Gertrude M. Cox
SOME TOPICS IN THE THEORY OF STOCHASTIC PROCESSES
by
GOPINATH B. KALLIANPUR
Special report to the Office of Naval
Research of work at Chapel Hill under
Project NR 042 031 for rosearch in
probability and stutistics o
Institute of Statistics
wurneo Series No. 45
F'or Limited Distribution
ii
And each man hears a,s the twilight nears
to the beat of his dying heart,
The Devil drum on the darkened pane:
"You did it, but was it Art?"
iii
ACKNOWLEDGMENT
I wish to express my deep gratitude above
a.ll to Professor Herbert Robbins whose penetrating
crtticism so often helped to clarify my own idea,s. To
Dr. Wassily Hoeffding who wen.t through part of the
manuscript and offered many useful
my heartfelt thanks.
suggestions, go
I am a.lso grateful to Professor
Michel Loave of the University of Ca.lifornia. a,t
Berkeley, for bringing to my a.ttention the problem
treated in
Cha~ter
II.
Fina.lly, thanks a.re due to the Inetitute
of Statistics and the Office of
Na~al
Research for
generous financial a.ssistance which ma.de this study
possible.
Gopina.th B. Kallianpur
iv
TABLE OF CONTENTS
Pa.ge
ACKNOWI.;EDGMENT
iii
vi
INTRODUCTION
Chapter
1.
ON A STOCHASTIC STIELTt.:""ES INTF.f}RAL A.TID THE REPRESENTABILI'l'Y OF A RANDOM SET J.'J"JCTION AS A KARHGl':EN
INTF,GRAL
1
1.
Preliminaries
1
2.
Additive Random Set Functions
3
3.
Definition of the Integral
f f(a.)
dZ
5
S
II.
III.
4.
Comparison with Older Definitions
11
5.
Representability of a. Random Set Function
as a. Stochastic Integra.l
17
INTEGRAL REPRESENTATION OF A RANDOM FUNCTION WHOSE
COVJ;RIANCE IS A CARL"7J;~ KE'RI'fEL
28
1.
Introduction
28
2.
The CarlelDEln Kernel of Class I
29
3.
Properties of Q(t,S;A.) a~d Some Auxiliary
Theorems
33
4.
Dafini tion of the Random Variable Acr(t,A.)
35
5.
Integra,l Representa.tion for X( t)
45
FlJr.TCTIONALS OF A lvT..ARKOV PROCESS - TEE DISCONTINUOUS C/SE
50
1.
Definitions and Nota.tions
50
2.
The Purely Discontinuous Ca.se
53
3.
On the Conditions (~) and (~ )
56
v
Ta~le
of Contents(Continued)
4.
Applications
57
5.
Another Form for Equation (2.3)
59
BIBLIOGRAPHY
61
INTRODUCTION
Kolmogorov ~9_7 was the first to make use of linear methods
in Proba.bility th'3ory, in the study of sta.tionary sequences of
random variables.
Later, Karhunen ~8_7, introduced these methods
in a systenia.tic Ii"~.nner, 81'.d 6!1lpJ oyed t.hem to give an elegant definition of the stocha.stic integral,
J
x(t) dm
,
S
where x(t) is a. second order process, and m an ordinary a-finite
mea.sure on the rea.! line.
(It ha.s been known since, that this is
the same as the integra,l due to Pettis,
£13_7.)
In Chapter I of
this work, we have utilized these concepts to give a definition
of the StieltJes stocha.stic integral, that is, an integral of the
form,
J
rea)
dZ ,
s
where Z is a random set function defined in a. certain manner.
This definition is seen to be a.t least a.s genera.l as the one given
by the usua.l method of a.pproximating sums.
Finally, a theorem is
proved concerning the representability of a. random set function
a.s a. Karhunen stocha.stic integral.
In Cha.pter II, an integra.l representa.tion is obtained for a.
random process of the second order whose covariance function is
vii
a. Carleman kernel.
This problem has been considered recently by
Dolph and Woodbury,
.L3_7 who
tion whic.h
The
::i18.1r:98 11813
ha.va obta.ined a. series representa.-
of Hellinger integra.ls.
thi~~d ch8.l,te:~
1s devoted to ae.i.itive fttnctiona.ls o"!: a.
purely discor..ti.;:mous Markov proc.eas.
by
So
recent,
(8.'3
This study vas motivated
Y8t unpublished.) pa.per of R. Fortet,
["5_7,
in
which he ha.s considered the continuous ca.se and has obta.ined
many interesting and remarkable results.
The principa.l result
of Chapter III is tha.t the conditiona.l cha.racteristic function
of such a. functional is the solution of a certa.in integrodifferentia.l equa.tion.
One of the a.pplica.tions of this theory
yields, a,s a. consequence, Feller I s equa.tion for the trans1tion
probabilities of a. discontinuous Markov process.
CHAPI'ER I.
ON A STOCHASTIC STIELTJES INTEGRAL AND THE REPRESENTABILITY OF A
RANDOM SET FUNCTION AS A KARHUNEN INTEGRAL.
1.
Preliminaries.
In this section we shall give one of the many definitions of
a stochastic process, (or random function), X(t) where we shall
assume that t ranges over the real line, R.
In most applications,
t plays the role of time.
Definition (1.1):
Let i~ be an abstract space whose points
or elements are denoted by wand let IT be a probability measure
defined over a a-algebra E of subsets of i~.
Consider a real or
complex-valued function X(t,w), such that, for every fixed t in R,
X(t,ID) is a IT-measurable function of roo
Then X(t,w) defines a
random variable X(t), and as t ranges over R, we thus obtain a
family of random variables depending on the parameter t.
This
family of random variables will be called a stochastic process
x(t).
We shall write
EX(t)
=
J
X(t,OO)dIT(oo);
E{X(t)X(ul!
il.
=
J
X(t,oo) ·X(u,OO)d IT (00),
[1
where X(U,ID) denotes the complex conjugate of X(u,w).
said to be of the second order if
p(t,u)
= E,LX(t)X(u)J
<
00,
for all t and u.
X(t) is
2
It will be assumed throughout this chapter that X(t) is of the
second order and that EX(t)
=0
for all t.
variance function of the process.
p(t,u) is the co-
In section (2) we shall give
I
Cramer's definition ~2-7 of an additive random set function,
(abbreviated as
r.s~f.)
and state some of its properties.
We shall
then use them to define the stochastic Stieltjes integral of a
function f(a) with respect to such a r.s.f.
This will be done in
section (3) by applying the "linear" methods, first used by
Kolmogorov ~9-7 in the study of stationary processes, and later
developed systematically by Karhunen, who applied them to define
the stochastic integral of the type,
JX(t)dm,
S
m being an ordinary a-finite measure on R, and S an m-measurable
set(see~8_7).
ThiS definition is also known to be the same as
that given by Pettis
LI3-7,
in a different connection.
In section
4, our definition of
If(Q)dZ
S
is compared with the definitions of Cramer and Karhunen(~2-Z,~8-7)•
Section 5 is devoted to the proof of a theorem on the representability of a r.B.f. as a Karhunen stochastic integral.
1.
Numbers in square brackets refer to bibliography.
3
The following theorem, well-known in the study of Hilbert
space theory 1s quoted below for convenience, as it is the main
tool employed in. obtaining the results of this chapter.
For the
terminology and notation used in this theorem, we refer to Nagy
Theorem 1.2.
