, and with
Proof:
P is separable with
T is a local! y compact ·'normal-
Borel measure on T.
set e
where
measurable process
Yt E: T
yet)
with continuous
such that
•
P is separable,
N(L(n,B,p))
N(L(n,B,p)) is separable by Theorem
has a quasibasis
is in the class S({e k })
ifz(t)
{e k}.
Let
us say a process
can be expressed in the form
00
z(t)
fixed
= I
ak(t)e k where the sequence {ak(t)} is ultimately zero for each
k=l
t and each function ak(t) is B(T)-measurable. No questions of
convergence arise in this definition.
Lemma 3.1.1
The proof will be done in two parts:
which shows that every process in the class S({e k}) satisfies theorem 3.1.3, and Lemma 3.1.2 which shows that every measurable
process
z (t)
with z (t)
44
Yt e.. T . can be approximated by
N(L(S'2, B,p»
€
a process in S({e }).
k
Lemma 3.1.1.
Every process in S({ek }) satisfies Theorem 3.1.3.
00
Proof:
Let z(t) =
ak(t)e k be a process in S({e k }). Let Ck
k=l
be the compact sets of Theorem 3.1.2 such that ak(t) is continuous on
r
Ck and ~(CCk)
extension of a
< 3e/w
2 2
k
k = 1,2,....
Let. bk(t)
be the continuous
00
k
to T.
Then if Co = n Ck '
k=l
00
~C
~( CCO) s 'k=Ll~. (. Ck)
00
<
~_3€
L
k=l w2k
2
= e/2
n
and Co is compact.
Let
n
(t)
= r bk(t)ek
• Then clearly y is a
k=l
n
measurable process with continuous paths, with y (t) € N(L(S'2,B,p»
for
y
n
each t.
It is clear that for each fixed
sufficiently large so PN(yn(t) - z(t»
Since z(t)
proc~s~es)
theorem.
t e Co yn(t)
converges to zero for all t
€
CO.
PN(yn(t) - z(t»
is measu~able by Fubini's
Hence by Egoroff'g theorem, there is a compact set COO £ Co
~(CO
By choosing
- COO)
< e/2
and
n large enough, and setting y = Yn
sup PN(y(t) - z(t»
t€C OO
~ (CC
for n
is also a measurable process (being the limit of measurable
it follows that
such that
and
= z(t)
oo ) ~ ~ (CO-COO)
+ ~ (CC ) <
O
e
<
Ii:
0
The next lemma will complete the proof of Theorem 3.1.3.
Lemma 3.1.2.
For every e
>
0 there exists a compact set C with
45
II (CC) <
e: and a process
z (t) in S ({ ek })
sup PN(x(t) - z(t))
teC
Proof:
Let 0 denote the class of all
such that
<
e: •
B(T) x B measurable processes
z, with z(t) e: N(L(n,B,p)L that can be approximated by a process in
S({ek }) in the sense of the lemma. If a,a are complex numbers and
x,y are in 0, then we will show that ax +. Sy is in O. For each
n
= 1,2, ...
let
x',
y'n
n
be processes
such that
.
sup PN(Y'(t) - yet))
te:O
n
<
21T
n
and O.
are compact sets such that
n
Let C
and
ll(CC ) <
n
3e: '
22
1T
~e:2
where C.n
n
C
3e: .
22
ll( On) <
n
1T
n
co
=
n C , 0 = n 0
Then for all
n=l n
n=l n
PN(y~(t) - yet))
converges to zero. By the properties of modulars,
PN(a(x~(t)
IlolI N
- x(t)))
andPN(a(y~(t)
also converge to zero.
If
where dN is the metric
associated with the modular PN as in (3.1.1), then Iia (x~ (t) - x(t) )II N
and Ila(y~(t) - y(t) I IN converge to zero. (Rolewicz (1972 p. l7).Thus
for all
denotes the F-norm
- yet)))
IIxllN = dN(x"O)
t e: C n 0
lim
n-+co
I la
x(t)
+.
a Yet) - (axl(t)
n
+.
Byl(t))11
n
. '-
<
=
lim Ila(x(t) - xl(t))11
n-+co
n
+
limI18(y(t) - r'(t))11
n-+co
n
so
lim Ilax(t)
n-+co
+.
By(t) - (axl(t)
n
+.
8y'(t)
n
II
=0
Vt e: C nO.
It follows (Rolewicz 1972 p. 17) that
lim PN(ax(t) '" ay(t) n-+co
(ax~ (t) + 8Y~ (t)))
= 0 •
46
Thus by Egoroff's theorem there exists a compact set
that
II (OC
O) <
Since
(ax~ (t) + 8Y~ (t))) <,.
c
ax'(t)
n
B(T) , B
By'(t)
n
+
is
in S({ek }), ax(t)
c"
B.
x
E
•
+
8y(t) c G.
B simple function is in G.
Let
Since
= N(l)P{B)
it is clear that
al, ••• ,an
D such
o
Next we show that every B(T)
A
n
and an integer n such that
E
sup PN(ax(t) .- ey(t) tcC
Co £ C
XB c N(L(Q,B;P)).
such that
<
ClO
,
Thus there exist complex numbers
o
t
~
A
n
so
sup PN(XAxB{t,o) tcT
F =
r ak(t)ek) <
E,
k=l
m
u A. x B. ,
. 1 1.
1.
1.=
so that
XAXB c G.
Moreover, if
,"
we can suppose without loss of generality that the
m
A." x B. are disjoint, so
X =
X
is in G by the previous para1.
1.
F
. 1 A.1. xB 1...
1.=
graph. • Sets of this type generate thecr-field . B(T) x B so given any set
r
E c B(T) x B, there is a sequence
that
ll"x
P(E
6
Fn) < lIn.
{F}
n
of sets of the above type such
It follows that
N(X E6F (t ,w)) is in
n
LICT x Q, B(T) x B,
II
x P)
and that the sequence N(x
E6Fn
(t,w)) converges
47
to 0 in this space.
Thus
E
N~EbF
(t,o)) converges to 0 in
Ll(T,B(T),~)
n
and a subsequence converges to 0
there is a compact subset Cl
a.e.[~],
and so by Egoroff's theorem
~(CCl) <
with
such that
8/2
Now Xp
is in G so there
and a process
z(t) in S({ek })
n
is a compact subset C2 , ~(CC2) < €/2
with z(t) € N(L(Q,B,P)) such that
sup PN(Xp (t,o) - z(t))
t€C 2
n
Set C = Cl n C2 , then by noting that
sup P (~X (t,o). tE:C N .E
<
8/2.
XEllP = Ix p - xE ' , we see that
n
n
~z(t))
sup PN(xE(t,o) - Xp (t,o))
tE:C
n
l
< 8/2 + 8/2 = 8 •
<
+
sup PN(X p (t,o) - z(t))
t C2
n
XE and thus any simple function is in G. Now consider a
measurable process {x(t) : t€T}, x(t) ~ N(L(Q,B,P)) , that is positive.
Hence
Let
to
xn be a sequence of simple functions such that xn(t,w) increases
x(t,w) pointwise.
Since N is increasing, NclxCt,w) - xn (t,w) I) ~
Nlx(t,w)
I.
Since
E N(x(t)) <
zero by dominated convergence.
of the proof,
x E: G.
E N( Ix(t ,w) -
00,
xn (t,w)
I)
converges to
By Egoroff's theorem and the first part
By expressing an arbitrary process in terms of its
positive parts we see that every. process
x that is measurable and has
x(t) e: N(L(Q,B,P) for each t € T is in G, and so can be approximated by
o
a process in S({ek }).
Corollary 3.1.1.
Let
{x(t), te:T}
be
~
measurable process, over the
48
probability space
(n,S,p) where P is separable ,with
Let T be a locally compact· normal.
measure on T.
jJ(CC)
< £
Then given
8
> 0,
and a measurable process
Set
N(t)
= t/1+t.
V~
> 0,
space,. jJ a finite positive Borel
Yet)
y(t,w)
= El +Ixl1xl
there exists a measurable proceSs
jJ(CC)
< d
sup E[ Ix(t)
te:C
Thus
sup
t€C
I
> E] < E •
with the topology of convergence in
Q
The modular PN(X)
a compact set C with
with continuous paths such that
Then, as mentioned in Remark 3.1.1, N(L(n,S,p»
is the space of all r.v.'s on
probability.
to S.
there exists.a compact set C with
sup p[w:lx(t,w) t€C
Proof:
r~spect
is an F norm.
By Theorem 3.1.3
y with continuous paths and
such that
y(t)I/(l+lx(t) - y(t)I)]
pew: Ix(t,w) -
y(t,w) I
<
0
> E] .
s 1+E sup E[ Ix(t) - Y(t) I/l + Ix (t) ... Y(t) I]
£ t€C
1+£
<""£
Choose
so that
Corollary 3.1..2.
1:
v
•
1+e:
£
0<
8
then
,
o<
8,
o
the result follows.
{x(t) : t€T} be a me.asurable process as in
Corollary 3.1.1 and suppose Elx(t)I P < 00 Vt € T for some p > o.
Then given
£ >
Let
0 there exists a measurable process yet) with contin-
uous paths and Ely(t)I P < ~ Vt , and a compact set C with
such that
Proof:
.
P
sup EIx(t) - yet) I <
te:C
Take N(u)
Theorem 3.1. 3.
= uP
, 'then
jJ(CC)
< e
\
£.
:P
= ~Ixl,the
result~follows
from
o
49
Remark 3.1.2.
If we restrict
T to be a real interval
I
= [a,b],
we can approximate a measurable process {x(t)tdR} by a process yet)
whose paths are infinitely differentiable, instead of merely continuous.
In the proof of Theorem 3.1.3, the process
y which approximates x
was obtained by approximating measurable functions on T by continuous
functions.
f
on
In the case
T =' [a,b]
[a,b], we can find a
given
~ >~
and a measurable function
COO function h, and a compact set C £ [a,b],
with the Lebesgue measure of [a,b] - C less than
f(x) on C.
e:, such that
This assertion may be proved as follows.
Let
measurable simple function, then there is a step function
cides with
except on a set of small measure, and a C
that coincides with
(1963) p. 58.)
=
be a
g that coin00
f
f
g(x)
g except on a set of small measure.
function h
(See Royden
Thus the assertion is true for simple functions.
Using
the technique of Halmos in his proof of Lusin's theorem (Halmos (1950)
p. 243), the assertion can be proved for general measurable functions.
Thus the proof of'Iheorem 3.1. 3 can be adapted to obtain an approximat ing
•
o
0:>
process w1th C paths.
We now turn to the approximation of second order processes.
Because
every separable Hilbert space has a basis, and not just a quasibasis,
we can prove a stronger result than Theorem 3.1.3.
Theorem 3. 1. 4 •
Let
{x(t), te:lR} be a second order measurable process
indexed by the real line m
~
a positive finite measure on
is absolutely continuous with respect to Lebesgue measure.
E >
o there
process
exists a compact set C with
yet)
~(CC) <
e:
R that
Then given
and a second order
with continuous paths, with yet) in the Hilbert space
so
spanned by x(t), such that
sup Elx(t) - y(t)l~
t€C
Proof:
on
<'€ •
Let R be the covariance function of x.
IR such that
f
R(t,t)v(dt)
<
-co
measure.
00
Let
and v is equivalent t~ Lebesgue
Such a measure exists (Cambanis (1973)).
Cambanis (1973) the process
be a measure
v
Then according to
x(t,w) has a representation of the form
00
(3.1.3)
where
the
are orthogonal random variables and the ak's are
~k's
measurable.
The process z(t) is orthogonal to ~k for each t and k,
2
and Elz(tJl = 0 a.e. [Lebesgue]. For each 'k, a k is continuous on
a compact set
lR.
Ck with ]..l(CC ) < 2E/~21T2.
Set C(O) = ~ C • Then
k
k=l k
Let bk be the continuous extension of ak on Ck to
N
.
Then for all t € C (0)
) = L b~(t)~k(w).
k=l
ElyN(t) - x(tJl
2
= Elz(t)1
2
00
+
L
lak(t)1
2
k=r~+l
00
But'I lak(t) 12
k=l
<
00
(Cambanis (1973)), so
~: EIYNCt) - x(t) 12= Elz(t) 12 Vt
Let A denote the set on which Elz(t)/2 -; O.
set C(l)
with A S CCCl)
and ]..l(c(l)
<
8/3
lim ElyN(t) - x(t)12
c(O) .
Then there is a compact
since
utely continuous with respect to Lebesgue measure.
(3.1.4)
Ii
=0
]..l is assumed absolThus on C(l) n c(O)
•
N~
By Egoroff's theorem, the result now follows from (3.1.4) by a familiar
51
argument.
0
Remark 3.1.3.
This theorem is not a special case of Theorem 3.1.3 since
we can assert that the approximating process
Yet)
is in the space
spanned by the process x(t), and thus, in theory at least,
yet)
can
be obtained by linear operations on x(t).This is not the case in the
general context.
Also, we do not require
P to be separable.
o
If the measurable process
Remark 3.1.4.
{x(t)t€R}
is stationary and
hence mean square continuous by Theorem 2.2.4, we can approximate x(t)
uniformly by processes with analytic paths.
representation
r
For if x(t)
has spectral
00
x(t)
=
e
21Tit
>'Z{d>'},
2
EI Z(ll) 1 = ).I(ll) ,
_00
n
t hen t he process xn (t)
= J e 2~it>'Z{d~}
A
satisfies
-n
Elx(t) - xn (t)1
The process
2
= ).I(C[-n,n])
xn (t)
and so
lim sup Elx(t) - xn (t)/2
n-+oo t€lR
=0
•
is stationary and has analytic paths (Belayev (1959)).
Such processes with spectral measures concentrated on compact sets are
called "band-limited;" we shall return to them in the sequel.
Remark 3.1.5.
