Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
529 Jan Grandell
Doubly Stochastic Poisson Processes
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
529 Jan Grandell
Doubly Stochastic Poisson Processes
Springer-Verlag Berlin. Heidelberg New York 1976
Author Jan Grandell Department of Mathematics The Royal Institute of Technology S-10044 Stockholm 70
Library of Congress Cataloging in Publication Data
Grandell, Jan, 194~iDoubly stochastic Poisson processes. (Lecture notes in mathematics ; 529) Bibliography: p. Includes index. 1. Poisson processes, Doubly stochastic. 2. Measure theory. 3. Prediction theory. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 529. QA3.L28 vol. 529 [QA274.42] 510'.8s [519.2'3]
76-20626
A M S Subject Classifications (1970): 60F05, 6 0 G 2 5 , 6 0 G 5 5 , 62M15
ISBN 3-540-0??95-2 ISBN 0 - 3 8 ? - 0 ? ? 9 5 - 2
Springer-Verlag Berlin 9 Heidelberg 9 N e w York Springer-Verlag N e w York 9 Heidelberg 9 Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin. Heidelberg 1976 Printed in Germany. Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
PREFACE
The doubly stochastic Poisson process is a generalization of the ordinary Poisson process in the sense that stochastic variation in the intensity is allowed. Some authors call these processes processes'
'Cox
since they were proposed by Cox (1955). Later on Mecke
(1968) studied doubly stochastic Poisson processes within the framework of the general theory of point processes and random measures.
Point processes have been studied from both a theoretical and a practical point of view. Good expositions of theoretical aspects are given by Daley and Vere-Jones
(1972), Jagers (1974), Kallenberg
(1975:2) and Kerstan~ Matthes and Mecke
(1974). Accounts of more
practical aspects are given by Cox and Lewis (1966) and Snyder (1975).
The exposition in this monograph is based on the general theory of point processes and random measures, but much of it can be read without knowledge of that theory. My objective is to place myself somewhere between the purely theoretical school and the more applied one, since doubly stochastic Poisson processes are of both theoretical and practical interest.
I am quite aware of the risk that some readers
will find this monograph rather shallow while others will find it too abstract. Of course I hope - although perhaps in vain - that a reader who is from the beginning only interested in applications will also find some of the more theoretical parts worth reading. I have, however, tried to make most of the more applied parts understandable without knowledge of the more abstract parts. Also in most of the more theoretical parts I have included examples and numerical illustrations.
JV
All readers are assumed to have a basic knowledge of the theory of probability and stochastic processes. The required knowledge above that basic level varies from section to section. The three appendices, in which I have collected most of the non-standard results needed, may be of some help.
In section 1.2 doubly stochastic Poisson processes are defined in terms of random measures. A reader not interested in the more theoretical aspects may leave that section after a cursory reading.
In sec-
tion 1.3.1 the same definition is given in terms of continuous parameter stochastic processes and finally in section 1.4 in terms of discrete parameter stochastic processes. Sometimes alternative definitions, given in sections 1.3.2 - 1.3.4 are convenient. Generally I have used the definition in section 1.2 in the more theoretical parts. Section 1.5 contains some fundamental theoretical properties of doubly stochastic Poisson processes and requires knowledge of random measures. In section 1.6 mean values, variances and covariances are discussed. Only the first part of it requires some knowledge of random measures.
In section 2 mainly special models are treated. In sections 2.2, 2.3.2 and 2.3.3 some knowledge of renewal theory is helpful.
In section 2.3
and 2.4 the distribution of the waiting time up to an event is considered. Palm probabilities, to which section 2.4 is devoted, belong to the difficult part of point process theory. I have tried to lighten the section by including a heuristic and very non-mathematical introduction to the subject.
Section 3 is purely theoretical and illustrates how doubly stochastic Poisson processes can be used as a tool in proving theorems about random measures.
In section 4 the behaviour of doubly stochastic Poisson processes after long 'time'
is considered.
In section 4.2 knowledge of weak
convergence of probability measures
in metric spaces is helpful.
Some of the required results are summarized in section At.
In section 5 'estimation of random variables'
is considered.
Here
estimation is meant in the sense of prediction and not in the sense of parameter estimation. ful. In section 5.1 tion 5.2 'linear'
Some knowledge of random measures
'non-linear'
is help-
estimation is treated and in sec-
estimation is treated. The main mathematical tools
used are, in section 5.1, the theory of conditional distributions and, in section 5.2, the theory of Hilbert spaces.
In section A2 the
required results of Hilbert spaces are summarized.
In sections 6 and 7 the discrete parameter case is treated. tion 6 'linear estimation of random variables' section 7 estimation of covariances treated.