If L(f) is a bounded linear operation on the Hilbert space
::R,
there exists a unique element g* in
P\"
such that,
L(f) = (f, g* ),
for every f
£
Pv
The results contained in sections 3 and 5
have been announced in a recent note by the author ~7-'.
2.
Additive Random Set Functions.
Starting from a stochastic process X(t), define,
where I is the half-open interval (t l ,t2 ]·
where the I
k
If I
= II
+ ... + In'
are finite disjoint half-open intervals, we may ex-
tend tqis definition by writing,
n
Z(I) =
~
k=l
Z(Ik ).
If the covariance function of X(t) has a certain property, to be
4
stated shortly, this definition can be extended, so tha.t we can
define the random variable Z(S), where S ia a bounded Borel set.
Strictly speaking, we should write Z(S,oo) to bring out the dependence on
00
00
explicitly.
Henceforth, we shall suppress the letter
and simply write Z(S).
Z is now a stocha.stic process whose argu-
ment is a bounded Borel set.
the integral over
1-1 with
8imilarly, by EZ(S) we shall mean
respect to the probability measure
TI .
With these brief explanatory remarks we can now (see ~~7) characterize an additive random set function as follows: --
An
a.dd1tive
random set function is a. family of random variables such tha.t,
(1)
For every bounded 'Borel set S of real m.unbers, Z(8) is a.
uniquely defined random variable.
(2)
If 81 , S2' ,., are disjoint Borel sets, such that
00
S
=
I:
i=l
S1
,
is bounded, then
Z(8)
=
00
I:
i==l
Z(8 i ) ,
where the series on the right converges in quadratic mean.
One
could conceive of such a process Z(8) a.s the "impulse" received
during an arbitrary Borel set 8 of time points.
that Z(S) is real valued, with EZ(S)
(for every bounded Borel set S).
We shall assume
= 0, and E~Z(s)_72<
00
Cramer ~2_7 has proved the
following theorem:
If the cova,riance function p (t, u) of the stochastic process
5
x(t) is of bounded variation in every finite doma.in (in the sense
defined in ~2_7), there exists an additive r.s.f Z(8), uniquely
defined for all bounded Borel sets 8 and such tha.t,
when 8 is a. half-open interva.l,
The covariance function of Z is given by
=
JJ
2
d p
(t,u) •
In particular, if
where F( t) is a. non-negative, never decrea.sing function, we obtain
the spectral random set function of Karhunen
J
18_7 ,
and we ha,ve,
dF( t) .
Definition of the Integral
In what followB it will be aSsumed that· we have a real-va.lued
additive r·s.r
Z(8) which is such tha,t,
6
E Z(S)
= 0,
and
where P 1s of bounded variation in every finite dome.in.
( e.)
The Linear Space of the Process Z ~
Let L(Z) be the set of all random variables of the form,
where the Sk are bounded Borel sets, and the c
stants.
k
are rea.l con-
Closing the set L(Z) with respect to convergence in the
mean, we obta.in an extended set
L (Z) . The elements of L (Z)
2
2
are either of the form (3.2) or are limits in the mean of ra.ndom
variables of the form (3.2) •
Define the inner product of two
a.rbitrary elements zl and z2 of L2 (we sha.ll write L2 for L (Z)
2
from now on), by the relation,
It is well known, (see ~2_7) tha.t the set L is a. Hilbert space
2
with the norm,
7
L2 is referred to a,s the linear space of Z •
(b)
Denote by z the elements of L
2
• For any z write,
Let
be a. bounded Borel set, the 8
1
being disjoint.
Then since,
CD
Z(8) = E Z(Si)'
i=l
(the series converging in qua.dra.tic mean), we ha.ve
Z(8) -
~
i=l
Z(8 i )
112~
ae
0
n~oo
Now,
=
,
fz(8)
(ELz·
II
z 11
2.
-
n
IZ(S) - E
i=l
~
1=1
Z(8 1 )}
Z(8 i )
_7 )2
1'2 , by the Schwarz
inequa.lity.
----)0
Hence,
..
0
a.S
n4
00
for every z.
8
Using the nota.tion of
(3.4)
p
z
(s)
for every z in L2 .
(3.3) we ha,va
=
Also, from
(3.1), taking S
= 0,
the empty set,
we obtain,
Again by Schwaol'z I B inequality,
for every z in L2 .
tha,t, for every
Z
Thus from relations
(3.4) and (3.5) we find
in L , Pz is e, signed mea.sure.
2
ma.y write (using the notation of Ha.lmos
Therefore we
£6_7),
where both P+ and P - aol'e measures which aol'e finite for bounded sets
z
S.
z
Actually, as is done in the theory of signed measures, we might
include sets S which ma,ke one of (but not both) P; or P~ infinite.
Write,
Theorem 3.1
Let f{ a.) be a real valued function such that,
9
the 1ntegra.l
( i)
exists and 1s finite for every z.
'J
( i1)
f(a) dP z
S
1s bounded.
Then there exists a, unique element, I(S}
tha.t
J
f(a) dP z '
S
for every z.
Proof:
Let
1
L S (z) =
I
f(a) dP z
S
For any two rea,l numbers a and
=a
~
, we have,
If,a)
S
S
in L2 , such
10
From (11) ,
,
m· II z 1/
Js
for all z in L2 • Therefore,
on the 811s.ce L2 "
clGmont in L2 ,
S!1y
(z) is
8.
bounded linea.r opera.tion
Hence from theorem (2.1), there exists a. unique
reS), such that for every z in L
2
'
r
( 3.6)
f(a)
\)
S
We she.ll define this element l(S) occurring in (3.6) as the stocha.stic St1eltJes integra.l of f(a) with respect to the process Z(S)
and write,
l(S) =
J
f(a) dZ.
S
Let Sl and S2 be two disjoint bounded Borel sets.
The integra.l
l(Sl + S2) is defined by,
E ~z • l(Sl +
s2)_7
Sl
=
J
8
1
f( e.)
J
I
f(e.) dP
=
z
+ Sf)
dP z +
<:.
f(a) dP z
82
11
Hence
for .all z in L2 • Therefore,
(in qua.dratic mean) •
Similarly onG can show that,
v
I
S
S
r
r
!
Lf(a.) + g(a·)_7 dZ =
J
S
I f(a.)
d.Z +
g(a) dZ ,
and
J
c •
f(a)
dZ
= c
S
4.
J
f(a) dZ,
c being a. constant.
S
C?~parison
with Older Definitions
The stocha.stic Stieltjea integral with respect to Z ha.s been
defined by Cramer
tion, by Karhunen
£2_7 , and, in the ca.se of the spectral func£8_7 by the method of a.pproxima.ting sums. We
sha.ll here consider only the case when Z is Karhunen's spectral
function.
Denote by
I
S
f(a)
dZ ,
the definition by means of
r
J
a~proxima.t1ng
Stuns and by
f(a.) dZ ,
S
the integra.l defined in the px-8coding section.
We prove the follow-
ing:
Theorem 4.1 :
If f(a) io B':.l.ch tha.t
exists, then
(D)
r ria)
dZ
J
S
exists and the two are equa.l.
Proof:
Since Z is a. spectra.l r • s . f,
where m is a. mea.surG.
Case (i):
f( a.) is e, simple function and m( R)
< co.
Here R
denotes the real line. Clearly, it suffices to consider integra.ls
over
R.
13
Let
r(e.)