If the measUrable process
ous, then the process
0
x(t) is mean square continu-
z(t) in the expansion (3.1.3) vanishes, and the
series (3.1.3) converges in the mean uniformly over compact subsets of
IR (Cambanis (1973)).
It follows that for any compact subset C of
lim ElY (t) - x(t)/2
n-+oo
n
=0
uniformly on C.
o
~
52
Remark 3;1.6.
If the process x(t) in .heorem
the random variables;k
3~1.4
is Gaussian, then
in (3.1.3) can be chosen tO'be Gaussian
(Cambanis and Rajput (1972)) and so the approximating process is
o
Gaussian.
3.2
Sufficient Conditions for Path Continuity of Stationary Processes
In this section we will discuss conditions on a stationary process
that are sufficient for path continuity with probability one.
We will
generalize known results (Kawata (1969), Belayev (1959)) to weakly stationary second order processes indexed by a certain type of LCA group, and
obtain conditions on the spectral measure of a mean square continuous
(m.s.c.) stationary process sufficient for path
con~inuity.
We also dis-
cuss band limited processes indexed by a group and show that such processes
satisfy a "sampling" theorem.
More specifically" many sufficient conditions for path continuity of
.a stationary process are known.
For example, if {x(t) t€lR}
stationary process on the real line
Kawata (1969) has proved that
I
is a m.s.c.
JR, with spectral measure
~,
then
x has continuous paths with probability
if
00
f
(3.2.1)
for
a>
1.
Ixl(log+lxl)'\J(dx) <
00
Kawata's proof depends heavily on the Borel-Cantelli lemma,
and his approach fails even for the case of a process indexed by
lR 2'
A different approach" using weak convergence of probability measures, is
used by Neuhaus (1972) to derive sufficient conditions for path continuity;
conditions which depend on the behavior of first order differences of the
process.
We will follow his technique in the sequel.
S3
As mentioned in section 3.1, band limited stationary processes are
those whose spectral measures are concentrated on a compact set, i.e.,
have compact support.
Such processes have analytic paths, and have the
important property that the value of the process at any point can be
k1T
obtained from its values at the "sampling points" -a , k = 0, ±l,±2, ••.•
More precisely, if x(t)
~
is a band limited process whose spectral measure
has support in the interval [-W,W]
x(t)
(3.2.2)
=
then
00
~
t.
k=_oo
x(k1T/a) sin a(t-k'IT/a)
a(t-k1T/a)
The series (3.2.2) converges uniformly on compact sets of
ability 1 for all
a
>
W.
IR
with prob-
(See e.g. Piranashvi'li (1967)).
We will consider m.s.c. stationary processes indexed by an LCA group
that is compactly generated, and obtain a condition similar to (3.2.1)
that is sufficient for path continuity with probability one.
We will also
discuss band limited processes on such groups and show that they can be
expressed in a sampling series similar to (3.2.2).
Every compactly generated r,:-:A group (i. e., an LCA group that has a
compact neighborhood
U of e such that
G=
00
U
n(Un(-U»
can be ex-
n=l
pressed in the form
(3.2.3)
where k and m are positive integers and
(Hewitt and Ross (1963 p. 90).
F is a compact abelian group.
We first obtain results for
m. k and then
combine lR k with the other factors to get the corresponding result for
G.
We will use the following result of Neuhaus (1972J:
Theorem 3.2.1.
Let
E denote the k-dimensiona1 unit cube and let
k
54
{x(t) : tEE } be a stochastic process. If there exist constants
k
y ~ 0, a > 1, K ~ 0 and continuous distribution functions F. corresJ.
EI~I
ponding to finite measures on
and satisfying
(3.2.4)
for all
i E I, Vt, t+h, h E Ek , VA > 0,
paths with probability 1.
Proof:
then
x(t)
has continuous
o
See Newhaus (1972).
A few remarks on the notation of the above theorem are necessary.
Let
I
= ~
I
p=l
Ii I = p.
and
p
function
~
= {i = (il, •.• ,ip )
Ip
Elil
= l, ... ,p
-+
and
IR
tv
If
: il ..• i p are integers, l s i S••• SipSk},
f
is a function
i
given by f (ul,· .• ,up )
=0
if v ~ i.
then [t].J.
E
p
-+
R" t
If t
= (t.J.
Ek
-+
= f(t)
R then
= (tl, ... ,tk )
, ... ,t.).
l
~p
ti~
where
If
€
f
fi
is a
= u~
for
mk and
is a function
and t.. t' E Ep then
L
I
(-l)~
°l'''''op
where the summation taken over all (ol""'op)
other words,
Remark 3.2.1.
~
with
°
~
E
{O, l} .
In
is a difference operator.
The theorem remains true with
E replaced by [_A,A]k
k
for A >0.
Remark 3.2.2.
By an application of the Tchebychev inequality, (3.2.4)
can be replaced by
55
Eld[t+h]i
(3.2.5)
[t].
1
x
il2 < Kld[t+h]i pJa
[t].
=
in the case of second order processes.
the process
i·
1
If we assume in add1tion that
x is m.s.c. stationary, we get the following corollary:
Corollary 3.2.1.
Let
{x(t), te:R k}
be a m. s. c. stationary stochastic
k with spectral measure ].l. Let f(xl""'~) be positive
and integrable on Ak = [-A,A] k . Then the paths of x(t) are continuous
process on
lR
with probability one on Ak if there exists
a·> 1
k
such that
> 0
(3.2.6)
~. K4-P
f
f(xl,···,xk)dxl,···,dx k
Ai(t,h)
where
Ai(t,h)
is the set {(xl'''' ,xk ) :t i . ;~Xjsti
].l
and
Proof:
i
€
I, t
Let
</>Ctl, ... ,t k)
= (tl, ••. ,tk)
= [exp(2'1fi
= exp
and h
t.1 A.1 )
P P
x
k
2'1fi
-Asx.SO
J
L Lt .•
j=l J J
+h.
jd, j=i ].l
].l
j~i}
if
Let
= (hl, .•• ,hk).
1
].l
i
=
•
(il, ... ,ip )
Then
exp(2ni(t.1 +h.1 )A.1 ]
P P P
;,
A. (0 t. + 1-0 (t. +h. ))
1
].l 1
].l].l
].l
1
1
].l].l
56
Proceeding in this way we see that the right hand side of (3.2.7) is
equal to
=
~
II e
2'1Ti t. A.
1
1
~. ~(1 - e
~=1
. 2 'ITh •
SIn
so
1
A.
]J
1
~
From the isomorphism theorem it follows that
t
F(tl, .. .,tk ) =
l
t
k
J. .. I f(xl,· .. ,xk)
-A
dXl' •.• ,dxk ,
-A
then
Thus by 3.2.5a sufficient condition for path continuity is (3.2.6).
o
If in Corql1ary 3.2.1 we set
f(xl""'~)
= 1,
then (3.2.6) becomes
(3.2.8)
~i €
I
and h
€
A
k
57
(3.2.8) is implied by
...:e.
.2
II s1n
Jk ]J=1
(3.2.9)
R
for some K'
~
O.
Since
2
sin 1f Ah:::: /1f Ah Ia:
for a < 2, it follows that
(3.2.9) is true if
.IT IA·l
J J=l
(3.2.10)
k
a
J
]J(dAl,···,dAk )<
00
for 1 <a <2.
R
We have proved
Theorem 3.2.2.
{x(t) t€lR } is a m.s.c. stationary stochastic prok
cess then x(t) has continuous paths on every bounded set with probability
If
one, and hence continuous paths with probability one, if (3.2.10) is
satisfied.
Remark 3.2.3.
k
= 1.
This result is not as strong as Kawata's for the case
However, as remarked before, his proof fails if k > 1.
o
Remark 3.2.4.
By a more judicious choice of the function
f
3.2.1 it might be possible to obtain a weaker condition on
we have not been able to find such an
in Corollary
]J.
However,
o
f.
We now turn to the general case, where {x(g) : g€G} is a m.s.c.
stationary process on
The dual
G of
G and
G is of the form ~k
group and D is discrete.
. dexedb yR kxF
Yn' 1n
process
Since
zm
G is a compactly generated LeA group.
x
D x~· where T is the circle
Define for each n
.-..
= (n 1 , •.• ,nm)
bY~Yn(t,g;w)=x(t,g,n,w)
is a discrete group,
whe~~
in
Zm a
. tERk , gEF.
x will have continuous paths with prob-
58
ability 1 iff Yn does for each n.
has the spectral representation
=J
x(t,g,n)
k
R
Let us fix an n.
J J e2ni tOA(a,g >e 2ni nos Z{dA,
g-g',O)
where
= E(x(t,g,n)x(t',g' ,n))
R is the covariance of x.
stationary with covariance Rn (t,g) = R(t,g,O).
is given by
where
product
a-algebra
(3.2.11)
of
n (t ,g)
Y
nf
x O.
Thus
Yn is
The spectral measure of Y
n
= ~(6 x trn)
~'(6)
is the spectral measure of x, and
~
da, ds } •
O-rn
Yn has covariance E yn(t,g) yn(t',g')
= R(t-t',
Suppose x
6
is a Borel set in the
Also,
= J e2ni
tOA( a,g> Z~ (dA, da)
RkXD
Z~(6)
where
J
e 2ni nos Z{dA, da, ds},
6 a Borel subset of RkxO.
6x-rn
Now
~'
is a finite measure; because
~'(Rkx{a})
we set
Z!
J,n
=0
D is discrete, it
£0110\\'5
except forcountab1y many a
say {a j lj=l' Thus if
E a Borel set of . Rk then
(E) = Z' (E x {a.}) for
n
J
is an orthogonal random measure on Rk and from (3.2.11)
Z!
J,n
(3.2.12)
;;
The series (3.2.12) converges in mean square since the series
00
j!l~'(Rk x {aj}) converges.
that
59
f
x. (t) = e 2ni tOA Zl {d~}, then x. (t) is a stationary
J,n
J,n
J,n
k
R
process on Rk and the x. (t) processes are mutually orthogonal, for
J,n
and fixed n.
The next result shows that the series (3.2.12)
Write
converges uniformly with probability one under certain conditions.
Theorem 3.2.3.
Let
~:
be a function such that
D + [1,00)
00
L (¢(a.))-l < 00, where {a.}oois the set described above.
j=l
J
. Jj=l
measure ~ of the process x satisfies
f ~(a)~(dA,
(i)
da, ds) <
If the spectral
and
00
"
G
f
(ii)
"
n-!A ,a
ll= 1
ll.
~(dA,
da, ds)
<00
for
1 < a
<
2
G
then the process x has continuous paths with probability 1.
Because of (ii) the processes _x.J,n defined above have continuous paths with probability 1 by Theorem 3.2.2,since the spectral
Proof:
rm) .
~.
of x.
is given by llJ'(O) = ~'(6 ~ {a .} ~
J,n
J
Thus we only have to prove that the series (3.2.12) converges uniformly
measure
J
with probability one.
00
It is enough to show that
r
00
Now E
Ix. n(t)t = E L ¢(a.)j=l J,
j=l
J
00
E L Ix. (t)1
j=l J,n
~
1:
~(a.)~ Ix.
J
J,n
converges uniformly.
(t)1
.
60
n
= (
l
j=l
~ (a . )
-l~.
) (
J
r~
n
k .
(a . ) II (R x{ a. } x
j=l
J
J -
~
rn )
so the series (3.2.12) converges uniformly with probability one.
Yn
and thus
0
x have· continuous paths with probability 1.
Corollary 3.2.2.
If the spectral measure
Hence
llhas compact support, the
paths of x are continuous with probability 1.
Proof:
If
II
has compact support, then the conditions of Theorem 3.2.3
are trivially·satisfied, since the support of
the. f~rm [_A,A]k. x DO x.
II
is contained in a set of
TIt where DO is a finite ~~bset of
We now consider band limited processes indexed by G.
o.
0
The spectral
measure of such a process is concentrated on a compact set Ak.x DO x ,.m
say.
Since the set
DO is finite, it follows that a band limited proN
=l:
{a. ,g)x. (t)
j=l J
J,n
cess is of the form x(t,g,n)
Since
II
has compact support, the
also have compact support.
~ectral
where DO
measures
We will show that a
b~nd
= {a l ,··· ,aN}.
of the x.
J,n
limited process can
ll.
J
be expressed in a sampling series similar to (3.2.2).
Since
each x.
has a spectral measure with compact support in
J,n
Ak , it follows as in the case k= I that the series
(3.2.13)
x. (t) =
J,n
II
I
=_00
JJ
·r
I
111'1f
JJ
k
k·
x. n ( PIl
a , .. .,-~
a IT
=-~ J ,
t=1 I<J
converges uniformly on compact sets of Rk with probability I, where
61
a > A and PJl. =
sin a(tJl. - VJ/.n/a)
a(tJl. - VJl.lT/a)
Thus
(3.2.14)
Now let
H
be a discrete subgroup of F
For lsi<jsN, a
(3.2.15)
- a
l
such that
t A , the annihilator of H in D.
j
A is discrete, since it is a subgroup of a discrete group.
n £ D that contains exactly one element from each coset a
a set
in D/A.
Without loss of generality we may assume that
j = 1, ... ,n.
Define
S(g) =
f
n
,.
<aj , g) j =1 , . . . , n.
fj
Then
is cont inuous .
f.
n,
vanishes off
J
f.
J
Hence by a theorem of
a.
€
J
+ A
n,
e is the Haar measure
(a,g) El(da) where
on D, the integral exists (see Kluvanek 1965).
and
Choose
Now let
f. (g) =
J
(D), f j € L 2 (F)
Kluvanek (1965) the
€
L
2
series expansion
(3.2.16)
L(a. ,h) S(g-h) = (a. ,g)
h€H J
J
converges uniformly on F to the limit
x(t,g,n)
= LL L (a. ,h)
Vj h
=
k
S(g-h)x.
rr x(Jl:,h,n)
Vh
We have proved
J
(aj,g).