In sec-
is considered.
In
and of the spectral density is
In both sections methods from the analysis of time series
are used. These sections require no knowledge of random measures depend only on section
1.4 and the last part of section
and
1.6. A rather
complete review of the required theory of time series are given in section A3.
All definitions,
theorems,
lemmata,
corollaries,
examples and remarks
are consecutively numbered within each main section. definition 5 in section
1.2 is referred to as 'definition
whole of section I and as 'definition the'List of definitions,
So, for example, 5' in the
1.5' in the other sections.
...' it is seen that definition
From
1.5 is given
on page 7. The end of each proof, example or remark is signaled by ~
.
VI
There are of course many topics related to doubly stochastic Poisson processes which are not treated in this monograph.
In particular we
shall not consider line processes, i.e. random systems of oriented lines in the plane, or their generalizations to flat (hyperplane) processes. A line process can be viewed as a point process on a cylinder by identifying lines with a pair of parameters which determine the line, e.g. the orientation and the signed distance to the origin. It turns out that 'well-behaved'
stationary line processes correspond
to doubly stochastic Poisson processes. What 'well-behaved'
shall really
mean is as yet not settled. To my knowledge the best results are due to Kallenberg (1976) where results of Davidson, Krickeberg and Papangelou are improved.
There are many persons to whom I am greatly indepted, but the space only allows me to mention a small number of them. In a lecture Harald Cram@r, see Cram@r (1969), gave me the idea of studying doubly stochastic Poisson processes.
In my first works on this subject I
received much help from Jan Gustavsson. Peter Jagers introduced me to the general theory of point processes and random measures.
From
many discussions with him and with Olav Kallenberg and Klaus Matthes I have learnt much about that theory. The extent to which I have benefitted from Mats Rudemo~s advice and comments on early versions of this monograph can hardly be overestimated.
In the preparation of
the final version I was much helped by Bengt yon Bahr, Georg Lindgren and Torbj6rn Thed@en. Finally, I am much indepted to Margit Holmberg for her excellent typing.
Stockholm, March 1976
Jan Grandell
LIST OF DEFINITIONS, THEOREMS, LEMMATA, COROLLARIES, EXAMPLES AND REMARKS number
page
number
page
number
page
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8
87 88 88 116 116 118 121 142
6.1
162
AI.1 AI.2 AI.3 A1.4
206 206 208 208
16
4 4 5 5 7 11 17 23
1.1 1.2 1.3 1.4 1.5 1.6 1.7
18 19 19 2O 21 25 28
4.1 4.2
69 81
5.1 5.2 5.3 5.4 5.5
89 116 118 123 141
AI.5 AI.6 AI.7 AI.8 AI.9 AI.10
207 209 209 2O9 210 211
2.1 2.2
35 57
A2.1 A2.2
212 214
7.1
196
3.1 3.2 3.3
65 66 68
AI.1 AI.2 AI.3 AI.4
205 206 207 207
A3.1 A3.2 A3.3 A3.4 A3.5
216 217 218 220 224
1.1 1.2 1.3a 1.3b 1.4
5 10 23 24 27
3.1
67
5.1
122
4.1 4.2 4.3
77 78 80
6.1
180
Corollaries
1.1
22
2.1
37
4.1 4.2
72 72
Examples
2.1 2.2 2.3 2.4
47 48 59 60
164 167 170 183 187
83 84
95 97 107 127 128 129 132 140
6.1 6.2 6.3 6.4 6.5
4.1 4.2
5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9
7.1 7.2
193 198
5.1
94
1.1 1.2 1.3
20 24 25
4.2 4.3
78 83
5.5
126
5.1 5.2 5.3 5.4
93 118 120 125
6.1 6.2 6.3 6.4
162 165 166 182
Definitions
I I 2 1 3 14 1 5 1 5' 1 5"
Theorems
L er~mata
Remarks
2.1
55
4.1
74
CONTENTS
I,
Definitions
and basic properties
1.1
A heuristic
introduction
1.2
The general definition
1.3
Doubly stochastic Poisson processes on the real line
9
1.3.1
Recapitulation of the definition
9
1.3.2
An alternative
1.3.3
Classes of doubly stochastic Poisson processes
12
1.3.4
A definition based on interoccurrence
15
1.4
Doubly stochastic Poisson sequences
17
1.5
Some basic properties
18
1.6
Second order properties
22
1.7
A characterization
of ergodicity
27
2.