= vk
if
where the Sk are diD Joint end
n
L
S ::: R
k
Then, by definition
I
f
=
r(a)
=
dZ
R
n
( 4.1)
E(zI)= E
k=l
£"'Z .
v
k
E
v
k
P z (Sk)
Z(Sk J_7
f
n
=E
k=l
=
\.1
f(a) d.P
z
R
by the definition of the ordina.ry Lebesgue-Stieltjes integral of
a. simple function.
I
Relation (4.1) show tha.t
f
:::
r(a) dZ •
R
Case (ii):
f( 8.) is bounc.ed and meR)
e,pnroxima.ted uniformly by
fn(a.) • (n
= 1,
2, ••. )
9.
<
00
•
Than, f( 8.) is
sequence of simple functions
(see
£"'8_7,
p. 37) •
Let ref) be the
14
integra.l defined a.ccording to (D ) •
l
I(f) = l ·
n
I( f ) •
• m
i
--7
By definition
n
00
Now from case (i), 1(1';;.) coincides with the (D2 ) integra.l of fn'
Hence,
(4.2)
E
LZ
I(fn )_7
=
fn(a.) dP z
R
Since
lim
n
f
- ? 00
n (8.)
f(a.)
=
uniformly,
(4.3)
lim
n-~
00
J
f (a.)
n
dP
z
=
R
r f( a.)
dP
J
z
R
Further, since
1 • 1 . m
n
-.-07
=
l(f)
l(f),
00
weak convergence holds, i. e.,
(4.4)
lim E;z . l(f n )- 7
-
n·~
=
00
Er-z
. l(f)_7
'L
From (4.2), (4.3), a.!'.d (4.4) "Te obt::.in
r
I
J
R
f( 8.) C.?
z
15
(for 0.11 z in Lt))'
... which proves that,
J
f(a} dZ ,
R
Ca.se (iii):
mea.sure.
f(a) bounded,
= 00
m(R)
, but m is a a-finite
Tha,t is, we can write
R
00
=
E
n=l
Rn ,
where the R are disjoint and m(R )
n
n
<
cD,
for every n.
Definition
(D ) gives
l
I
=
I
00
f(a} dZ
=
E
n=l
I
f(a.) dZ,
R
i f the series converges in quadra.tic mean.
I
n
=
f(a.) dZ
sa.tisfies the relation,
f(a) dP
Let
J
'"
=
z
From ca.se (11),
16
e.nd
n
101
n
=
~
k=l
R
k
Then,
(4.5)
E
(z I )
n
:::::
f( s.) dP
I
z
W
n
Since
1 • 1 . m
n ~ 00
(4.6)
In
=I
E(z . I n) : : :
lim
n-?oo
for every z in L
2
.
E (z I),
Also,
(\
J
,
f(a) dP
z
~
W
f
f(a) dP z
R
n
Hence,
E ( z I )
f
:::::
f(e.) dP
z
R
which proves 8.gs.in ths.t
f
R
f(a) d7. •
,
,
8.S n
~
00
•
17
The case when f(a) is unbounded can be treated similarly.
Thus
we havo sho'WIl tha,"G "Ghe :!.ntogra.l defined in section 3 is at lea.st
as general as the integra.l given by the older dofinition.
As a motiva.tion for this section, consider a. stochastic
process x(t) of the second order with covariance function r{t,s}.
Then we easily see that the Karhunen integral,
r
xes) =
j
x(t) dm ,
'J
S
(when it eXists) is an example of an additive
in (2).
e,
r . e • f • defined
The quostion na.tura.lly arises: under what condj.tions can
r . s • f • Z(S) be expressed in the form of such an integral?
We are able to prove the follovTing result:
Suppose that the following conditions hold:
The linear space
L (Z) is separable.
2
18
Defino the set function,
F(S)
=
BUp
1
z
pz Ii
The total variation (see
for every bounded S.
1,
I: E Lr z • Z (s) _Ii
/;0 a.-7,
p. 278),
I)
z
#0
.
(S) of F(S) is finite
Then, there exists an element wet) in L2 , for
almost a.ll t, BUCn tha.t,
z(s)
=
J
",{t) d 11
,
S
the integra.l being taken in the sense of Karhunen.
Proof:
First of a.ll, we need the result tha.t
set function ( in fact, a. mea.sure).
V
is a non-nega.tive additive
This is proved
s~para,tely
as a.
lemma. and is not new.
Now by definition,
2. This condition, as well a.s the lemma. that foJ.lows, has
been used by Dunford and Pettis in a similar situatior. (Transactions of the American Mathema.tical Society 47, JQ"~). A1~Y;lier
""'proof" of the -a:uthor .without conditiona of' -ut11S sort, but mere. ~
with the condition (1) E £Z(S) ]2 = 0, whenever ~(S) = 0, ~
being a given mea.sure, wa.s found-to N incorrect. Consequently,
the theorem in its present for.m owes much to this condition of Dunford and. Pettis. In
7, the result. is sta.ted with both (i)
and (5.3), but (i) is unnecessary.
£7
19
=
Tote.l va.ria.tion of P over S
=
sup
Z
~ I Pz
1=1
(Si)
II;
over a.ll finite
I
sequences of bounded seta S1 contained
in S.
:$
II Z Ii
~
liZ!!
=
Therefore,
j PZ!
liz
sup
. Total variation of F over S
V
il
(f V
n
E F(S1)
1=1
,
(S)
for every z, and hence
for every z. Here the symbol
(:I:
is
Ul~ed
to
mean lI absolutely continuous with respect to" •
Applying the Radon-Nikodym. theorem to the family of signed
mea,sures Pz ' we obta.in, for overy bounded Borel sct S,
(5.4)
P (s)
z
=
r
Gz (t) dJ
\I
S
Further, i P
i Z
I (8) which is the tota.l varia.tion of P
1s given by,
(I
(5.5)
'p
I
:j z II (8)
I
;
=
,
1
Js
IGz(t>/
d-J
Z
over S
20
Thus we have,
r
y
(8) ,
Js
for a.ll
Z
in L2 • A'ld this
h~lds
for every boundl.>d S.
Thorefore
,
for almost all t.
If a, b are rea.l numbers
I
J
8
=
aP
zl
(8) + bP
z2
(8)
["
=
G
zl
(t)
dV
+ b • G
\
z~
~
J
8
Thus we get,
.- 0 ,
8
and this holds for
a~l
bounded Borel sets S.
Bunce,
(t)
=
a G (t) + b G (t)
zl
z2
Now,the exceptiona.l set of t-points in (5.6)
for almost all t.
depends, in genera.l on z, and. in (5.7) the exceptional set
depend on the numbers a"
b as well as on zl and z2.
lIJ8,y
It is this
fact which prosents the greatest difficulty in the proof and
necessitates the imposing of the restriction
(5.2) •
Since the spa.ce L2 is separa.ble, there exists a(denumerable)
fUndamental set, r in L2 • Let La bo the linear set which spans
r.
That is, La consists of all finite linear combinations
rational coefficients of elements of
If z
€.
LO only,
(SO
sets ofY -mea.sure zero.)
€
r,
and La is dense in L2 •
one can find a. single exceptiona,l set So
for both (5.6) and (5.7).
z in La and for t
S - SO.
= union
of a.
d~numerable
number of
Thus (5.6) and (5.7) hold for every
We shall now ma.ke use of (5.2) and
(5.3) to show tha.t the same set So serves for every z
Lot z be any element of L •
2
l zn f
with
~hen
L2 •
there exists a. soquence
€
of elements in La, such that,
asn~oo.