J,n
(Vlr) lTPn
k
S(g-h)
a
IT PJ/.
JI.=l
JI.=l~
Hence by (3.2.14)
62
Theorem 3.2.4.
If {x(g) : g€G} is a band limited m.s.c. stationary
process indexed by a compactly generated group
G, and H is a discrete
subgroup of the compact factor of G with the property (3.2.15), then
x(g) has the sampling expansion
(3.2.17)
l:
x(g) =
jl€
where
l:
x(Jl~,h,n) S(f-h)
n h€H
Z
k
IT
t=l
sina{t t -jl R, -rr/ a)
(tj/, -Jlj/, 'IT/a)
g = (t,f,n), and the convergence is with probability 1.
Remark 3.2.5.
The series (3.2.17) converges with probability one uni-
formly on compact sets of G.
This follows from the uniform convergence
0
of the series (3.2.13) and (3.2.16).
3.3 General Band Limited Processes on
IR and the Sampling Theorem.
In the previous two sections of this chapter we have alluded to the
concept of a band limited m. s.c. stationary process, which on
m
is a
process having a covariance R of the form
W
(3.3.1)
R(t) =
f
-w
e21riAt Jl(dA)
and noted that the importance of these processes lies in the fact that
almost all paths are analytic functions and that their paths may be
reconstructed by a knowledge of the values of the process at the
"sampling points"
k1r/a, k€Z, a>W.
It is of interest to consider pro-
cesses that are not stationary, and to find conditions that such processes
;
satisfy a sampling expansion, and have analytic paths.
Piranashvili (1967) proved that the sampling expansion (3.2.2) is
satisfied by second order processes having covariance functions of the
63
form
R(t,s)
=
I
~(dA,
f(t,A) f(s,n)
dn)·
AxA
where
~
is of finite variation on the bounded set AxA, and for each
.A€A, f(o,A) is a function of exponential type
<
W, say and
In a different approach, Zakai (1965) considersup If(t,A)1 < ~.
telR A€A
ed second order processes x(t) such that
2
co
I ~dt
(3.3.2)
1+t
-co
<
2
(lO
..
and found that such processes satisfy the sampling theorem (and have
almost all paths analytic) if they can be reproduced without distortion·
when passed through a certain filter.
Zakai's condition is "temporal"
in contrast to the "spectral" condition of Piranashvili.
Below, we briefly describe Zakai's results and then use his methods
to obtain similar results for more general processes than those satisfying (3.3.2).
Zakai considers' functions
f
~
f
(3.3.3)
satisfying
2
.1.f.(lli dt
-co
l+t
2
<
00
and defines such a function to be band limited (W,a) if f(t) is reproduced without distortion when passed through the filter
is the inverse Fourier transform of the function
That is to say,
f
h where
h
H(N,e) in Fig. 3.3.1.
is band. limited (N,a) if
co
f*h(t) =
f f(s)h(t-s)ds = f(t)
a.e.
_00
;;
Zakai then defines a process to be band limited (N,e) if almost all of
64
-\11-0
-w
Fig. 3.3.1
its sample paths are band limited (W,O) and it satisfies (3.3.2).
He-
characterizes such functions and processes and shows that the sampling.
"."
theorem is valid for them.
We will consider the class of all second order measurable processes
{x(t),te:W such that there exists a polynomial
2
Elx(t) 1 < pet)
(3.3.4)
If R(t,s)
= Ex(t)x(s)
p with
Vt e: 1R.
, then (3.3.4) implies that there exists a positive
integer k such that
(3.3.5)
We will denote by ~k the measure with density (1+t 2)-k i.e.,
. dPk = (1+t 2)-k dt and by L2 (P k ) the Hilbert space of all functions
with
f
65
We make the following definition:
Definition 3.3.1.
Let
$(t)
= ~(t;
co
be a C (i.e. infinitely dif-
W,o)
= $(x;W,o)
ferentiable) function whose Fourier transform ~(x)
CO
and is a C function such that
~(x;W,o)
=1
~(x;W,o)
=0
for
x ( [-W-o,W+o],
x ~ [-W,W] and is real and symmetric about
for
will say that a function
f
exists
is band limited (W,o)
(b~(W,o))
O.
We
if
o
co
Remark 3.3.1.
We should point out that if $ is a C function with com-
pact support [-A,A] then there exists a rapidly decreasing function
~
which is CCO, and in Ll n L2(~ such that ~ = $ and $ can be extended to an entire function on ~ (complex numbers) that satisfies an
inequality of the form (Donaghue (1969) p. 212):
(3.3.6)
!$(z)1
<
n Al1m
Cn(l+lzl)- e
for each positive integer n.
i
I
We will consider $ to be such a function
0
in the sequel.
Lemma 3.3.1.
~ L2(~k)
For f
exists, is in
L2(~k)
continuous function
and
$
as in Definition 3.3.1, f * $
and is continuous.
f
~~
$
may be extended to a
u on C by
co
(3.3.7)
u(z) :
J f(t)~(t-t;W,o)dt
.
-co
U is entire if
(3.3.8)
f
is continuous and U satisfies
66
Proof:
C
n
From Remark 3.3.1 for every integer
n ~ 1
there is a constant
such that
Thus
00
f
00
I I~(x-t)ldt ~Ck+l f
If(t)
If(t)l(l+lx-tl)-k-l dt
_00
inequality (3.3.9) is less than
and so the convolution exists.
To see that
f -'1'
cf> E:
L (lJ ), we proceed
Z k
as follows:
00
f
_00
2
000000
Jf*cP(x) L d
2 k
x
(l+x )
=f f f
_00
_W
00
00
00
-00
-00
00
(3.3.11)
< (C~) 2
I
_~
Jf(t)fIS);(x-t)$(x-s)
2 k
, (l+x)
I
dt ds dx
-00
00
fI
f(t)
12 (l+x)
Z -k (1+ I,x-t)
I-n
dt dx
since
21~llbl ~
2
lal
2
+ Ibl ,where
CC~)2= C~ j{1flsl)-nds,
67
n > 1.
-co
co
Define
J(1+x2)-k(1+(X_t)2)~kdX.
=
IkCt)
_00
Since 1 + t 2 ~2(1+x2)(1+(X-t)~), it can be seen that
Ik(t) < K(k).Ik.~1 (t)!(1+t 2)
and so
IkCt)
K(I)
I (t)- for some constant
(1+t 2)k-1 1
<
=
to Zakai (1965),
constant
2
< ~/l+t.
I 1 Ct)
K(O).
= 2k
Set n
co
(C 2k )2
for k > 1 and some constant K(k)
Thus
But according
2 ~
K )C1+t)
for some
CO
Then (3.3.11) is less than
Ik(t)
in (3.3.11).
K(l).
<
00
f J If(t)1 2c1+x2)-k C1 +(x_t)2)-k dx dt
_co
_00
co
=(C
Zk )2f
2
If(t)1 I k (t) dt
co
~(C2k)2K(0) f
2
2
If(t)1 /(l+t )k dt
<
00
•
_00
Hence
To see that
f(t)$Cxn-t)
U(x) is continuous, fiote that if'x'o+',x we hav.e
n
+
f(t)$(x-t)
for each t
If(t) Hx
I
n
-t)
<
CZk If(t)
2 k
(l+t )
since
I
~
00
is C.
(l+t 2)k
(1+lxn -tl)
Also
2k
Now C' ( 1+lx l)2k = 0(1) since xn is a convergent sequence, and
n
f(t) (1+t 2) -k € L (lR ) since
1
68
00
•
Thus we can apply the dominated convergence theorem to assert that
00
f
00
* ~(xn) = ff(t)$(Xn-t)dt
converges to
=f
*
~(x) •
_00
. . 'X)
Thus
ff(t)$(X-t)dt
f * $ = U is continuous.
If f
by Theorem 2.84 of Titchmarsh (1939).
is continuous then
U is entire
Finally, (3.3.8) is proved in a
similar way to the proof that the convolution exists; we omit the details.
o
Remark 3.3.2.
If f
is
bt(W,o) in our sense, then
to a continuous function.
Since
measure zero, we take for
f
the lemma,
we say that
f
£
L2 ( lJ k)
is equal a.e.
is defined only up to a set of
its continuous versionf * $.
f * $ has an entire extension of € -.
version of f.
(coR.(W)) •
f
f
Thus by
In the sequel, when
is bt(W,o) we shall always mean this entire
.0
Such a function is bounded by a constant (polynomial),
and is in
L2 (lR). If $ is as in Definition 3.3.1 then (f*$)A(S)=
f(s) $(s) = ~(s) since $= 1 on [-W,W] and ~(t) = 0 off [-W,W]. Thus
A
A
our definition contains the "conventional" definition as a special case.
o
69
Our next result shows how a
terms of
cb~(W+o)
Theorem 3.3.1 (A)
b~(W,o)
function can be expressed in
functions.
If f
E:
L2 (llk)
is
bt(W, 0) then its entire extension
can be wri tten
fez) =
(3.3.13)
k-l
I
fen) (0) In! /z 1) +·z
k
g(z)
n=O
where
g(z) is the (entire) extension of a
(B)
.
Let
k-l
k
f(t) =
c t n + t get)
n=O n
r
f(t) is bt(W,o) for W > A and 0
Proof:
Define functions
>
cb~(W+o)
where
g is
function .
cbt(A).
Then
O.
go, ... ,gk recursively by
(3.3.14)
n=l, ... ,k-l.
We claim that each
~
is entire, in
L2 (llk_n)
and that
for constants
(3.3.15)
The claim is obviously true for n = 0 by Lemma 3.3.1.
true for n
~(z)
so
<
k.
Expanding
= gn(O) + zw(z)
gn+ 1 is entire.
Now consider Izi
1.
n
•
Suppose it is
g (z) in its Taylor series about 0 we obtain
n
for some entire function
~
K
~.
But
~(z)
= gn+ l(z)
Then
K
!gn+1 (z)!
~ (!~(z)1 + Ign(o)I)/lzl ~~I{(1+lz1)k-ne(W+~)lIm
~ ~(l+lzl)k-ne(w+o)IIm
zl
,
= 2K (l+/zl/- n - l e(W+o) lIm zl (l+l z l
n
---rzr )
zl+11
70
For Izi s I
/g~~I/ is bounded by K~+l say, since ~+l is entire.
Set Kn +1 =
max(4Kn,K~+l}'
Then (3.3.15) is satisfied" for n + 1.
~+l E: . L2 (llk-n-1)'
see that
To
it suffices to show that
f /gn+l (t)
/2 11k _ _1 (dt) <
n
OC)
••
It/~l
since ~+1 is continuous and hence integrable over every finite
interval.
Now
~ E:
which is finite since
all
n, n=O, .•• ,k.
L2 (llk-n)'
From~.3.14)it
Thus the claim is satisfied for
follows that
(3.3.16)
Differentiating (3.3.l6) successively and setting
..
.g-n (0)
.
and so (3.3.13) is proved.
jgk(z)/
<
e(W+o)/Im z/
(1954) p. 103) that
B.
Since
f
.
= f (n)··
(O)/nl
Lastly, since
Ii
= 0, ••• ,k-1
gk
€
L2 (lR)
Since
and
it follows by the Paley-Weiner theorem (Boas
g is cbt(W+o).
is not in general Lebesgue integrable, it is the Fourier
transform not of a function but of a distribution.
1969).
z = 0, we obtain
fez) is entire and
(See e.g. Donoghue
71
(3.3.17)
f
is the Fourier transform of a distribution with compact support
[-A,A] (Donoghue (1969) p. 213).
So
bution T, with support [-A,A].
with W > A and
<5
>
0, and
l/J
f
=f
for some tempered distri-
~
Now if
is as in Definition 3.3.1
is any test function
00
(ia e. C with
A
compact support) set
from [-A,A].
p
= (~-l)l/J,
then
p
has compact support disjoint
Then (for the symbols used see Donoghue
(f*~)~(l/J)
(1969) §30)
= (f*~)l/J
= f($l/J)
~
= T(p)
A
+ T(l/J)
~
= T(liJ)
T= T,
since
~
A
and so has the same support as
suppT n supp p
Thus (f*~)
~
T, hence T(p)
=0
as
= ~.
=f
and so
f*~
=f
and f
is band1imited.
0
We now turn to the sampling representation ofa bt(W,o) function
f.
Such a function is entire and satisfies (3.3.17) for some k, it follows
that (Piranashvi1i (1967))
fez) has the sampling expansion
00
sin ci(z-mr/ll)
fez) = l.\ f(n~"
j
a (z-mr/CL)
n=-oo
(3.3.18)
where
a
>
W+o
and
13
<
k
'
sin l3(z-mr/a)
k
k
13 (z-mr/CL)
(a-W-o)/k.
If SN(z) denotes the partial sum of (3.3.l8) obtained by summing
from -N to N then
((NCL)k+l+(Na,,)
K L(z)
If(z) - SN(z) I< -:k:---'--''-'}
13 (CL-W-o-kl3)
..
72
~
where
L(z) is bounded on every bounded subset of C.
We will use this
result to derive a sampling theorem for second order processes that are
"
bounded in the following sense:
Let
Definition 3.3.3.
{x(t),t~IR}
he a m.s.c. second order measurable
process that satisfies (3.3.4) and hence satisfies (3.3.5) for some
positive .integer k.
Since x(t) is assumed measurable, almost all
sample paths are in
We will say x(t)
L2(~k).
-if almost all of its paths are
is band limited (W,o)
b~(W,o).
In the same manner as Zakai (1965) we can prove the following result,
whose proof is omitted.
Theorem 3.3.2.
Let
whose covariance
is
b~(W,o)
{X(t),tElR}
be a second order measurable process
R is continuous and satisfies (3.3.4).