Some miscellaneous
results
31
2.1
The weighted Poisson process
31
2.2
Doubly stochastic Poisson processes and renewal processes
33
2.3
Some reliability models
4O
2.3.1
An application on precipitation aerosol particle
10
definition
times
scavenging of an 4O
A model with an intensity generated by a renewal process
44
A model with an intensity generated by an alternating renewal process
5O
2.4
Palm probabilities
53
2.4.1
Palm probabilities for doubly stochastic Poisson processes in the general case
53
2.4.2
Some special models
58
2.5
Some random generations
63
2.3.2
2.3.3
Characterization and convergence of non-atomic random measures
65
4.
Limit theorems
68
4.1
0he-dimensional limit theorems
69
4.2
A functional limit theorem
74
5.
Estimation of random variables
86
5.1
Non-linear estimation
87
5.2
Linear estimation
115
5.3
Some empirical comparisons between non-linear and linear estimation
143
Linear estimation of random variables in stationary doubly stochastic Poisson sequences
158
6.1
Finite number of observations
158
6.2
Asymptotic results
161
7.
Estimation of second order properties of stationary doubly stochastic Poisson sequences
190
7.1
Estimation of covariances
192
7.2
Estimation of the spectral density
195
A1
Point processes and random measures
2o5
A2
Hilbert space and random variables
212
A3
Some time series analysis
214
3.
6.
References
226
Index
232
I.
DEFINITIONS
I. I
A heu~tic
AND BASIC PROPERTIES
introduction
We will start the discussion
of doubly stochastic
Poisson procecesses
in a very informal way~ in order not to hide simple ideas behind notations
and terminology.
mathematical points
Consider therefore
model is needed for the description
in some space.
To be concrete,
in time and assume that multiple model describing The simplest
a situation where
a situation
such model,
events
of the location events
do not occur.
except perhaps
for a deterministic
intensity
in each time interval
dent.
Depending
in disjoint
occurring
A mathematical
intervals
is Poisson
distributed
different
with
Further,
are stochastically
of course on the situation
one, is
X. In this model the
mean value equal to X times the length of the interval. number of events
of
of this kind is called a point process.
the Poisson process with constant number of events
we consider
a
the
indepen-
objections
may
be raised against the use of this simple model. We will here discuss some objections
in such a way that we are led to a doubly stochastic
Poisson process.
(i)
Assume that the model seems realistic
know the value of the parameter
~, a rather
except that we do not common situation.
then natural to use some estimate
of ~. There exist, however,
tions where this is not possible.
Consider
insurance business dent pattern
and suppose that
follows
situa-
an automobile
for each policy-holder
the acci-
a Poisson process but that each policy-holder
has his own value of ~. The insurance knowledge
for example
It is
of how ~ varies
company may have a rather good
among its policy-holders.
For a new policy-
h o l d e r it may therefore as a constant
be reasonable
to treat his value
but as a r a n d o m variable.
w e i g h t e d P o i s s o n process is frequently
(ii)
In both the P o i s s o n
In many
variations
~. The number
situations
or other trends.
of events
in a time
perform
Formally this
is not a serious
a transformation
the m o d e l with constant
of the time X (cf Cram$r
this
complication
is more
X(t)
is required.
Thus
Suppose plays
S t a r t i n g with the ~(t) i n s t e a d of the con-
interval
is then P o i s s o n
a model
an important
variation,
complication
over the
since we may
scale which leads us b a c k to (1955, p 19)).
since k n o w l e d g e
for ~(t)
In practice
of the
function
is needed.
role.
variation.
There may of course be different To he concrete
at least partly,
depends
again we assume
on w e a t h e r
and the weather.
In spite of this
cessary to use a stochastic model In such a situation
it is thus
tion of a stochastic process. then led to a doubly
dependenc@
natural to regard
stochastic
that the In m a n y
b e t w e e n the time of
in order to describe
As indicated
reasons
conditions.
of the w o r l d there is a strong dependence
the y e a r
of ~(t)
now that we are in a s i t u a t i o n where the seasonal v a r i a t i o n
for a seasonal
parts
serious
~ was
~ will vary with time
d i s t r i b u t e d with m e a n value equal to the integral interval.
In fact this
and the w e i g h t e d P o i s s o n model
P o i s s o n model we are led to use a function stant
model.
a
used in insurance m a t h e m a t i c s .
a s s u m e d to be constant. due to seasonal
We are then led to use
as our m a t h e m a t i c a l
model
of ~ not
~(t)
it may be nethe weather. as a realiza-
in the p r e f a c e we are
Poisson process.
1.2
The general d e f i n i t i o n
In this section a general process will be given.
definition
of a doubly stochastic
The definition will be based on the theory of
random measures
and point processes.
are e~g. Jagers
(1974) and Kerstan, Matthes
vey is, however,
In section
Sometimes
As in Jagers
in time were considered,
there is a need for more general
state
i.e. R 2, is often
(1974) X will be assumed to be a
compact Hausdorff topological else is stated.