Hence,
"z
- zmII-~
0 ,
U n
I
From (5.. 6) and (5.7) we obtain,
I
=
-z (t)
n m
Gz
I~
I
22
for
._~
0
a.s m,
n~co
•
Hence
lim
n ..--::,;CX)
8a~,
for t
E
G (t)
zn
S - SO'
exists
=
GO(t),
Since,
lim
=
n-~oo
I z II ,
it follows tha.t
Also, for every n,
I
G (t)
Zn
\ ~
C
,
00
(C is some constant, independent of n.)
Hence by the Lebesgue
convergence theorem ,
(5.10)
lim
n--31CD
r
I
I
....
S
Aleo, since
G (t)
Z
n
dv
=
J
S
GO(t) d-J
23
lim
n---';)CD
we obta.in,
Pz(S)
=
lim
n-)oo
P
Z
n
(S)
r
=
I
lim
I
I
I
n··-~oo
G (t) dv
Z
n
'J
S
"
I
=
Go(t) d).)
I
I
v
S
Thus,
(5.11)
('
J
Oz(t)
dY
=
r
I
II
Go(t) d V
J
S
S
On account of
(5.9) and (5.11) we may, for our purposes repla.ce
and a,b
rea,l numbers.
----3'8.
are
and b respectively, as n---:XJQ.
sequences in La
which~ zl
a.s n-- ~OO.
..
zn
II
----7 0,
Then from the relation,
number sequences
l 3,
Zln
and z2 respectively.
we soe tha.t
II Z
ra~ion8~
(
""
t Z2n ~
Writing,
arc
a.
n
G
zln
+b
G
n
z2n
r~ 00) ,
we obtain the lim!ting rela.t! on (a.s
for t € S-SO' whore arguing a.s above we may identify
G with
2
Gz(t), G (t) and G (t) respectivoly.
zl
z2
the relation
holds for any zl' z2
€
=
a, G
L2
and
zl
(t) + b G
z2
In other words,
(t)
From (5.9) and
te S-SO'
(5,12), it 1s cloar tha.t for overy
G (t) is a. bounded
t€ s-So,
z
linear oporation defined on the 1111bort spa.ce L2 ,
oxists an element
z (t)
(5.13)
2
=E
(z, w( t) )
/
1z
\.
for z in L2 a.nd
Honco there
wet) in L , such tha,t,
=
G
GO' Gl and
t € S-So'
E['"z
• z(s)
.
'1
,
w(t)J(
Finally,
-
7 = p Z (s)
1\
I
=I
J
S
for a.ll z in L ,
2
integral,
G (t) d~
Z
=
JE f z w( t) f d ~
\-
~~.
s
The relation (5.13) defines Z(S) a.s the Karhunen
,
I
w( t) d
V
J
S
Proof of Lemma.:
Clearly v(O) = 0, and
oJ (8)
V is
non-nogative by definition. Now,
sup
E
n
=
F(E ) ,
i
1=1
for a.ll finite sets El , ••. , En
in S. Suppose
of disjoint bounded sets conta.ined
CD
S
=
E Sk'
k=l
(disjoint, bounded sets), and let E , .•• En be any finite number
l
of disjoint sets contained in S. Then,
=
By
the definition of total varia.tion, we he.ve,
))(8)
}.
n
N
E
Z
F(E
i=l k=l
i
· Sk) ,
where N is any positive integer.
=
N
E
n
E
k=l 1=1
F(E . • S )
l
k
Since this is true for any sequence of sets E , ••• , En
l
,
for any N.
As
<
)) (E)
we ha.ve,
00,
(5.16)
By definition of
V,
we can find a finite sequence
El, .•. ,En
a.ll contained in S, such tha.t
-0 (8)-
(a)
where
€
>0
<
€
n
1: F(E ) ,
i
1=1
is a.rbitra.ry.
Also for each E1 (by dafinition of
F) we can find a. zi in L such tha.t,
2
(b)
(,
\)
Hence combining (a.) and (b), by choosing the
sma.ll, we get,
~) (8) -
Further,
€
I
<.
.11
E
1=1
1
6'i sufficiently
2'7
=
and
I
Pz (E i )
:I.
I
~t
k
I
PZ1
(E i . Sk)
Thereforo}
»(8) -
t·
,
-
~
n
E
1=1
t
k
1
P
I\zi~
zi
(E
• S
k
i
)I
i
I
n
~
t
k
E
1=1
F(E
1
• 8 ) ,
k
and this 1a true for every finite sequence
Combining (5.16) and (5.17), we have,
V(8)
=
\ E. J'" ,
L
1
CHAPTER II.
INTEGRAL REPRESENTATION OF A RANDOM FUNCTION
WHOSE COVARIANCE IS A CARLEMAN KERNEL.
1.
Introduction.
Suppose X(t) is a. second order process whose pa,rameter t
ranges over the finite closed interva.l, La, b_7.
When the co-
variance function R(t,s) is continuous in the square,
{a. ~ t.~
a, ~ s <,
bj
b}
,
a. representation for X(t) in seri-es he.s been obtained by severa,l
writers, (see e.g. Loave
L10_7
and
L3_7
for reference to Ker-
hunen) by using the theory of integral equations with a. continuous
kerneL
This representation, however, is not genera.lly va.lid if
a. :: -
,
00
b = +
00
The rea,son for this breakdown is that
•
R(t, s) does not now sa.tisfy the conditions of the Hilbert-Schmidt
theory.
For example, if
R( t, s)
= e- It- s/
,
the integra,l equa.tion)
co
¢(t) •
h
J .- It-al ¢(a) da
-00
is known to ha.ve a, non-denumera.ble set of eigenva.lues.
every- value of ~> ~
In fact,
is an eigenva.lue and the equa.tion ha.s the
29
solutions
.;- iQ t
e
To consider this
£3_7,
,
cB,se,
where
A.
=
Dolph and Woodbury ha.ve in
B
recent paper
employed the theory of singular integral equations developed
by Cerleman
£1_7.
They ha.ve obtained a. series representation for
X(t) Which makes use of random Hellinger integrale.
It is the a,im of this che.pter to derive a representB,tion for
x(t) in the form of a random integral.
This, we shall do in the
following sections, ma.king extensive use of Carleman's generalization of Mercer's theorem on positive definite kernels.
2.
The C13rleman Kernel of Class I.
Let x(t),
a ( t ( b
be a second order process whose co-
variance function, R(t,s) satisfies the follOWing conditions of
Cerleman:
( 2.1)
e,
exists and is finite,
I
b
(2.2)
11m
t'--;) t
["R(t,.) - R(t' ,.)J2 d.
=0
,
B.
where both (2.1) and (2.2) hold for every t except on a
Fl9t
Q,£
30
~ 0'
denumera.ble points
••• ,
~.
which ha.ve only a. fini te num-
bel' of limit points.
It is possible to select a. finite number of points,
, such that the integra.l,
'11 m from among the
'1 1 '
J
exists and is finite, for an
I&
aX~itrary
£>
0, for evelydamain
obta.ined by excluding fromF8., b_7, the interva.ls,
t -
'1 y
<&
( V = 1,
2, ... , m).
The conditions (2.1) - (2.3) which define a. Carleman kernel
a.re the origina.l ones st1pula.ted by Carleman and can be considera.bly simplified.