Then
x(t)
iff
00
JR(t,V)~(S-V;W,O)dV = R(t,s)
(3.3.19)
_00
o
for all t,s in lR.
It is clear that a m.s.c. stationary process is
b~(W,o)
if its
spectral measure has compact support in [-W,W], so that this definition
of a band limited process extends
~he
definition it follows that if x(t) is
·00
(3.3.20)
x(t)
= I
n=--oo
X
~1T)
a
usual definition.
b~(W,o)
sin a(t-n1T/et)
et(t-n1T/a)
From the
then
sink act-n1T/a)
k
k
13 (t -n1T/a)
with probability one, for a,13 as in (3.3.18) so our band limited processes ;admit a sampling expansion.
We now turn to the problem of characterizing the band limited processes
73
in terms of simpler processes.
Theorem 3.3.3.
Let
{x(t)t€m} be
fying (3.3.5).
Then
(3.3.21)
where
with covariance R satis-
b~(W,Q)
•
x (t) =
k-l x(n) (0) n
l
It
n=O
n.
+
k
t Y(t) ,
x(n) is the n-th mean square derivative of x(t), yet) is a
harmonizableprocess whose spectral measure is concentrated on
[_W_o,W+o]2 and
Ely(t)1 2 = Ry (t,t) € qlR).
.
~
Proof:
Since
x is bt(W,o) its covariance function
R satisfies
(3.3.19) and hence by another application of (3.3.19)
co
co
J J Rtu ,v)
Ht-u)
</>
(s-v) dv du = R(t ,s) .
-co _co
Using the methods of Lemma 3.3.1, R can be extended to an entire
function
R(~,n)
which satisfies
Since the covariance
R has partial derivatives of all orders, the
mean square derivatives of the process
Define new processes
(3.3.21)
x(t) all exist.
vn(t) recursively by
vO(t) = x(t)
{ vn + 1 (t) = (vn(t) - vn(O)/t
whose covariances
Rn
n=l,.~.,k-l
are given by
RO(t,S) :: R(t,s)
(3.3.22)
Rn+ l (t,s)
= (Rn(t,s)
- Rn(t,O) - Rn(O,s)
+
Rn (O,O))/ts
74
As in Theorem 3.3.1 we can show that for each n = O, •.. ,k R is
n
IRnCt,t)I < C C1+/tl)2 Ck-n)
n
for some constants Cn' Thus
x(t) =
= Vk(t).
where Yet)
k-1
Cn)CO) n
k
x n.,
t + t Yet)
n= O
I
It thus remains to prove' that· yet) is a process
Since Ry = Rkis bounded, and RyCt,t) € L1(R),
2 .
L2Crn. ) and hence its Fourier transform ljJ(x,y) exists.
of the required type.
Ry(t,s)
€
By copying the proof of Theorem 6 in Zakai (1965) we can see that
.
off[-(W+~),W+o]
ljJ(x,y) is zero
Remark 3.3.3.
2
and so the result follows.
A measurable harmonizab1e process {x(t), t€ lR} whose
spectral measure is concentrated on [_A,A]2
~ >
O.
is bi(W,~) for W > A and
To see this, consider
A A
= J J e2~i(tu-sv)~(du,dv)
R(t,s)
-A -A
Then
0
00
J R(t,A)~(S-A,Wi~)dA
_00
00
= J ~(S-A)
J J e2~i(tu-AV)~(du,dv) dA
-A -A
_00
J J e2~itu I e-2~iAV41(S_A)d.A ll-(du.dv}.
A
=
A A
A
-A -A
A
A
00
_00
= J Je e2~i(tu~sv)$(_v) ~(du,dv)
-A -A
A
=
I
75
A
f f e 2ni (tu-sv) ~(du,dv) = R(t,s)
-A -A
00
since
=
~(-v)
A
. equal to 1 on [-A,AJ for W > A.
e 2niAv ~(A)dA 1S
Thus
_00
by (3.3.19)
x(t) is bR,(W ,0).
If in addition
R(t ,t)
then
l
almost all sample paths of x(t) are in L (R) and so by Theorem 3.3.1
l
almost all sample paths are cbR,(W') for W' > A .
€
L (R)
An alternative proof which gives
Alternative proof to Theorem 3.3.3.
some insight into the form of the series expansion of a bR,(W,o) process
is presented below.
Since
x(t) is m.s.c. and satisfies 3.3.5, it can
be expanded (Cambanis and Masry (1971». in a series.
00
(3.3.23)
= I
x(t ,w)
n=l
a
n
(t)~
n
(w)
where the series converges in mean square, the orthonormal random
variables
~n(w)
are given by
00
(3.3.24)
E;n (w) =
f
x (t ,w)
r;rn JJk (dt)
_00
almost surely (the functions
functions
an (t)
f
n
are complete in
»
2 k
L (JJ
and the
are given by
00
(3.3.25)
We need the following lemma.
Lemma 3.3.2.
If
x is bR,(W,o) the functions
a (t)
n
are also
b~(W,o).
76
Also
I
00
(3.3.26)
00
an (s)cj> (t-s)ds =
JE(x(s) E;n) cjl(t-s)ds
00
•
00
I Elx(s)"~nllcjl(t-s)ldS~C JR(s,S)~(l+lt_sl)-(k+l)
But
ds
_0>
which is finite by a now familiar argument.
Thus we can apply Fubini's theorem to the right side of (3.3.26)
to get
00
f x(s,w)cjl(t-s)ds) = E
E(E;n
~ x(t) = a (t)
n
n
_00
o
since x(t) is band limited (W,o).
Returning to the alternate proof of Theorem 3.3.3:
are a c.o.n.s (complete orthonormal system) for
spanned by the
cess
r E(x
0>
in mean square, i. e. ,
verify that
H(x), the Hilbert space
x(t), and because each derivative
x(t) is in H(x), we can express
x (j) (t) =
n=l
x(j)(t)
because the E:n's
x(j)(t) of the proas a series converging
(j) (t) ~ H:
It is easy to
n n
E x(j)(t)~ = a{j)(t), and so
n
(3.3.27)
j = 0,1,2, ...
Note that each an is bt(W,o) and hence analytic, and in
L2(~k)'
Theorem 3.3.1,
(3.3.28)
;:;
..
M
cbt(W+o). From (3.3.27) and (3.3.28) we get
00
k-l 00 an0) (0) j
k
+
t
E;
t
x(t) =
L bn(tHn
L J.. I
n
n=l
j=O n=l
where each bn is
r
so by
77
00
where yet) = L b (t)~ •
n=l n
n
by summing (3.3.28).
The convergence of this series is deduced
As in the first proof we deduce that
E/y(t)/2
is Lebesgue integrable, hence,
II
(3.3.29)
00
n=l _00
00
/bn (t)/2 dt = f Ely(t)12 dt
_00
<
00
•
Noweach bn is cbt(W+o) so for each n there is a function
e: L (-W-c ,W+O) such that
2
W+o
bn(t) =
e2~ixt~(x)dX
- (W+o)
~n(x)
I
By the Parseval theorem,
W+O
00
I/bn (t)/2 dt = J l~n(x)12 dx .
_00
-W-o
Hence by (3.3.29)
WJ+o
2
/~n(x)/ dx
n-l_ w_o
00
~
(3.3.30)
<
00
•
00
Thus
R_(t,S)
-Y
= Lb
n=l n
(t) b (5)
+W+o
= n~l f
n
W+o
f e2ni(tx-SY)~n(X)
~n(Y) dx dy •
-W-d -W-o
Now
n~l
W+o
W+o
J
J
-W-O -W-o
I~n(x)/ l~n(y)1
dx dy
78
W+o
00
~ 2 (W+o)
J l~n(x)12dX
l:
n=l
<
00
-W-o
by the Cauchy-Schwartz inequality and (3.3.30).
00
Thus
l: ~ (x)~)
n=l n
n
(-W-o,W+o]2
~(x,y) integrable on
converges to a function
and
W+o W+o
00
r
n=l
I I
-l~-o
-W-O
W+o
W+o
J
e2~i(tx-sy)W (x)~ (y) dx dy
n
n
converges to
J e2~i(tx-sy)~(x,y) dx dy •
-W-o -W-o
Thus
Ry
is harmonizable with a a.c. spectral measure concentrated on
(-1'1-0 ,W+o].
0
The next result is a converse to Theorem 3.3.3.
Theorem 3.3.4.
Let
y
bea harmonizable process whose spectral meas-
ure is absolutely continuous and concentrated on [_A,A]2
and whose
Ry satisfies Ry(t, t) e: LI (m). If CO',,· ,C k_l are
random variables with E Icn 12 < 00, n = 0, ..• ,k-l then the process
covariance
x(t)
k-l
=r
n=O
Cn t
is b1(W,o) for all W > A and 0
n
>
k
+ t yet)
O.
Proof:
By Remark 3.3. 3, almost every sample function of y is
cb1(W).
From Theorem 3.3.l(B) it follows that almost every sample path
79
L2(~k)'
of x is bt(W,o) and in
Thus
x(t) is bt(W,o), W > A,
o > o.
0
Remark 3.3.4.
If x(t) is bt(W,o) with covariance
R(t,s) satisfying
(3.3.5), then the sample paths of x are bt(W,o) and in
L2(~k)'
The
same is true for the expansion coefficient functions
the functions
subspaces of
a and also for
n
In fact we can say more; the closed
R(' ,s) for each s.
L2(~k)
generated by these three sets of functions coincide.
This is a consequence of the following general theorem.
Theorem 3.3.5.
Let
{x(t),uIR}
be a measurable m.s.c. second order
process whose covariance R satisfies
00
f R(t,t)~(dt)
(3.3.31)
<
00
.00
for some finite measure
~,
with
mutually absolutely continuous with
~
respect to Lebesgue measure and let x(t) have the expansion (3.3.23).
If Sea), Sex)
and S(R)
denote the closed subspaces of
ated by the coefficient functions
the functions
Proof:
{R(·,s)
:~m}
First we show that
L2(~)
gener-
a, almost all the sample paths and
n
respectively, then Sea)
Sea) = S(R).
= Sex) = S(R).
For each fixed
~R
00
(3.3.32)
R(t,s)
=L
n=l
a (t) a (s)
n
the series (3.3.32) converges absolutely.
n
Let
-N
N
L a (t) an (s).
n=l n
~-(t) =
Then
the integrand of the right side converges pointwise everywhere to zero.
80
Also
N
IR(t,S) -
co
r
ra (t) a (sll < IR(t,sll +
la (t)1 la (5)/
n=l n
n
=
n=l n
n
~ (R(t,t)R(s,s))~
-
co
.
+ (
co
r /a (t)/2)*( r la (S)/2)~
n=l
n
n=l
n
= (R(t,t)R(s,S))2k
N
2
and thus
S(R) £ Sea).
r an(t) an(s) I <4R(t,t)R(s,s) € L1 (~). So by the dominn=l
ated convergence theorem it follows that ~ (0) converges to R(. ,5) in
so /R(t,s) -
L2(~)
a
€
To prove equality in this inclusion, let
Sea) be orthogonal to each R(o ,s).
Then
co
I R(t,s)
which implies
co
=
J
_co
But R(t,s)
am-
f n (5)
E:
E:
~(dt) = 0
Vs
IR
E:
co
=J an(t)
(3.3.33)
art)
art)
~(dt)
co
f R(t,s)fn(s)
art) lJ(ds) lJ(dt) •
_00
L2 (lR X ffi, ~"'.~ lJ) since R(t,t) E: L1(IR,~).
Also
L2 (R x R, ~ x lJ) since each factor is in L2 (lJ). Thus by
Holder's inequali ty R(t ,s) l;n (s) a (t)
L1 (R x R, lJ x ll) and we can
appeal to Fubini's theorem to assert that (3.3.33) is equal to
co
J
€
co
I R(t,s)
art) ll(dt) fn(s) lJ(ds) = O.
Thus (a ,a) = 0 Vn
n
.
and
_co -co
;:
so a = 0, and S(R) = Sea).
Next we show that
Sex) = S(R).
By
theorem 8 of Cambanis and Masry (1971) the series (3.3.23) converges
81
in
L2(~)
almost surely.
Thus if
O denotes the set of probability
x(t,w) E:: Sea) Yw,~ Qo' Thus if
zero where convergence fails,
= sp{x(· ,w),
Sex)
W
W E::
Q-QO'
f
E::
= S(R).
S(x) So Sea)
E::Q-Qo}
inclusion, suppose that
Q
To prove the reverse
S(R) is orthogonal to each -x(·,w)
Then
I
00
x(t,w)~(dt) =
f(t)
Yw
0
E:: Q-Qo
_00
so
I
00
o=
E
f(t) x(t,w)
~(dt)
2
_00
lXl
(3.3.34)
=
lXl
I J f(t) R(t,~)f(s) ~(dt) ~(ds)
_'lXl
.
_lXl
L2(~)
Now let R be the operator defined on
by
I
lXl
(3.3.35)
Rf(t)
= f(s)R(t,s)~(ds).
R is a compact symmetric operator on
L2(~)
L2i~)
so every element
f
of
can be represented as
(lC)
(3.3.36)
r
f =h +
(f,e )e
n=l
n n
where h is the projection of' f
on the subspace {f: Rf = O}
and
the e 's are eigenvectors forming a c.o.n.s. for the closure of the
n
-
'.
range of R, which is. the closed subspace spanned by these eigenvectors
of R corresponding to non-zero eigenvalues {An} of R.
Sz-Nagy (1955) p. 242.
(See Reisz and
Then by (3.3.35) and (3.3.36) (Rf,f) =
00
l
Ani (f,eJI 2 and so by (3.3.34) and (3.3.35) (f,en ) = O.
n=l
.
in the null space of R. Also (Cambanis and Masry 1971)
Thus
f
is
82
00
(3.3.37)
R(t,s)
= LA e
n=l n n
the series (3.3.37) converging in
(t) en(s)
L2(~k)
is in the closure of the range of R, and
f
~
-:=-----=::L
Range R , f
~
S(R) £ Range R so
in each variable, so R(·,s)
S(R) £ Range R.
f = O.
Hence
But
S(R) = S(x).
o
Remark 3.3.5.
As
note~
S(R) ,S (x) and. S(a)
process
in Remark 3.3.4, the
thre~
equal subspaces
are generated by band limited functions if the
x(t) is band limited.
In fact, all the functions in these
three subspaces are band limited.
To prove this, we argue as follows.
The linear manifold of finite linear combinations of the
a 's
n
consists
of band limited functions, since
Now if h
€
Sea), there exists a sequence of functions
hn converging
a linear combination of the a 's
n
and hence each hn is band limited, and
llh - h*4>1'-< Ilh - hnll
= II h
.
Thus
+
- hn II +
<
IIh - hnll
+
<
IIh - hnll
(1
Ilhn - h*<I>11
II (hn
- h) *<1>11
a2k(K(O))~llhn-hll
+
by (3.3.12)
C2k(K(O))~)
h is band limited and Sea) contains only band limited functions .
CHAPTER IV
STOCHASTIC PROCESSES TAKING VALUES
IN AHILBERT SPACE
In this
process.
chapte~
we consider a further generalization of a stochastic
Instead of considering processes that are 'collections of
complex-valued random variables, we now consider collections of random
elements that take values in a separable Hilbert space.
~
These
include as a special case the concept of a multivariate stochastic
process, where the Hilbert space is the n-dimensional Euclidean space
m.
n
Hilbert space valued processes have been considered by several
authors.
A basic source is the paper of Payen [1967], who defines
second order Hilbert space valued processes, and shows how such a process may be realized as a family of operators.
He defines the co-
variance operator of such a process and characterizes this class of
operators.
He also gives a definition of a stationary Hilbert space
valued process and obtains a spectral representation and a moving
average representation.
We will use several of Payen's results in the sequel.
TIle integration
of operator-valued functions will also playa role in what follows; for
the basic facts of this theory, see e.g. Hille and Phillips (1957).
Several results of Mandrekar and· Salehi (1970) are also used.
In section 4:1 we introduce the concept of Hilbert space valued
(H-va1ued) random elements, and show how they can be identified with a
83
84
class of Hilbert-Schmidt operators in the case when they have second
absolute moments.
We have relied heavily on Payen (1967), and for the
theory of Hilbert-Schmidt operators, on Shatten (1960).
In section
4.2 we define a second order H-valued stochastic process, and show how
it can be realized as a family of Hilbert-Schmidt operators.
We define
the covariance operatQr of such a process, and show that if the process
is stationary in a sense to be defined, then the covariance operator
can be decomposed into weakly continuous and lo.cally a. e. zero components'as" in Theorem 2.2.2.
for processes.
~
stationary
The corresponding decomposition is given
In section 4.3 we give the spectral representation of
p~ocess.
Our
apprQ~ch
is
pas~9
on Bochner's theorem, in
~ontr~st
with that of payen, who uses Stone's theorem to obtain his
sp~ctra1
representation.
We prove a generalized Bochner's theorem,
using a result of Mandrekar and Salehi (1970) to extend a theorem of
Palb (1969).
We rely heavily on the concept of the trace measure, as
in Mandrekar and Salehi, to prove a spectral representation.
Finally,
in section 4.4, we take up the questions of path continuity and sampling
considered in Chapter III, and give Hilbert space analogues of theorems
proved there.
4.1.
Random Elements Taking Values in a Hilbert Space
Much of this section is an elaboration of Payen (1967), and the
mat.erial on. tensor products. is found in Schatten (1950).
Let
H be a separable Hilbert space, H* its dual.
-:H ~ H* by f(h)
between
= (h,f).
Define a map
- is a conjugate linear isomorphism
H and H* (Schatten (1950)).
Let
K be another Hi lbert
space, not necessarily separable.
map H*
+
For (f€H,keK) denote by f 0 k the
85
K. given by
(4.1.1)
f ~ k(h) = (f,h)k
It is easily seen that
f
e k is a finite rank bounded linear operator
from H* to K with norm I If 1I I Ikl I (see e.g. Scha~ten (1959».
Let
H @ K denote the vector subspace of L(H*,K) (the space of bounded
linear maps
H
@
H*
K ) generated by f ek
+
for f € H, k € K.
On
K . we may define an inner product by
n
(4.1.2)
(
<
I
i=l
h.
e
k. ,
1
1
m
I
h!
j=l J
e
k!
J
)
n m
= I I (h., h ! ) H(k. ,k!) K
i=lj=l 1 J
1
J
where the inner products on the right are those of the spaces H,and
The completion of H
@
K.
K with respe etto this inner product' is' a Hilbert
space which we denote by He K -:thetensor product of Hand K.
Now
H @ K is a'vector space of finite rank operators. Since H is reflexive,
it follows (see Schatten (1950) p. 25) that H @ K is precisely the
class of all finite rank operators from H* to K.
complete orthonormal system (c.o.n.s.) for
c.Q.n.s. for
operators
H*
H*.
+
H.· Then
00
K i. e. the set', of' ~al i bounded operatorsA : H*
00
\.
I IIA$nll~
<
+00.
n=l
.
product (o;o)Hs'given by
(4.1.3)
'"
{~n}n=l
is a
Denote by HS(H*,K) the set of al1'Hilb'ert-SCJ:unidt
.," ;_.
for which
00
Let now {~nb=l be a
_"
+' K
_."• • 0
~HS(H* ,Kf is a Hilbert space with inner
'i.~
•
00
(A,B)HS = n!l (A$n'~~n)
The finite rank operators H*
+
'k' ar'e dens'e in . HS(H* ,K) in the topology
induced by the Hilbert-Schmidt inner product (o,o)HS'
Moreover on
..
86
H @ K the inner product (4.1.2) agrees with the inner product
(4~1.3).
H @K and HS(H*,K) are equal up to isomorphism.
It follows that
In the sequel we will identify HS(H*,K) with
H @ K.
We now turn to a discussion of H-valued random elements.
Definition 4.1.1.
x: n
~
Let cn,B,p)
be a probability space.
H is weakly measurable if (xCw),h)H is measurable in the usual
sense for all
h
€
H, strongly measurable if it is the limit a.e. [p]
"
n
I
"of" a sequence of simple fimcti"ons of the form
h. XB
. 11.
1=
where h.
1
1
E:
H,
being taken in the topology of H.
1, ••. ,n, the limit
Since
A function
H is assumed separable, the notions of strong and weak measura-
bility coincide (see Hille and Phillips (1957) p. 73) so we will just
speak of measurable maps.
A measurable map
Q ~
H is a random element.
o
Let
fl
L2Cn,H)
/xCw)! ,2dP <
be the set of all random elements
x such that
L2 (n,H) is a Hilbert space under the inner product
+00.
n
(x,y)
= JCX(w),yCW))
dp •
n
The following theorem is fundamental Csee Payen (1967)).
x be a random element inL 2 Cn,H). Then there
exists a unique operator X in HSCH,L 2 cn,C)) such that Xh = (h,x).
Theorem 4. 1. 1.
Let
Conversely, if X € HSCH,L2Cn,C))
L2(n~H)
and
such that
there exists a unique element
x in
Xh = (h,x). This correspondence between HSCH,L2 (n,C))
L2 (n,H) is an isometry.
n
r
with
87
First we note that random elements 6f the form
~.(w)h.
,
'. i=l 1. . 1.
Proof:
;i
L2 (O,C) are dense in LZ(O,H). Define a map
T : ,L2 (O,H) + H 9 LZ(O,C) (LZ(O,C) is the space of all complex r.v.'s
~ on
0
€
n
r
;.
(w)h.) = r h.
i=l
i=l
n
with EI~12 < 00)
by 'I(
1
1
9 ;..
1.
T is well
1
defined and preserves inner products, so can be extended to an isometry
between
L2 (O,H) and H @: LZ(O,C).
By the previous remarks of this
section, we can regard Tx as a Hilbert Schmidt operator H*+LZ(O,H).
Let
X be the functionH
LZCO,C) defined by Xh = TX(h)
+
We now must prove that Xh
bar denotes complex conjugate.
where
(h,x)
where the
~
(h,x)
is the random variable (h,x)(w) = (h,x(w)),' and that
X € HS(H,LZ(O,C)).
Since
is measurable, (h,x(w)) is measurable
2
Z
LZ(O,C) since EI (h,x(w)) I < Ellx(w) 11 'llhll <
for each h, and is in
x(w)
n
First suppose that
x(w) =
.
t 1;.(w)h.,
1.1
then
1=
,...
Tx(h) =(
n
n
r h.
@ ;.)(h) =
• 11.1.
1=
so
Xh = (h,x).
Now let
•
r (h.ill);.
1=
x
1
1.
LZCO,H),
€
1.
=(Lh.;.,h) = (x,h)
1. 1.
then there exists a sequence
x of random elements of the above form such that xn + x in
n
LZ(o,H). It follows that TXnh converges to Tx h in L2 (O,C)
each h
since
j ITXnh - Tx
hi IL
Z
(O,e) < IITxn -
Also (xn,h) converges to (x,h) in LZ(O,C).
for
Txll Ilh II
Hence
Tx h = lim T~n h = lim(xn ;h) = (x,h)
and so Xh = (h,x) for all x
€
L2 (O,H), Vh
€
H.
This representation
00.
88
also shows that .X
is unique.
Finally we must show that
X € HS(H,L 2 (Q,C)). It is easy to see that
2
2 EITx~12 .;.
II Tx~1I2
=
/IX<!Jn 11 =EIX(<!J)1
n
X is linear, and
so
X is
HS
since
since Tx is.
Conversely, if X € HS(H,L 2 (Q,C)) then the map
defined by
is in
HS(H*,L2 (n,C)), and so there exists an
and so
4.2
X(h) = X(h)
Xh
= (h,x).
x € L2 (Q,H) with Tx
0
....
= X,
H-valued Stochastic Processes
Definition 4.2.1.
Let G be an LCA group and {x(g) : g€ G}
a family
of random elements with values in the separable Hilbert space ~ If
for each g ~ G we have EI Ix(g) I 12 <
we will say that x(g) is a
00
second order H-valued process.
Remark 4. 2. 1.
For each g € G, let
associated with
Xg be the Hilbert-Schmidt operator
x(g) as in Theorem 4.1.1. It is clear that the closed
subspace of L2 (Q,H) generated by the x(g) and the closed subspace of
HS(H,L2 (Q,C)) generated by the Xg are isometric. Then nothing is lost
by identifying the process x(g) with the family of operators X.
g
Definition 4.2.2.
Let
{x(g) : g € G}
be a second order H-valued
process, and
{Xg
g € G} the corresponding family of operators.
The covariance operator of the process x(g) is the operator valued
function
V(g,g')
= X*,X
g g
, where
* denotes the adjoint.
V(g,g') is
89
an operatorH -+ H and since X,
and Xg are Hilbert-Schmidt, V(g,g')
g
is trace class. If the function V(g,g') depends only on g-g', the
process
x(g) is said to be weakly stationary.
For brevity, we will
use the term "stationary" in the sequel, and we shall also refer to
the operators
Xg as a "stochastic process."
Definition 4.2.3.
A map V: G x G + L(H,H) (the class of bounded
linear operators H + H ) is of positive type if for any finite subset
n
n
1=
J=
I I (V(g.~ ,g.)h.,h;)
. 1 . 1
J ~ J
(4.2.1)
Theorem 4.2.1.
(Payen)
~ 0 •
The following are equivalent.
(a)
The map V : G x G + L(H,H)
(b)
There exists a probability space (n,B,p)
ators
is
~f
positive type.
and a family of oper-
H + L2 (n,H) such that X*,X
= V(g,g').
g g
Definition 4.2.4.
A map V: G + L(H,H)
n
I
(4.2.2)
n
I
i=l j=l
is ,positive definite if for
(V(g.-g.)h.,h.) ~ 0
. 1 J 1 J
The following result is a corollary of Payen's theorem.
Theorem 4.2.2.
(a)
The following are equivalent.
V is a positive definite map G +
T(H,H)~.(T(H,H)
denotes the
set of trace class operators H.+ H.)
(b)
There exists a probability space (n,B,p) and a stationary process {X g } of Hilbert-Schmidt
X~,Xg
= V(g-g').
operators H + L2Cn,C) such that
0
90
Our next result is an analog of Theorem 2.2.2.
We first need a
definition.
Definition 4.2.5.
Following Hille and Phillips (1957) p. 74, we say that
a map Y : G + L(HtH)
the map
g
+
is weakly measurable if for every h, h'
€
H
(Y(g)h,h') is measurable in the usual sense, and strongly
measurable if Y(g)h is strongly measurable in the sense of Definition
4.1.1 for all h
€
o
H.
We can "now-state our theorem.
Theorem 4.2.3.
Let Y: G + T(H,H) be a weakly measurable trace class
operator valued function on. G.
Then there exist unique weakly measurable
trace class operator valued functions' Vel)
(i)
Y(g)
= y(l) (g)
(li)
The map
g
(iii)
y(2) (g)
=0
+
y(2) (g) :
y{l) (g)
+
and y(2) on G such that
is weakly continuous.
locally almost everywhere with respect to the
Haar measure on G.
Proof:
The proof is similar to that of Theorem 2.2.2.
Let
m denote
the vector space of all maps G + H that are identically zero except
at a finite number of points
gl"" ,gn'
Every element
f
of
lE:
has
a canonical representation
n
(4.2.1)
f(g)
=l
i=l
h.e
1
gi
(g)
where
..
e:
..
gi
.
(g)
={ 1
.'
0
g = g.
1
g -F g.
1
and the
gi
are distinct elements of G.