For instance it is sufficient only to a.ssume
tha,t (2.1) holds for a.lmost a.ll t in
Fa.,
b_7, sa.y for t
€
T.
If we define,
Rn (t,s)
=
R(t,s)
whenever t and s
elsewhere, where En is the set of points for which R(t)~ n, then
31
b
b
I Ir
I J
1\
~Rn(t,s)_72 dt ds
I
/
"
<D
,
\)
a.
a
and the Hilber-Schmidt theory 1s valid for the kernel Rn(t,s).
(2.4)
R(t,s) 1s of cla.ss I.
Tha.t 1s, the equa.tion,
b
¢(t)
I"\
= ~
R(t,s) ¢(s) de
Ja
has no solution
other than the trivial one for non-real A.
It is known that this
is equivalent to saying that the integra.l operator
b
H(
) =
J
R (t, a)
(
) da
a.
is self-a.djoint.
(2.5)
Let D be the domain TxT where T is the set defined
a.bove of t va.lues for which R(t) is finite.
Then R(t,s) is
continuous for (t,s) in D.
Conditions (2.1) and (2.4) do not seem to a.dmit of any ea.sy
statistics.l interpreta,tion but are sufficiently general as to
32
include a. la.l'ge class of covaria.nces.
Before procet:lding further,
we remark tha.t here we ha.va taken a., b to be fini to.
This is no
restriction since, if the limits are infinite, they can be made
finite by a. change of sca.le.
The important point is the non-ful-
fillment of the Hilbert-Schmidt assumptions.
be the eigenva.lues and eigenfunctions respectively of the equa.tion,
b
¢(n)(t) = A(n)
r
R (t,s) ¢(n)(s) ds •
n
J
a.
Since R(t,s) is a. function of non-nega.tive type, so is Rn(t,S) by
its definition.
Hence a.ll tne
A. (n)} 0.
'lI
~
Let,
c1(n)
(S )
!jJ -y
9 (t,S;A.)
:::l
c,
,
if A.) 0,
if A.
~
0 •
Thon 9 (t,s;A.) is a funct:'.on of bounded varia.tion with respect to
n
A. in every finite interva.l, () , f3 ) , ex
>
0.
Carleman has
shown tha.t
lim
n~oo
9n (t,S;A.)
=
Q(t,S;A.) ,
exists for every rea.l va.lue of A. a.nd for every point (t, s) in D.
~';'
?,
In the next section, we sta.tt' a. numbe:r'
g(t,s;~) and
~f
properties of
some auxiliary theorems relevant to our purpose.
Most of these results are to be found in CarlemanLl,
stone
1:15_7,
18'.-7
and
but the notation of the former author is a,dopted
throughout this chapter.
3.
(3.1)
Proper~..£f Q( t , a ; ~) ~nd Some
For every (t,s) in D,
varia.tion in
~
in every finite
O(t,s;~)
is
Auxiliary Theorems.
Q(t,B;~)
is a. function of bounded
~- interva.l,
s~atric
( a , i3 ) with
in t and s and continuous with
respect to (t,s) in D for every fixed va.lue of
tJ.O(t,B;~)
LlO(t,a;A) hIt) h(a) dt da )
0,
e,
a
(3.4)
That is,
b
JI
~L
for every h(t)
~.
=
is a. function of non-nega.tive type.
b
C{:'> O.
2
(e.,b).
If 61' A 2 are disjoint interva.ls not ha.ving 0 as an
end point, then
b
I
a
6 10(t,u;A) LI,O(u,a;A) du • 0,
34
and
b
f'bO(t,UjA.) [\Q(u,SjA.) du =
AQ(t,sjA.) •
Ja.
The orthogona.l1ty property of Q(t,SjA.) sta.ted in (3.4) is of
great importance in deriving our result.
It is here
the~
the
necessity for restricting ourselves to kernels of cla.ss I becomes
clear, for (3.4) doee not necessB.rlly hold if R(t, s) 1s not a. kernel of cla.ss I.
b
J
R(t,u) !iO(U,8jA.) du
=
d O(t)s;A.)
a
Since R(t,s) is of non-nega.t1ve type and continuous in
D,
00
R(t,s) =
J
P
1
dA.Q(t,SjA.)
A.
::I
lim
€.~
+0
p~
(J)
I
l
d Q(t,SjA.)
A.
€
0
0.7)
The following theorem will a.lso be needed.
Sn,(X)
(n
= 1,
2, ... ) ,
is a, sequence of square integrable functions over (e., b), such tha,t
( a)
lim
n~
g (x)
n
OJ
= g(x) ,
a.lmost everywhere, and
for all n,
(b)
C being a constant independent of n.
gra~le
Then if f(x) is square inte-
over (a,b),
b
b
J
lim
n-7CO
r fix)
f(x} g" (x):
J
a.
a.
This is a specie.l case of Theorem IV* in
(3.8)
g( x) dx.
£1_7.
Part of Loeve I s Fundamental Lemma,: (L10_7, p 315) .
If a random function, X
t'
a,s. l' - - ) -1' 0"
(t), (t
€
A) tends in qua.dratic mean to X(t)
the covariance of X(t} 1s given by
r(t,t')
=
E
LX t
(t)
X
't
(tl}J
-
(t,t' €A) •
(We are. concerned here with only rea.l- va.lued random varia.bles.)
4.
Def~tion of _~he Random Va.,ria,ble fj" o(t,A.).
Define the random variable,
b
fio.>(t,A.) =
J x(u)
p
t:. Q(t,UjA.) du,
a.
where the Ki1rhunen integre.l on the right side exists if
b
b
JJ
B.
exists.
~
R(u,v) L}Q(t,UjA.) l\Q(s,VjA.) du dv ,
a
18 a. finite A.-interval.
b
( 4.1)
f
=
R(t,u) t::. Q(B,U;") du
e,
I~
=
a.,.Q(t,B;")
,
6
from (3.5).
Now
b
ELACJl(t,,,) •.
f. J
lJ.ai(B,~) 7 =
J
b
R(u,v)IIQ(t,u;")i:>Q(e,v;,,)dudv
a.
a,
b
=
lIQ(t,u;,,)
6
e,
(4- 1)
=
·I~
J~
'"
6.
dA.Q(t,sjA.)
dQ (U,B;") du
Proof of the last assertion in (4.2):
~
into n sub-intervale
~i
ILl i I
denote the length of
iJ
D1 vide
and let
(i
i.
= 1,
••• , n).
Then,
(\
I~
J
dQ(u,s;~) =
lim
max Ibil-~ 0
A
Here s is fiXed and
T.
€
Write
n
1:
1
1=1
11
= E
i=l
Hence
b
lim
n~<D
I ~Jn(u)-72dU =
a.
Therefore,
"12
A.
J
dQ(s,sjA.)
<
CD
•
38
b
;'\
JL'Jn(ulf du <
c
a.
for a.ll n, the consta.nt C being independent of n.
Also,
r
lim
n~
In(u)
j ~ dQ(u,8;~)
exists =
CD
,
,..
~
for a.lmost B.ll u.
Write,
f(u)
(t a.B well a.s
B
=
!1Q
(t,u;~)
is kept fixed in T~)
Since
b
J
2
L'bQ(t,U;'A)J du
< ro
,
a
for t in T, we ma.ya.pply 0.7) and obta.in,
b
lim
n-)oo
du
=J f(u} ~lim
n~co
a.
The rj.ght hand side is
=
J
a
6Q(t,u;'A)
J~
6
d O{u,s;'A) du
In(u}_1 du .
39
The left hand sido 1s
b
n
lim
=
n~oo
L'i=l
r.
>..
1
i +
l!
i
J
II. Q( t, u;X)
~ig(u,s;~)
du_7
a
b
n
=
lim
n~oo
L'r.
1=1
1
~1 + € 1
f
lIiQ(t,u;X)
6ig(u,s;~) du_7
a.
from (3.4)
=
n
1
r 1=1
~ -:;::--i +
1
lim
n,~
(D
€
-
f ~ dQ(t,s;X)
=
7
6 .g(t,s;~)
1.
-
= II r(t,s;x)
,
~
say.
By an exactly similar a.rgument 'We have,
b
J
1
-
'\Q(t,u;X)
dO(u,s;~)du=O
~
a.
if 6
1
and
6
2
a.re disjoint
in~erva.ls
(not ha.ving 0 a.s an end
point) •
Now consider any interva.l
!:l,
a ,
~
~
13, where
,
40
>
a
13.'>
O.
Let
TI
... <
0:
=
where
~1-1 ~ ~i ~ ~i·
of the subinterva.ls
( 0:
,13
denote a. partition,
tJ i'
) we can choose e.
n
Z ¢(~i) ~iro(t,~) ,
1=1
m.OT )
Let
d.enote the maximum. length
¢ (X.) ia uniformly continuous in
Since
b> 0,
such tha.t
,
I~i
when
BO
- A.1 _1
sma,ll tha.t m(
1< b,
TI )
if ~i-l~ ~ ~ :\1.
a{
for
A.
~ 13
~
(1=1, ••• , n).
b.
Make the divisions
Define tho function
We ha.ve then
,since m(
TI ) -( be
If
no 1s a. "finer"
. subdivision of (a ,13 ) whose points of di vi sion j.nclude those of
TI,
3.
~le notati(;n used hero is taken from ~11_7, p.
68.
41
!:
TI
¢
I
I:
o
¢
I
TI
Tt'
t:. m(t,h)
•
Hence,
1:
¢
I
"IT
~m(t,h) - E
¢ Am(t,h)
I
"IT
o
E ,¢
=
11 "IT
o
I:11'
=
,~m(t,h)
I:
"IT
,¢ A m(t,h)
o
(¢TI t
••
¢) ~ m( t, h) •
o
Therefore,
E r!:
L
"IT
t
¢
,6m(t,h)
-
n ,¢
E
/).m(t,h)
-7 2
o
=
~. 0
, (¢n' -
I:
n
CD
in such a. way the.t m(
Therefore for every tinT,
n
t
A r(t,tjh) ,
o
as m, n~
!:
¢)2
¢ 6 m(t,h)
TI },
m(
T(o ) ~ O.
42
tends in quadra.tic mean to a random. variable, which we wr1to as
J~(~)
d£ll(t,>') •
!:'J..
Taking
define,
J
,h d<Jl
(t,~).
A
Now
"
l.Ji dtJ.>{t,~)
vA
In
=
{
by using
(3.8)
~i
Ef
!1j(1){s,~)_7 = 0,
for
11m
I:
n~ 00
i=l
•
fb~ dt.rJ(B,~)_7
A
l::.1(1)(t,~) ~i(J){s,~}_7,
and because,
EL L\ 1(1)( t,~)
=
lim
n
- 4 00
[n
I:
1=1
1rj
,..
~
1
.
J
61
·1
~ d9(t'B;~l
43
Since Q(t,BjJ.,1) is a. function of bounded variation of J.,1, we may
write
f'
i
1 dQ(t,Bj!J.)
J.,1
~i
=
J
1
j.l
+
dQ (t,Sj!J.) -
r
J
1- dQ(t,8jJ.,1)
j.l
Ai
.61
whore,
and
are non-decreasing functions of
j.l.
Then,
n
E A.
1
i=l
n
= E A.i
i=l
J
!
I..l
Ai
=
Sa.y.
An =
by the mean value theorem.
dQ+ (t,SjJ.,1) -
n
E
1=1
A.
i
J
t. i
!j.l
dQ- (t,SjJ.,1)
,
44
n
t1
E -,-.;:..1=1 /\.i + € i
::
1.6.
i!::
"'I ,
-
'l]
say.
n
E
ex 1=1
a"1
=
-~
0 a.s n
~ <Xl
11m
n~oo
A
::
n
II.
+ (t,s;/\.) ,
uQ
Hence,
D. 1
+
9 (t,s;/\.)
since ~1Q+ is non-nega.t1ve.
1n such a. way tha.t ., - 7
o.
Therefore,
~ g+(t,a;/\.) ,
and similarly,
11m
n~oo
Finally lole obta.1n,
n
lim
n--7'
1
I:
00
1=1
/\.i
J
-1I..l
dQ (t,S;f.L)
bi
Thus we he.ve esta.blished the rela.tion,
E
L
6.C1(t,/\.).
= 6 Q(t,Sj/\.).
By a. simila.r argument, if
6
1
and
D. 2 8J:'e disjoint ;\-intervals,
we got
(4.it)
It is not surprising tha.t we should obta.in relation
~
for from (3.2) and (3.3)
(4.3),
Q(t,e;;\) is a. eynnnetric function of
non-nege,tiva type and 1s henc& the covariance function of a. random
function ~a(t,~).
This 1s precisely what we have in
(4,3),
but this argument a.lone would not he.va given us the orthogona.lity
relation (4.4) •
5.
I~tegre,l Re;present~.tion
for X(t) •
Exa.ctly a.s a.t the beginning of Section 4, we define the
stoche.stic integral,
f3
J
¢C"-) daCt,"-)
=
U¢C(;
Ct,
~)
a
for every t in T.
The covariance
is givon by Lo~ve' s fundamenta.l lemma.
(3.8) •
46
U~(8; a , f3
ELU¢(t j 0, t' )
=
lim
n
~
(I)
r~[~ ¢(~j) ~1o(t,~)
~ 1=1
n
:::
11m
t
n··-? co
=
11m
n--l>
1=1
n
t
00
1=1
I
~
=
)_7
~ ¢(~j) lija(BI~)_.;l
-J
•
J=l
¢2(~j,>. EL ~1a(t,~) 6ia(e,')...)_7 by (4.4),
¢2(~1) AiO(t,s;~)
~2(~) d~Q(t,.;~)
I
,
o
Tak:1118
~(~)
1
::::r;
,
and writing,
f'
U(t;
Ct ,
~ ~
)
J
a
WO
obta.in
da( t,;>,.)
,
13
ELU(t;" ,tl ) U(a;" ,
(5.3)
tl ) ]
=
J -{-
dQ(t,s;l..) ,
0:
from
(5.1) .
<
Let 0
0:'
<
0;
(3
ELU(t; a',
I )
and
(3')
(3,
- u(tj
0:,
t3
}_72
(31
0:
=
1
X dG(t,tjA.)
J
a'
~
0
8.S
0:,
0:
I
~
I"
+
I
J
1
';'.
dG(t,tj"')
A.
,
(3
+ 0
and
(31'
(3 '~ + CD, since
00
J~
dQ(t,s;l..)
o
exists a,a an improper integra.l for every (t, 8) in D and equa.ls
R(t,s).