Define a map p,
Ex lE+
C
by
p
91
I
n
m
]
n
f4
Lho'e:
I
L
h!e:,
::
L
L
(V(g.-g'.)h.lh!)
i=1 1 gi j=l J gj
i=l j=l
1
J 1 J
ll(,
As in the scalar case it can be shown that
is a subspace of
lE
I
the completion of
-
,~-
.
.'
E : p(flf) =
E:
o}
and that the equation
defines an inner product on the quotient space
'.
...............
_.-
= {f
Let H
IE/E •
v
denote
v
lE IlEv, with respect
to this inner product.
,
On lE/IE
the equation
= fg
U (f + lE)
v
g
where
fg(g') = f(g' - g)
+ IE
v
lE/~
defines a unitary operator on
which
may be extended to a unitary operator on Hv • We thus obtain a unitary
group {Ug : g € G} of,unitary operators on H . As in the scalar, case,
v
since V is weakly measurable l we can find a closed subspace S of
such that
H
v
USc S. U S~ c S~
g - • g -
(i)
for all n
(ii)
€
V
g
HV the map g
and locallya.e. zero if ~
(U g ~In) is continuous if
+
€
~ E:
S
S~.
PIQ be the projection operators on S and S~ so that Q = I-PI
Let
PQ
G
=
QP
= O.
[h] = he: e
+
If e denotes the identity of GI and h
lEv
I
then [h]
€
Hv .
Denote by
E:
HI let
W
g the map H x H + C
;?i:v~n; ~~y:
= (UgP[h], P[h'])H
v
It is easy to see that
W
is a bilinear functional l moreover
g
v
92
11/1 (h,h') I ~ lIu P[h] II H IIp[h'] II H
g
g
v
v
~ Il£h]
v
v
= 1, II Ug /I = l.
since
" P II
But
II [h] II~
v
Thus
II H II [h' ] II H
= p(h€e,he: e ) = (Y(e)h,h)
ItPg(h,h') I
<
~ IIY (e) II IIhl1 2
IIY(e) II Ilhll Ilh'll
bilinear fWlctional.
so that
tP g is a bOWlded
Thus there exists an operator y(l) (g)
such that
= (UgP[h],P[h']).
(y(l)(g) h,h')
That y(l) is weakly continuous follows from the properties of the subspace S.
To see that
n
l
n
l
i=l j=l
y(l) is positive definite,
(V
(1)
n
(g. - g . ) h. , h. ) =
1.J
J
1
n
I
l
i=l j=l
(U
gi
P[h.], U P[h.])
1
gj
.J
n
=
II . l 1Ug. P [h 1. ] 11 2 ~
1=
0 •
1
In a similar manner we can show that there is a positive definite operator
function y(2)
such that
y(2)
=0
locally a.e. and
(V(2)(g)h,h') = (U Q[h], Q[h'])
g
Moreover
=
(Ug[h],[h'])
= (UgP[h],P[h'])
+
(UgQ[h],Q[h'])
= (V(l)(g)h,h') + (y(2)(g)h,h') •
It only remains to prove that "yO}' and y~2~
.
that the decomposition is Wlique.
,
are trace cIa'ss valued!'. and
The latter is proved by the method used
93
in the scalar case; the former is a consequence of the next theorem.
o
Let X
Theorem 4.2.4.
g
be a stationary process of
operators corresponding to the stationary
H~valued
Hilbert~SChmidt
process {x(g),g
€
G}.
Suppose V, the covariance operator of the process is weakly measurable.
If V = vel)
+
V(2)
is the decomposition of V described in the previous
theorem, then there exist stationary operator processes X(l), X(2)
g
corresponding to stationary
(i)
X
g
= X(l)
g
+
processes xl(g), x (g) such that
2
x(2) and x(g) = x (g) + x (g) •
H~valued
I
g
x(i)*x(i) = V(i) (g~g') where Veil
g'
g
of x.,
i = 1,2 .
J.
(ii)
x(l) *X(2) = E(x (g'),x (g)) = 0
2
g'
g
l
(iii)
(iv)
g
2
is the covariance operator
Vg, g'
€
G •
If H(X), H(x(l), H(X(2)) denote the closed subspaces of
HS(H,L (f2,C)) generated by the processes X, x(l) and x(2)
2
and H(x), H(x l ) H(x2) denote the closed subspaces of
L2 (f2,H) generated by the processes
Proof:
Consider the map
n
L(
I h. €
. I 1. g.
1.=
J.
L
lE / lEv
-+
x, xl and x2 then
L2 (0, C) given by
n
+ lE)
V
= • I I Xg. h.1.
1.=
L is well defined and
pres~rves
inner
1.
products, and can be extended to an isometry from Hv to the closed
Let
subspace M of L2 (f2,C) generated by {Xgh : g € G, h E: tI}
p = LPL~l, where P is the projection onto S of the last theorem.
.
\0
,
,
94
,...
The
P is a projection.
Define xCI)
g
= PXg .
Then
XCI) is Hilbert-Schmidt since it is the
g
composition of a bdunded operator with a Hilbert-Schmidt operator.
Moreover
h') = CPX h, PX ,hl)L
CXCI)*X(I)h
C)
g'
g ,
H
g
g
2'
en
= CLPL -1 Xgh,
C4.2.2)
LPL
-1
Xgl h)L cn,C)
2
= (PU [h], PU I [h I J) H
g
g
v
Now S reduces
Ug
for each g, so
P commutes with Ug and
~.2.2)
equals
(U
=
g-g IP[h], P[h'])Hv
(y(l) (g-g')h,h')H
l
xi;)*xi ) = y(l)(g_g'). Similarly let
so
Q=
projection onto Sol of the last theorem.
y(2)(g-gl.)=Xi~)*Xi2). Thus x~l)
and y(2)
are trace-class valued.
,...
P
,...
+
Q = I, Xg
g
g
P and
+
g
and
are stationary, and Vel)
,...
Since
= X(l)
Q is the
Define X(2) = QX
X~2)
aQd
LQL- l where
x(2)
g
,...
Q satisfy
V
g
€
G •
x CI )*x C2 ) = X* ,...,...
The decomposition is
g'
g
g' PQXg = 0 since PQ = O.
mique since it is comprised of projections. The last statement
Also
follows as in the scalar case.
The assertions for the process
o
follow from the above and Theorem 4.1.1.
As
x(g)
in the scalar case we use these results to obtain sufficient
conditions for a stationary H-valued process to be weakly continuous,
i.e. for (V(g)h,h ' ) to be continuous.
First we note that, exactly as
.. (2)
in Chapter I, if Hv is separable, then V . fg)= 0 Le. YCg) is
weakly continuous. In terms of the process, Y(g) is weakly continuous
95
if H(X) is separable.
Definition 4.2.6.
h
€
An H-valued process
xis measurable if for every
H, (x(g,w),h) is measurable with respect to the product a-field of
0
subsets of G x n.
It is clear that if x is measurable and
process corresponding to it, then
Xg is the "operator"
Xgh (regarded as a complex stochastic
process) is product measurable for each h
€
H and conversely, since
Xgh(w) =' (h,x{g,w))L (n,C) .
2
It follows from Theorem 2.2.4 that if
M{h)
denotes the closed linear
subspace of L2 (n,C) generated by {Xgh : g € G}, then M(h) is separable.
Let D be a countable dense set in H , and let M be as in the last
theorem.
It is obvious that
M(h)
M for all
c
the closed linear manifold generated by a subset
[ u M(h)]
h€D
h € H, so if
TAT
is
A of M we have
c M
Conversely if X;
€ M, let hn be a sequence in 0 converging to
g
; € H.
Then X
h converges to Xg~ in L2 (n,C) as n ~ 0 0 . Since
-g n
each X h
gn
€
u M(h), it follows that
h€D
and so M= 1 u M(h)] .
h€D
Each M(h)
But
is separable and 0
M is isomorphic to
continuous.
H
v
so
is countable, so M is separable.
Hv is separable, hence
V(g)
is weakly
We have proved:
Theorem 4.2.5.
Let
Schmidt operators.
{X
g
: g € G}
be a stationary process of Hilbert-
If X / is weakly measurable (i.e. if the processes
g
96
Xg h
are measurable for all h
€
H) then the covariance V(g) of the
process is weakly continuous.
4.3.
The Spectral Representation of Weakly Continuous Stationary H-valued
Processes.
In this section we derive the spectral representation of a weakly
continuous process using methods different from those of Payen (1967).
We use a verion of Bochner's theorem which is an extension of a result
of Falb (1969), and draw on results of Mandrekar and Salehi (1970).
We
prove a general representation theorem to obtain the final result, that
is more general than Payen's representation theorem.
Our first result
is a theorem of Mandrekar and Salehi; we give a proof based on a technique
of Dincleanu (1967) p. 263 since this is not given in the paper of
Mandrekar and Salehi.
M be a set function on a measurable
space (S,S)
,
Let
Theorem 4.3.1.
that takes values in the set of positive trace class operators T+(H,H).
00
Suppose also that M is weakly countably additive,i.e. if {6n }n=1
a sequence of disjoint sets in S, then
is
00
(M( U 6n )x,y) = I Q4(6n )x,y)
n=l
n=l
(4.3.1)
for all x,y
(1)
€
H. Then
If
T
is the set function defined on the sets in S by
T(6) = trace M(6) then
(2)
T is a positive finite measure on S.
There exists a strongly measurable operator valued function
M'(s) on S such that M'(s) is positive and trace class a.e. [T]
and
97
M(~)
(4.3.2)
=
f M'
(s) T(ds) •
~
Remark 4.3.1.
sense.
The integral (4.3.2) is interpreted in the following
Since M'(s) is strongly measurable,
H-va1ued function on
S.
T
..
M'(s)h is a measurable
is a positive measure, so the integral
[M'(s)hT(ds) exists as a Bochner integral (see e.g. Hille and Phillips
li
(l957) p. 79) iff
filM' (s)h II HT(ds)
<
Moreover, if jM'(s)hT(ds)
00.
~
,
~
exists as a Bochner integral for each h
(1957) p. 85) the operator
E
H then (Hille and Phillips
A given by
Ah = fM'(S)hT(dS)
6
is a bounded operator.
The theorem asserts this operator is just M(6).
o
Proof of Theorem 4.3.1:
S
and
00
{~k}k=l
00
{6n }n=1 be a disjoint sequence in
a c.o.n.s. in H. Then
(1)
T(
Let
U 6 ) = trace M(U6 )
n=l n
n
00
00
00
00
(4.3.3)
But M(6n ) is a positive operator for each n so (M(6n)~k'~k) ~ 0,
Vn, \fk. Thus we may rearrange the order of summation (4.3.3) and obtain
.
..
98
so that
T is countably additive.
follows from the fact that M(6)
.
for each 6
T is positive and finite
is a positive trace class operator
S•
E:
For each x,y
(2)
That
H let
E:
~xy(6)
= (M(6)x,y).
weakly countably additive, ~xy is a complex measure.
denote the variation of
~
I~xyl (6) = sup{
xy .
Since M is
Let
I~xyl
Then
n
L I~ xy (6k) I
k=l
: {6k}~=1 is a partition of 6}
n
~ sup L IIM(6k) Ilullxll Ilyll
{6 }k=1
k 00
<
(4.3.4)
since
sup L T(6k) lIxll llyll
{6 }k=1
k
= T(6) lIxll lIyll
I IAI Iu
(the uniform norm of
T.
complex function
~ xy (6)
=
is less than T(A) for all positive
By (4.3.4) I~ xy I is absolutely continuous with
Hence by the Radon-Nikodym theorem there exists a
trace class operators
respect to
A)
A.
gxy such that for all 6
f gx(s)T(ds)
y·
and
€
l~xyl(6) = JlgXy(S)/T(dS)
6
Since
S
6
I~xyl (6) ~ T(6) Ilxll Ilyll, it follows that
a.e. [T].
(4.3.5)
Without loss of generality we can modify gxy on a set of zero
T-measure and assume that
(4.3.6)
Igx/s)I ~ Ilxll Ilyll
for all
5 €
S.
99
It is easy to see that the map
on
H for each fixed
functional.
<x,y)
-+
i~ a bilinear function
gxy (s.)
(4.3.6) shows that it is a bounded bilinear
s~
Hence there is a botmded operator HI (s) on H to H
such that
(M'(s)x,y) = gxy (s)
11M' (5)1 I
(4.3.6) shows that
~ I
Vs
€
S.
Vs
Since
€
S, Vx,y
€
gxy is measurable,
M'(s) is weakly measurable and hence strongly measurable since
separable.
11M' (s)xll
Now
... ~q.Jlac:(s)xrh(ds)<
<
'1M' (s) II
H.
H is
Ilxll ~ Ilxll and T is finite
Thus the equation A6 (xJ= JM' (s)xT(ds)
6
defines a bounded operator A6 for each 6 € S.
00
Yx
€
H.
6
But
x,y) =
6
(A
J(M' (s)x,Y)T(ds)
Is
= f gxy (s) T(ds)
6
= lJxy (6)
= (M(A)X,y)
so M(6) =
fM' (s) T(ds)
6
in the sense of Remark 4.3.1.
It remains to
.
prove that M'(s) is positive and trace class.
Now
~t'(s)x,x)
for all 6
€
S.
= gxx(s)
~
0 a.e. [T]
lJ
Similarly
00
l:
k=l
and
since
00
(lvl' (sHk ,If>k) =
l:
k=l
gl/>
If> (5)
k' k
xx (6)
= (M(6)x,x)
~
0
100
fI::,.k=lI g~
~
~k'~k
(s)T(ds) =
fg~ ~ (s)T(ds)
k=l I: ,. ~k'o/~
Y
00
=
trace M(I::,.)
00
so 0 S
Lg
(s)
k=l <Pk,<P k
<
00
a.e.[T] since 0 S T(I::,.) <
00
VI::,.
and so M' (s)
is trace class a.e. [T].