Hence,
1 .
q •
o:~
U(t;
m
0: ,
p )
= U(t)
+0
( 3 - +00
exists, (for every t in T) and we ma,y writE>,
co
U(t)
=
J
o
Alao,
00
J°
Remembering tho ma.nner in which
1
~O'(t,A.)
dQ ( t , 8.: f..) •
was defined, we obtain
by repeating the procedure used in the preceding section,
By utilizing (5.6) we have
E £X(t) U(t; " ,
~
f3
r
)J =
J
1
d.9(t, tjA.) •
A.
a
Making
a~ + 0,
f3
~
+00 in (5.7)
WI]
get,
00
1
J
°
Finally, for evory t in T,
A.
dQ (t, t; A.)
•
00
:=
R(t , t) -
r~
J
dg ( t, t; A.) )
A.
o
=0
because of Cerleman IS reprosents.tion theorem (3.6).
Thus we he.ve the integral representa.tion
x(t)
for every t in T.
It must be remarked that tho th00ram of Carleman quoted in
(3.6) a.SSUInOS tha.t R(t, s) fulfills the condHion,
b
JIg(t)1
R(t)dt ,
a.
is convergent for every
get)
€
2
L (a.,b) •
CHAPTER III.
FUNCTIONALS OF A M!L'ltKOV PROCESS - THE
1.
DISCONTI~UOUS
CASE.
Definit. ions al::(l Not8tion .
X(t), (T
l
~
t
~ T )
is a Markov proce3s with the transition
2
probabilities,
F(t,Xj
1"
,e) = Pr ["X(r: )€ejX(t) =
x_7,
where
and
e
E
B
which is a a-algebra of subsets of
;£.
:*~
i tsdf may be taken
to be the real line, though our theory applies to more general
spaces.
["5_7,
we give below the definition of an
additive function£1,l of X(t).
The terminology used below is the
Following
Fortet
~)
To every interval (t,
(t~: ~
real numerical random 'Te,riable L(t,
't' )
) let there correspond a
which hriS the following
properties:
(1.1)
L(t,
l'
for every t ot:. t
)
I
= L(t,
.c 1:';
t
l
)
+ L(t',
~
),
if we write
(i.~ (t)
= L(t o'
t),
to being some fixed value, ,~"':: (t) is a rand.om function of t and
L (t, 't ) is the increment of
(
.t~ t) for the interval (t,l" ) .
51
For every t, "C' , t', with t <: l'
( 1.2)
and t'c.;· t or t' :::. 1:' }
and for ever:'>, x arKL E ,L(t, 1:" ) ard y(tl) ere independ~nt, ~iv~
that X(t) = x and X(ZJ) -=
~
This
mc~ns,
G(t,x; '1.:' , § ; ex) :: Pr ["L(t, 't: ) <:. ex
I x(t)
that if
= x,
X( t
)
= ~ _7
and
G(t'.x'; t,x;
't"'G ;
ex)
= P-:-'["L(t,r
I x(t)
) <:. ex
= x,
x( 't' )
=
s
X(t') = x I _7
then,
G(t' .x'; t,x; 't',
~
;
= G(t,x;
ex)
1:' , ~ ; ex).
, and for every
(1.3)
·x,
Xl,
S
,L(t,t') andL(t',t') are indep€:Ddent ranccomvariables,
given that
X(t) = x)
w~ sh~ll
=x
X( t')
then say that L(t,
~
I ,
x(
-r ) ::
~
) is an additive functional of X(t) .
Notations.
¢(t,x; ~ ; v)
= E~eivt(t, ~
::
I -+
~
(t ,x;
)
-r , \ ;
v) d
~
F(t,x;
't'
,2).
52
Define
"(t,x; 't' ,e; v)
=
J 'V
(t,x; T ,x'; v) dx,F(t,x; 't' ,e')
e
1.1.
is not a characteristic function (c.f.), but
l.I.{t,x;
is.
t" , '£'
= ¢(t,x;
; v)
v)
t'
Fortet, in the paper cited above, has derived the following
functional equation for
(1. 4)
l.I.(t,x;
=
J
~
1.1..
,e; v)
l1(t',x';
't' , e i v)
~(t' ,x';
1:
1
(t, x it' ,x' i v) d x IF (t , Xi t ' ,e ' )
X
or
(1.5)
=
f
,e;v)dx,~(t,x;t' ,e').
..~
The form (1.·5) 1s interesting, as it resembles the equation
satisfied by the transition probabilities:
(1.6)
F(t,x; 't' ,e} =
I'
F'{t' ,x I
;
't' ,e)d ,F(t,x;t I ,e'}.
x
The study of additive functionals of X(t) when X(t} is a
continuous Markov process has been carried out by Fortet in the
paper referred to above.
In what follows, we shall be concerned
with a purely discontinuous process X(t). (see Feller,
£"4_7>.
53
2.
The Purely Discontinuous Case.
Assumptions on X(t):
(a)
For every fixed t and x,
+ ( 't - t) 'p(t,x) 'P(t,x,
where
t"
>
+ o( 't
-
t);
t and,
b( 5 - x) = 0
(b)
§ )
~
if
(' x, 1 if
p(t,x) is non-negative, continuous in t, a measurable
function of x and is bounded in every finite t interval.
(e)
in
P(t,x,
~
~
) is, for fixed t and x, a distribution function
,continuous in t, and a measurable function of x.
These are the usual hypotheses made in the derivation of Feller's
equation
[4J.
~umptions
(a)
on L(t,
1;'
).
For fixed t, x, and
~
'-.jJ (t,x; t + A t,
lim
6t--:,
lim
~
+0
'\V
(t,x; t -
~
t, ~ )
==
) =
H(t,x; ~
)
6.t-~+O
exists and is finite.
(We have momentarily suppressed v which is
kept fixed throughout the following argument.)
For fixed t, x,
(~)
'-V (t,x;t
+ ..1 t,x)_1
-
lim
At --;> + 0
=
j
t
_
,,¥ft
x;t ~
lim
6.t-~+O
exists and is finite.
b t,x}
- - - 1
ut
From (a) and
H(t,x,x)
(~)
= G( t,x,x )
it follows that
= 1.
We shall return to consider conditions (a) and
(~)
later on.
From equation (1.4) we have, ( 6 u :> 0)
=
ft
J lJ.(u+
t:. u,x I;
t' ,e)
·dXl F(u ,,,,
v· u+
A ,
~
U
.\f.t (u,Y;""l+ D. u,x')
e')
('
+ p(u,y) b. u
Hence
J
~(u+
A u,x'; ~ ,e)
'\jJ (u, Y; u+ t':.. u, x ' )
55
(2.2 )
,,=~U+ ..1
u,yj 't' ,e) 'V (u,y;u+ ti. u,y) - j..l(u,y; 1.' ,e)
!.\ u
= p(u,y)
.r
~(u+
il(U+ ~ u,yj t ,e)
"if
(u,Yju+ 6. u,y) - p(u,y)
'If (u,y;u+
b u,x'; l' ,e)
.1 u,x') dx'P(u,y,x')
~~
T
The left-hand side of
IJ. ( u+ f). u,y;
r ,e ) .
0(;)
(2,2) can be written as
'\j/
[
(u,Yju+
~ u,y)
-
~u
II
-.;
+ j.!(u+ ~ u,y; l' ,e) - 1.1(u,y;t ,e)
[1 u
Making
l'.J U --~ 0, (a similar argument holds if
of conditions (a) and (13), we find that
o I-l(t,x;ot t
~ ~.
,ejv) + G(t ,x,x;v ) 'j.! (t ,x; t' ,e;v )
D U < 0), because
exists and we obtain,
I:
p ( t,x)
('
I H(t,x,x';v)/-L(t,x';
- p(t,x)
't ,e;v)dx,P(t,x,x')'
J
Thus j.! satisfies the integra-differential equation
(2,3)
oO~ = LP(t,x)-G(t,x,xjv)_7·j..l(t,x;
rI H(t,x,x'
v
X
t ,ejv)-p(t,x)
jV)/-L(t,x'; 't' ,e;v)d ,P(t,x,x').
x
~ (t ,x;
1:' ,ejv)
56
We lmow that
J~. (
¢(t,Xjt+i\ t) :::
t, X; t+ LI t,
~1.
d
~
F(t,xjt+ t:. t,e)
~
+
p(t,x)
+
O(
~
j
t
'V(t,Xjt+.} t,
§)
d~ p(t,x, ~ )
t)
Hence,
(3.1)
9-(t..L.x jt+ 6tl-l:::
bt
~(t,Xjt+6t,x)-1 _ p(t,x),(t,x,t+~t,x)
At
+ p(t,x) }v<t,x;t+ A t, \ )d\ p(t,x, \
'};'
+
O( J )
From (3.1) it follows tha.t we may replB.ce (/3) by the condition
(/3*):
lim
!(tzxjt+6t)-l:::
6t-4,O
~t
exists and is finite.
A(t,x)
1
57
4.
Applications.
(1)
Take
L(t,t')
+,"(t,x;
= X(T)
t',!
- X(t),
;v)
H (t,Xj ~; v)
= e iV( ~
-x)
'
Since
'V (t,Xjt+ D t,XjV)
= e iV(x-x) = 1
,
=0 .
G (t,x,Xjv)
Equa.tion (2.3) becomes,
(4.1)
a~ = p(t,x)~
at
n
- p(t,x)
I, e iV(x'-x)
~
(t ,x I
t
) dx' pet ,x,x I) .
j
,e;v
If we now put
F(t,Xj t'
,~ )
e = ( -
00
= F(t,xj l' ,e) ,
where
<
x ,
~
),
a.nd if we a.ssume that
~~
exists for some fixed
~
, 't , then
equation.(4.l) holds for v=o and we have for any t,x, and for the
va.lues of
~
, "t: , considered,
~(t,x;
(4.2)
'L, ~ )
at
co
= p(t,x)[""F(t,x;
1: , \ ) -
J
F(t,x'; t',
f
)dx,P(t,x,x')J ,
00
which is Feller a equa.tion.
I
Let V(tjx) be any function of t and x.' (-CD (x .( 00).
(ii)
1:"'
> t, then L
is an a.dditive functions.l of X(t).
aV
tha.t V is a. continuous function of t and tha.t at
Assume further
exists for fixed
x and is finite.
The conditional c.f. of
L(t,t+6,t), given X(t) = x, and
X(t+bt) = ~, is
"'f" (t,x;t+ t::. t, ~ ;v)
= eiVL"v(t+6 t, ~ )-V(t,x)_7 •
Therefore,
H ( t,x,
Alao,
~
; v)
= eivLV(t,
f
If
)-V(t,x)_7.
59
2
,[V(t+ t:. t,x) - V(t,X).?
,
lit
where
a.s
\ 0
I~ 1 .
b. t ---?'-
°
The last term on the right hand side .. ---j 0,
and we get,
G(t ,x,x;v ) -- i v dV(t,X)
at
•
Thus conditions (0 ) and (f3 ) are both ea.tisfied and I.l. in this
ca.se sa.tisf1es the equB.tion
00
-
p(t,x)
ri
!
e iv £V(t,
S)-
V(t,x).? .j.l. (t '1;
~
t'
,e v ) •
J
-00
.d p(t,x, ~ ) •
i
Note tha.t for v=O, we again obtain Feller I s equation.
5.
Another Form for Equa.tion (2.3).
In some applications it ma~ be desira~le to have (2.3) in a
form which involves
making
6 t
A(t,x)
-~
A(t,x;v). (see section
3.) In equation (3.1),
0, we obta.in
= G(t,x,x)-p(t,x)+p(t,x) •
f
%
R(t,x; ~)d p(t,x, ~) •
~
60
ha~e,
SUbstituting in (2.}), we
~~
+ A(t,Xi V )
= p(t,x)
~(t,xi ~.eiv)
J~(t,x,x, ;v) L .(t,.x; 't ,e;v)-.(t,x'; r ,e;v)J•.
x
Many interesting problema remain to be considered.
For
instance, one might consider cases which are neither continuous
nor purely discontinuous.
Also, one could consider applications
of this theory which are different fram the classical ones.
hope to return to these questions a.t a. later de.te.
We
61
BIBLIOGRAPHY
1.
Cerleman, T.,
Sur les Equa.tions Integra.les Singulieres
a Noya,u Reel et 8,:-;aetrique, U:p:psa,la, Uni-
v:e:.cB'1'ty,
192~-"----
lao
Carlemen, T.,
"La. Theorie des Equations Integrales Singulieres et ses Appliea,tions," Anna.les de
l'Institut Henri Poincare, I ~193i}, 401 430.
2.
" H.,
Cramer,
"A Contribution to the Theory of Stoeha.stic Processes," '~fOS.ium on Probab5.lit¥
Theory a,t Berkeley, 1950r,-(To be published. )
3.
Dolph, C. L. and Woodbury, M. A., "Re]?reaenta.tion Theory
and Pl'edietion of Stocha.stic Processes,"
EMF - 11, (1950).
4.
Feller, W.,
"Zur Theorie der Stoehastischen Prozesse,"
Ma.thema.tisehe Anna,len, exIIl (1936), 113 -
160.
5.
Fortet, R.,
6.
Ha,lmos, P. R., Mea.sure Theory, New Yo!'k, D. Van Nostra.nd
COLlP8 .:1Y, 1950.
"Addi tive Functions,1s of a V..arkoff Process,"
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Ka.l1ianpur , G.,"Integrale de Stie1tjes Stoehastique et un
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d'Ensembles, " Comptes Pendus des seances
------.'---,
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922 :.. 923.
K.,
"
"Uber
11neare Methoden in der Wahrscheinlichkeitsrechnung," Annales Academiae
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8.
Karhunen,
9.
Kolmogorov, A. N., "Stationary Soquences in Hilbert
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Go.SlJ,18rstvennogo U:: i veY~;ltefs.T~;':.Dt ike.,
Il"0-941)-;--'
.
-
10.
Loave, M.,
AppendiX in P. Levy J Processus Stocha.stiques et Mouvement Brov.'llien, Paris J
Ga.uthier-Villars, 19~:
62
lOa. McShane, E. J.,
Integration, Princeton, Princeton University
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11.
Murray, F. J.,
An Introduction to Linear Transforms.tions in
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12.
Nagy, B. v. Sz,
Spektraldarstellung Linearer Transforma,tione~
des Hilbertschen Raumes, Berlin, Springer,
1942.
13.
Pettis, B. J.,
l'On Integration in Vector Spa.ces, II Transa.ctiona of the American Mathematical Society,
44 (1938), 277 - 304.
14.
Sa.ks, S.,
Theory of the Integral, New York, Hafner
Publishing Compa~, 1937.
l5~
Stone, M. H.,
Linear Transformations in Hilbert S ace,
See chapter 10., ew York, American
Ma,thematica.l Society ColloqUium Publications, XV, 1932.
.