0
We now turn to a discussion of orthogonal operator valued measures
and integrals.
We first make a definition.
Definition 4.3.1.
A set function
Z defined on the measurable space
(S,S) is an orthogonal Hilbert-Schmidt (H.-S.) measure if for every set
I::,.
E
S, Z(I::,.)
(i)
(ii)
E
HS(H,L (Q,C)) and
2
For any sequence
If I: ,. and
1::,.1
{l::,.n}~=l of disjoint sets of S
are disjoint sets in S then
Z(I::,.) *Z(I::,.)
= O.
o
There exists a correspondence between orthogonal H.-S. measures and
measures of the type considered in Theorem 4.3.1 which is described in
the following theorem:
Theorem 4.3.2.
space (S,S).
Let
Z be an orthogonal
Then the set function
H.-S. measure9n a measurable
M defined on (S,S) by
valued set function satisfying the
;
hypotheses of Theorem 4.3.1.
Conversely if M is a T+ (H,H) valued
set function satisfying the hypotheses of Theorem 4.3.1, there exists
101
an orthogonal H.-S. measure
Proof:
Z with
Z(6)*Z(6') = M(6 n 6').
If M is a set function on (S,S) defined by M(6) = Z(6)*Z(6),
it is clear that for each
6
€
S, M(6)
€
T
+
(H,H).
Moreover,
M
is
weakly countably additive.
of sets of S.
Since
To see this, let {6n } be a disjoint sequence
Z is strongly countably additive, for x,y €H we
have
co
co
co
N
N
= lim( 2 Z(6n )X, 2 Z(6n )y)
N+co n=l
n=1
N N
= lim 2 2(Z(6 )x, Z(6 )y)
N+co n=l m=l
n
m
N
= lim 2 (Z(6 )x, Z(6 )y)
N+co n=l
n
n
co
= 2 (M(6n )x,y)
•
n=l
Thus the set function M satisfies the requirements of Theorem
4.3.1 and so M(6) =
f
M'
(s)'t(ds) for some strongly measurable T+(H,H)
6
valued function
M'.
Conversely, ifM
is a set function on (S,S) satisfying the hypoth-
eses of Theorem 4.3.1, define a map
F on the Cartesian product of S
with itself by F(6,6') = M(6 n 6').
F takes values in L(H,H), the
space of all bounded linear operators
and hI' ... ,hn
n n
2
€
H then
n
n
LCM(6. n 6.)h. ,h.)
i=1 j=1
1
J 1 J
IC F (6. ,6.)h. ,h.) = r
i=1 j=1
H ~ H. Moreover if 61 , ..• ,6n
1
J
1
J
€
S
102
n
=
~ J
I
'-1
i=l JM6.
(M'(s)h.,h.)T(ds)
J
1
J
=,
f(!vl' (s)
S
I
I
X (s)h.,
X (s)h.)T(ds)
1
i=l 6i
j=l 6j
J
since M'(s) is a positive operator.
co
{~k}k=1
Let
0
Thus we can apply a theorem of
Payen (1967) and assert that there exists a set function
taking values in
~
Z on (S,S)
L(H,L (n,C)) such that
2
be a c.o.n.s. in
H, then
= trace M(6)
Using the property that
<
co
Z(6) * ZO(6') = M(6 n 6') it is easy to see
that
so
~
Z is countably additive in H.-S.norm, and Z(6)*Z(6') = 0 if
n 6'
=~.
Thus
Z is an orthogonal H.-S. measure.
Let us now define an operator valued integral of the form
Jf(S)Z(dS)
S
n
If f
is a simple function,
f(s) =
r
C,XA
. 1
1=
1
u.
1
(s) say, define
o
103
n
L c.Z(~.)
ff(S)Z(dS) =
Let
. 11.1.
1.=
m
f'(s) = L c!x~, (s).
j=l J j
fl be another simple function
Then
(ff(s) Zeds) ) * (ff' (s) Z(ds) )
S
S
n
*
m _
r
(4.3. 7)
=
I c.c~Z(~.) Z(~.)
i=l j=l 1. J
1
J
It is easy to see that property (ii) of Definition 4.3.1 implies that
'Z(~) * Z(~I) = Z(~
n ~')*Z(~ n ~') = M(~ n ~') so that (4.3.7) is equal to
n
L
m
i=l
n
=
=
r C:-c!
j=l 1. J
I r
L . 1 c.1
i=l J=
S
f.f (~.
1. n ~n
J
c!x~ (S)X~I(S) M'
J.
.
J
1.
(s) T(ds)
ff(S) f' (s) M' (s) T(ds)
S
Thus for all simple functions
(4.3.8)
Let now
f
and
f'
[ ff(S)Z(dS)]*( If'(S)Z(dS)] = ff(S) f'(s) W(s)T(ds).
S
S
S
~
be any set in S.
J trace
~
Then if
M' (s) (ds) =
J
~
~k
is a c.o.n.s. in
H
I (M' (S)~k'~k)T(ds)
k=l
=kt (
JM' (S)T(dSHk'~k)
~
= trace
M(~)
;,
= T(~)
104
so that
trace M' (s) = 1 a. e •. [T].
(4.3.9)
From (4.3.8) we obtain
(J f(s)Z(ds),
Jf l (s)Z(ds) )HS
S
S
=
tracer If(S)Z(dS)r( Jf' (S)Z(dS))
S
S
= trace If(S) f' (5) Mt (s)'r (ds) •
(4.3.10)
S
Let now
00
{~k}k=l
be a c.o.n.s. for H•. Then (4.3.10) is equal to
I
(4.3.11)
If(S)f' (5) (M' (5) cflk ,cflk) T(ds).
k=l
S
.-
Since Mt(s) is a positive operator for each
5 €
I
S and since
I(M'
Jlf(s)llf'(s)I(M'(S)cflk,cflk)T(dS) = flf(S)1 If'(s)1
(S)cflk,cflk)T(ds)
k=l S
. S
k=l.
= Ilf(S) I/f ' (s)ltrace M'(s)T(ds) .
= Jlf(s)1 If'(s)IT(ds) <
00
by (4.3.9), we may interchange sum and integral in (4.3.11) which is
now equal to
If,S) f' (5) trace W (s)T(ds)
S
= If(S) f'(s)T(ds).
S
.
Thus
(4.3.12)
(ff(S)Z(dS), ff'(S)Z(ds))HS = If(S) f'(S)T(ds) •
S
S
S
105
Now let
f
be any function in
L2 (8,S,T), then there is a sequence {fn}
{fn} converges to f in L2 (8,S,L);
of simple functions such that
(4.3.12) shows that the sequence
in HS(H,L2 (Q,C)).
{If (s)Z(ds)}
8 n
We define the integral
is a Cauchy sequence
!f(s)Z(ds) to be the limit
8
of this Cauchy sequence; it can be shown that the limit is independent
of the particular sequence chosen.
The sequence also converges in the
uniform norm, consequently if f, f' e L2 (S,S,T) and {fn},{f'n} are sequences of simple functions converging in L2 eS,S, T) to
f
and f', then
e!f(s)Z(ds)) * (!f'(s)Zeds)) is the limit uniform norm of the sequence
S
S
{(If (s)Z(ds))*e!f'(s)Z(ds))}oo_l which by (4.3.8) is the same sequence
S n
as
S n
n-
{fl:YS)f'(s) M'(s)T(ds)}.
Now
S n
I 'ffneS) f~(s)M'es)L(ds) - J'f'[SY f' (s)M' (s) .(ds) Il u
S
S
<
flfn(s)f~(S)
- f(s)f'(s)IT(ds)
8
whicb converges to zero as n
for all functions in
~
00.
Thus (4.3.8) and (4.3.12) are true
L2 (8,S,.).
We now state and prove a general representation theorem from which
we will derive a spectral representation of stationary processes.
Theorem 4.3.3.
Let
space (S,S) and let
Z be an orthogonal H.-8. measure on a measurable
M, M' and T be as above.
Let
f(g,s) be a
function on G x 8 such that for each g e G, f(g,o) e L2 (S,S,T).
Then if
-.
106
Xg
= If(g,S)Z(dS)
s
is a process of Hilbert-Schmidt operators, the covariance operator of
X is given by
g
X*g ,X g
(4.3.13)
= If(g,S)f(g' ,s)M'(s)T(ds)
S
+
Conversely, if M is a measure on (S,S) taking values in T (H,H) and
satisfying the hypotheses of theorem 4.3.1, and
X is a process of
g
Hilbert-Schmidt operators having covariance (4.3.13) then there exists
an orthogonal H. -So measure
and
Proof:
Xg = !f(g,s)Z(ds).
The first assertion of the theorem follows from the proceeding
remarks.
that
Z on (S,S) such that Z(6') * Z(6) = M(6' n 6)
For the converse, let
Zo
ZO(6) *ZO(6') = M(6 n 6').
Define a process
be an orthogonal H. -So measure such
Such a measure exists by theorem 4.3.2.
Y byY = !f(g,s)Zo(ds).
g
g
S
Then by the first part of
X*,X = y*,y = !f(g,s)f(g',s)M'(s)T(ds).
g g
g g s
.
Define now a map T from the manifold generated by the Yg to that
the theorem we see that
n
n
generated by the X by T( I c.Y ) = L C.X
• T is clearly linear
g
i=l 1 gi
i=l 1 gi
and
n
2
n
2
II T ( . I 1c.1 Yg.) II HS = II . I 1c.1 Xg. III.JsI
1=
1
1=
=
1
n
n
n
n
c. c:- trace X X
i=l j=l 1 J
gj gi
=l
I
i=l j=l
n
=
*
L l
c.c. trace Y *Yg.
1 J
gJ'
1
2
II i~l CiY gi II HS
107
Thus
T preserves
norms and is onto, so T may be extended to an
isometry between H(Y) and H(X) , the closed subspaces generated by
the
Yg'S and Xg's.
Define
Z(6)
= TZ O(6).
Then
Z(6)
V6€S, and is countab1y additive in H.-S. norm since
since
T*T
=I
because
H.-S. measure.
T is an isometry; thus
Moreoever, if
f
€
HS(H,LZ(Q,C))
Zo is.
Also
Z is an orthogonal
is a simple function, it is easy to
see that
(4.3.14)
T ff(S)ZO(dS)
S
= ff(S)Z(dS)
.
S
A simple passage to the limit shows that (4.3.14) is true for all
f
€
LZ(S,S,T), hence
ff(g,S)Z(dS)
= T !f(g,S)ZO(dS) = T(Yg) = xg .
o
Our next result is a Hilbert-space version of Bochner's theorem
which is an extension of a theorem of Falb (1969).
Theorem 4.3.4.
V: G + T(H,H) be weakly continuous.
Let
Then
V
is positive definite iff it has the representation
(4.3.15)
V(g)
=
f <a,g)M' (a)T(da)
A
G
for some strongly measurable function
Proof:
If V(g)
=[
G
M' :
G+
T+(H,H).
\a,g)MI(a)T(da), then for elements gl, •.• ,gn of G
108
n
n
r r
i=l j=l
(V(g.-g.)h.~h.) =
1
J
1
J
I I J<~,g.-g.)(M' (a)h. ,h.).(da)
i=1 j =1",
1
J
1
J
G
n
n
J.: l{a,gi)(a,gJ-) (M' (a)h. ,h.).(da)
f i=lr r
=
J
1
A
G
f
=
A
2
II .r1
(a,g.)B(a)h. II
1
1
n
.(da)
0
~
1=
G
where
B(a) is the positive
s~lare
root of the positive operator M'(a).
Conversely, if V(g) is positive definite, then the function
g
~
(V(g)h,h) is positive definite.
Also
(V(g)h,h') = ~[{(V(g)h+h' ,h+h') - (V(g)h-h' ,h-h')}
+ i{(V(g)h+ih' ,h+ih') - (V(g)h-ih,h-ih)}]
so the function (V(g)h,h') is
functions.
a linear
combination of positive definite
Thus there exists a complex measure ]Jh,h' of finite
variation such that
(V(g)h,h') =
f <a,g)]Jhh'
. (da)
•
'"
G
Now the map {h,h')
+
]Jhh,(6)
for each fixed Borel set
is a bilinear functional on
Hx H
6 and is bounded since ]Jhh is a positive
measure and
Thus there exists a bounded linear operator M(ll) such that
04(1l)h,~')
= ]Jhh,(6) for all Borel sets
6 and all h,h'
€
H.
Since
]Jhh' is a measure, it is clear that M(~) is weakly countabiyadditive,and
109
M(6) is a positive operator for each
measure.
To see that
a c.o.n.s. for
H.
6 since
~hh
is a positive
M(6) is trace class for each
6 let {<I>k} be
Then
00
00
00
00
=
L(V(eHk ,4>k)
k=l
~
Thus
trace Vee)
<
00
•
M(6) satisfies the requirements of Theorem 4.3.l'land so there
+
A
exists a function MI : G + T (H,H) such that
M(6)
= IM'(a)i(da)
•
6
f \a,g)(M'(a)h,h')i(da).
Hence (V(g)h,h) =
But the integral
A
G
I<a,g) M' (a)hi(da)
exists as a Bochner integral for all h
€
H since
<
00
,..
G
fll{o.,g)M'(a)hlluiCda)
A
~IlIhlliCdo.) =
IlhlhCG)
A
G
so
G
V(g) = I<a,g)M'(a)i(da) in the sense of Remark 4.3.1.
0
A
G
With the aid of Theorems 4.3.2, 4.3.3 and 4.3.4, we can now easilyprove the spectral representation of a stationary process with weakly
continuous covariance.
Theorem 4.3.5.
Let
Xg be a weakly continuous stationary process of
Hilbert-Schmidt operators.
Then there exists an H.-S. orthogonal
.
-
110
measure
Z on the Borel sets of G such that
(4.3.16)
Xg =
f(a ,g)Z(da)
,.
G
and Z* (6)Z(6)
=
M(6)
for all Borel sets
6 where M is the measure
appearing in the Bochner representation of the covariance operator V
of the process
Proof:
X.
g
By Theorem 4.3.4, the covariance operator V has the represent-
ation (4.3.15) and thus by Theorems 4.3.2 and 4.3.3 the process has the
representation (4.3.16) and the measure
Z satisfies
Z(6) * Z(6) = M(6).
o
We conclude this section wi tha few remarks on Payen' s approach to
the,sp~c~ra~ .~;ept~,sent~tion
(4. 3.16)',;Which we. will use in section 4.4.
M be the subspace ofL2 (62,C) generated by the random variables
{X h : g ~ G, h ~ H}. We may define an oper.ator U by Ug (X g ,h) =
g
g
Let
Xg+g
. ,h; Ug may be extended to a tmitary operator on M. The family
of operators {Ug : g € G} constitutes a tmitary group of operators,
which by Stone's theorem (see e.g. Hille ~d Phillips (1957) p. 598)
admits a representation
f
Ug = (a,g)E(da)
,.
G
where E is a projection valued spectral measure on the Borel subsets
A
of G , and the integral is defined in the sense of Halmos (1951) p. 60.
Hence
Xg
= l<a,g
) Z(da)
where
Z(6)
= E(6)Xe
since Xg = UgXe .
G
The measure M of Theorem 4.3.4 satisfies M(6) = Xe* E(6)X.
e
this, consider the ftmction (V(g)x,y)
To see
= (X;Xgx,y) = (UgXeX,Xe~)'
111
(V(g)x,y) is the inverse Fourier transform of the measure
~xy
of
Theorem 4.3.3, and by Stone's theorem
(U g
Xe
x,X
e y)
=
I(a ,g}
"xy(,JcL)
G
where
It follows that v xy = ~xy i.e. that
(M(A)x,y) Yx, y e H and so Xe *E(A)Xe = M(A) for all
Vxy(A) = (E(A)XeX,Xey)·
=
(E (A)XeX,Xey)
Borel sets
A.
Our final result is a lemma that will be useful in section 4.4.
" 3:~ L
Lemina' 4.
on G.
Let
f(a) bea bounded measurable' complex valued function
Then
Trace X*e Jf(a)ECda)Xe
= ff(a)'r(da).
...
G
Proof:
G.
It is enough to prove the lemma for the case f
can then express a general
f
0, since we
in terms of its positive and negative
parts and apply the result for positive
general case.
~
f
to obtain the proof of the
First we prove the result for simple functions
n
f
=l
c'X
.
i=l]. Ai
Then
n
c.X *E(A.)X·
trace X; If(a)E(dOl.)Xe = trace
.1= 1 1 e 1 e
r
A
G
n
= . r1c.1
1.=
trace M(A.)
1
n
-.,--.
= . LI C.'r(A.)
1
.1,
1=
-;.
=
f
f(a)'r (dOl.)
A
~
G
Now let
f
be a positive bounded measurable function.
There exists a
m
~
112
sequence
f n of simple ftmctions slich"thatfn ' converges to
uniformly, and the sequence f
p. 86).
n is increasing.
f
(See e.g. Halmos' (1950)
Thus by monotone convergence
!f(a) 1: (da)
= lim Ifn(a)1:(d~)
"
"
G
G
= lim trace X* ·If (a)E(da)X
e
n
e
"
G
But
Itrace x; If(a)E(da)Xe - trace
,..
G
= Itrace
where
I IAI lu
But
x;
Ifn(a)E(da)Xel
"'"-
G
x; J(f(a)-fn Ca))E(da)Xel
=
I (Xe '
<
I IXellHsl I
ff(a)-fn(a}E(da)XeIIHS
<
Ilxell~sll
!Cf(ll)-fn(a})E(da)ll u
{(f(a)-fn(a))E(da)Xe)Hsl
denotes the uniform norm of the operator A on L2 (n,C).
II
!(f(a)-f (a))E(da) II
n
"
G
u
=
sUE If(a)-£ (a) I
a€
G
n
+ 0
as n
+
co
(see e.g. Ha1mos (1951) p. 62), so
lim trace x* If (a)E(da)X
n-+co
e
n
"
G
e
= trace
x; !f(<l}E(da)Xe
"
G
which proves the lemma.
4.4.
o
Path Properties and Sampling of H-valued Processes.
In this section we generalize some of the results proved for complex
valued stationary processes in Chapter III relating to sampling and path
continuity.
We
win assume throughout this section
113
that the process in question is indexed by the real line lR, rather
than an arbitrary LCA group G.
We first give definitions of path
continuity, and then generalize a result of Kawata (1969) to the Hvalued case.
Finally, we show that a band limited H-valUed process
satisfies a sampling theorem.
Definition 4.4.1.
Let
stationary process, and
{x(t)
t
€~
{X t : t
€
R}
Hilbert-Schmidt operators.
be a weakly continuous H-valued
the corresponding process of
We will.say that
x(t) has strongly contin-
uous paths with probability 1 if for almost all w , the function
x(o,w) is continuous as a map from
uous with probability I for all h
IR to
€
H.
If (xCo,w),h)H is contin-
H, then we say that x(t) has weakly
continuous paths with probability 1.
Clearly x(t) has weakly continuous
paths with probability 1 iff the complex valued process Xth has
continuous paths with probability 1 for each h
€
H.
0
We now give a sufficient condition on a weakly continuous stationary
process
x (t) for strong path continuity.
of Kawata (1969).
Let
In
= [2nn/T,
We rely heavily on a paper
We first define a "periodic" process based on x(t}.
2(n+l)n/T), and let
E be the spectral measure in
Stone representation of the unitary group Ug
section 4.3.
Let
Qri
= E(I n).
introduced at the end of
Define
\
(4.4.1)
The series (4.4.1) converges in Hilbert-Schmidt norm s.ince the
are orthogonal, i.e. ~~
= 6nm~
~
, and if {~k} is a c.o.n.s. for
H
.
..
114
~
~
III e21Tint/T~xoII ~s =
n=-~
I
n=-~
~
=
I
k=l
~
~
I (~xOepk'
k=l
~XO</>k) L
2
(Q C)
'
2
~
L II~xo</>k"L 2 (Q ' C)
n=-~
~
= k~1IA=~~QnXO</>kll~2(Q,C)
(4.4.2)
00
since the projections
~
~
equals
L I IXO~kl IE
k=l
Theorem 4.4.1.
(Q C) = trace
2
'
x~xo
<
"
~
•
Xt(T) converges to X in Hilbert-Schmidt norm as
t
Proof:
and
co
=
=
~.
Since L On = I, (4.4.2)
n=-oo
are orthogonal.
!(~XO,~XO)HS
n=_10
=
I
e-2~int/T JeitAT(dA)
n=-oo
115
(by Lemma 4.3.1)
In
= n=~oo
J eit(A-2~/T)T(dA)
In
Thus
IIXt(T)-Xtll~S
J(l-cost(A-2~/T))T(dA)
2 I_oo
n
=
"I
... s,2 I_oo
n
,
n
J 2 sin2
i'T(dA) = 4T(lR)sin
Ilxt
2
- Xt(T) IIHS
A
~~
In
since 0 < 1 - cos t (x - 2~~ ) < 2 sin 2 ~~ on
Thus
2
converges to 0 as
T+
In if T >
00
21 t I.
o
•
A
Now corresponding to the operators
second order random elements
~
"
Xt(T) and QriX o we have H-valued
x(t,T) and ~n , given by
(~n,h)H
(x(t,T),h)H = Xt(T)h
and
(4.4.3)
x(t,T) = L e2~lnt/T~
n
n=-oo
00
= QriXoh and the series
•
converges in L2 (n,H) norm.
We wish to yrove that the series (4.4.3) converges uniformly in
H-norm with probability 1, tmder certain conditions.
It is enough to
00
E( I II ~n II H) < 00 • If there exists an even ftmction
n=-oo
00
g(O)
>
0,
Ig(n)-l<
g : IR+ R which increases for t >,0, satisfies
n=O
00
prove that
and
fg(A)T(dA)<
00
then
00
116
~
90
( .l
n=O
90
g (2~1T) -1)
J g (A) T (dA)
<
90
0
Also
E(
-1
l II ~n II )
-1.
r g(2 (n;l) 1T) -1)
n=_9O
«
n=_9O
-1
r
(. g(2 (~+1)1T) T(I ))
n
n=_9O
90
E l lI~nllH
n=_9O
so
<
90.
'"
Hence if such a g exists,
x(t,T) has almost all paths strongly
continuous.
The proof of the next result is the same as the corresponding result
of Kawata (1969).
Theorem 4.4.2.
H-valued process.
Let {x(t) : t
E:
IR} be a weakly continuous stationary
If there exists a function
even and increasing for A.
~
g(A.) that is positive,
0 satisfying
90
(i)
l
g(n) -1 <
90
n=l
and
(ii)
90
J g(A)T(dA.)
<
00
_90
then
Proof:
x(t) has almost all paths strongly continuous.
By the above remarks, the processes
A
k
x(t,2) have strongly
117
continuous paths.
we see that as
By copying the- proof of Kawata' s theorem 9
k ~ co , x(t,2 k) tends W1iformly to x(t) on every
d
compact interval with probability 1, and hence
x(t)
has strongly
continuous paths.
Definition 4.4.2.
Let
{x(t) : t
uous stationary process.
E:
IR} be an H-valued weakly contin-
Let
X be the corresponding operator process
t
the W1itary group associated with the Xt • If the spectral
and Ut
measure M of Theorem 4.3.2 is concentrated on a bounded set, we shall
say that the process Xt is band-limited. Equivalently, if the
spectral measure E in the Ston·e representation of the W1itary group
Ut
is concentrated on a bOW1ded set, then
Xt
is band limited.
o
Our next theorem shows that the sampling expansion (3.2.2) is valid
for H-valued processes.
Theorem 4.4.3.
With the notation of Definition 4.4.2, if E is
concentrated on the interval [-W, W]
co
(4.4.4)
~
Xt = lX(kn/a)
. k=_co
then
sin a(t-klT/a)
(t-kn/a)
the series (4.4.4) .converging uniformly in Hilbert-Schmidt norm for
a > W;
co
(4.4.5)
= ~ x(k"'/a) sin a(t-kn/a)
x (t ) . l "
a(t-kn/a)
k=_co
the series (4.4.5) converging in H-topology with probability 1 uniformly on compact sets for
~alytic
with probability 1.
a
>
W.
Finally, the paths of x(t) are
Proof:
Consider
II
sin a(t-kn/a) "
k=:n-1<n/a· a(t..;kn/a)
HS
Xt
I
U
sin a(t-kn/a)
- k=-n kn/a
a(t-kn/a)
(4.4.6)
Now
118
~ x_
w
I
U
U
sin a(t-kn/a)
t -k=-n kn/a
a(t-kn!a) =
f(e itA
nt
e
n=-n
-
L
-w
'/ a
~ok nA
°
( k n/ a ) )E(dA)
Sin
ata(t-kn/a)
so
I
U
sin a(t-kn/a)I I
-k=-n kn/a
a (t-kn/a) u
=
(4.4.7)
Now
e
iAz
I
sup le itA eiknA/a sin a(t-kn/a)I
-WSASW
k=-n
a(t-kn/a)
is an entire function of exponential type
IAI
so by a result
of Piranashvi1i (1967)
l
e itA _
for all
t
~ eiknA/a sina(t-kn/a)
k=:n
. a(t-kn/a)
and some constant
K.
I
<
=
Ka
ca:lTfTn
Thus the left side of (4.4.7) is
less than
(4.4.8)
-W~~~W
Ka
Ka
(a-/TI )n = (a-W)n
Thus by (4.4.6) and (4.4.8) the series
~.4.4)converges
~
sin a(t-kn/a)
To prove (4.4.5), let Yn(t) =k~_nXkn/a
a(t-kn/a)'
•i
E
I /IX(t)
n=l
¥
sin
k=:nX(k~/a).
in H.-S. norm.
Then
2
2
a(t-kn/a)
11
~
II
11
(t-kn/a)
H =n~l Xt-Yn(t) HS <
00 •
119
Hence the series (4.4.5) converges in
H~topology
with probability one.
The convergence is miforrn on compact sets since the convergence of
00
HX
L -Y (t) II~s is miforrn. Finally, the analyticity of
n=l t n
the paths can be established by the methods of Be1ayev (1959) as in
the series
the
cas~
of complex valued processes.
o
APPENDIX
SOME FACTS FROM' ABSTRACT HARMONIC ANALYSIS
A.l. LCA Groups. A topological group is a Hausdorff space G that
is also a group, where the group structure and the topology are related
.~, ,~y~_~~~,-,-.r.z~H~~:r~~A!~
.. 1:h~t... ~~.$,ml}'p~ ... <g"h) ..~~g+~~cl
ous.
g.. ~ -g be continu-
If G is locally compact as a topological space and abelian as
a group, it is an LCA group.
All groups considered in this work are LCA,
the binary operation is always written
is written as
-g.
The symbol
+
and the inverse of g
a of G such that
e will mean the identity of G.
G.
ing Rudin (1962) we have throughout denoted the value of a
at
{a,g).
a is continuous and
It is clear that
operation
+
{a,e)
= I Va
€
la(g)
I =I
€
G.
Vg
Followg by
If we define a binary
on G by
is an abblian groop with identity the constant homomorphism 1,
and the in~erse -a of a is given by <-a,g)
'.
G
Characters. Denote by G the set of all complex homomorphisms
A.2.
G
€
= {a,g)
.
".
"G.,,_~SS,~I.lJ'.¥,J:J:l~.~hll.J:~9~~r gI:9uP40~gua.l,~~;lJ.pJo£
G, its
A
elements are characters.
in such a way that
It is possible to define a topology on G
G becomes an LCA group.
120
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