(1974). A sur-
are located will be called the state
In e.g. ecological models the plane,
natural.
for that theory
and Mecke
1.1~ where point processes
X was the real line. spaces.
Good references
given in section At.
The space X where the points space.
Poisson
locally
space with countable basis when nothing
A reader not interested
in topological
think of X as the real line or, perhaps better,
concepts may
as R 2" Often, how-
ever~ we will consider X = R when its natural order of real numbers is convenient
or X = Z where
Z is the set of integers.
Let B(X) be the Borel algebra on X, i.e. the a-algebra open sets. A Borel measure negative measure
that is finite on compact
all Borel measures. space.
(or Radon measure)
on (X,B(X))
sets.
Endowed with the vague topology M is a Polish
advised to turn to the beginning
Borel algebra on M. Let
N@B(M)
concepts
is
of section AI for definitions.
may also be helpful to read section
and B(N)
is a non-
Let M be the set of
(A reader not familiar with these topological
valued elements
generated by
1.3.1
first.)
Denote by
It
B(M)
the
be the set of all integer or infinite
of M. Endowed with the relative
denotes the Borel algebra on N. Usually
will be denoted by ~ and ~ respectively.
topology elements
N is Polish in M and N
Definition
I
A random measure
is a measurable mapping from some probability
(W, W, ~) into (M,
space
B(M)).
Usually a random measure will be denoted by A. The distribution is the probability measure H on (M, B(M)) H(B M) = ~ ( w ~ W
; A ( w ) & B M) for B M E
on (M, B(M)) we may take (W, mapping,
B(M).
For any probability measure H) and A as the identity
i.e. A(~) = ~. Thus any probability measure
talk about a random measure
random measure,
on
(M, B(M))
We may, and shall, thus
A with distribution
ference to an underlying probability
known,
induced by A, i.e.
W, ~) = (M, B(M),
the distribution of some random measure.
of A
H without any re-
space. When we talk about a
it is tacitly understood that its distribution
is
and it is often convenient to use the notation P r ( A E B M)
instead of H(B M) for B M ~ B ( M ) .
Let a random measure
A with distribution
H and a set B ~ B ( X )
be
given. We will talk about the random variable A(B), which is nonnegative
and possibly extended,
see theorem AI.2.
Similarly
for
given B I .... ,Bn~B(X) we talk about the random vector
(A(B 1 ) . . . . .
A[Bn)).
Definition
2
A random measure with distribution H is called a point process if ~(N) = I.
Usually a point process will be denoted by N. We will, whenever convenient
and without comments,
of a point process
assume that all realizations
are in N and interpret its distribution
probability measure on
(N, B(N)).
as a
is
Definition 3
Ar a n d o m ~ measure A is completely random if A{BI},...,A{Bn } are independent random variables whenever B I , . . . , B n ~ B ( x )
In e.g. Kerstan, Matthes
and Mecke
are disjoint.
(1974, p 24) it is shown that for
every ~ E M there exists exactly one p r o b a b i l i t y measure H
(N, B(N))
which is the distribution
on
of a completely random point pro-
cess N with
Pr{N(B} = k} = P{B)k e-P(B}
kl for all k = 0,1,...
and all bounded B ~ ( X ) .
I n this paper a set is
called bounded if it has compact closure.
Definition 4
A point process N with distribution
H
is called a Poisson process
with intensity measure ~.
We note that if N is a Poisson process with intensity measure ~ and if B is an unbounded set in B(X) with ~{B} = ~ then Pr{N{B}
We will now give the general definition Poisson process.
= ~} = I.
of a doubly stochastic
In order to justify the definition the following
lemma is needed.
Lemma
I
For every B e B ( N ) measurable.
the function ~ ~ H {B} from M into E0,1]
is
B(M)-
Proof
This lemma is a consequence
of 1.6.2 in Kerstan,
(1974, pp 64-65).
We will, however,
that the function
U ~ H {BN} is B(M)-measurable
form {v~ N; v{B I} = k I
~"
..,v{B
n
Matthes
give a proof.
and Mecke
We will first show
for sets B N of the
) = k } where BI,...,B n are disjoint n
sets in B(X) and kl,..,,k n are finite nonnegative
integers.
In this
case we have k.
n
~{9.i } ~
i=I
-~{B i } if all ~{B.}
e
I
0 and almost all
(Lebesgue measure) (iii)
(F).
xER.
f E 1(x)dx < co for all bounded B @ B(R). B
From (ii) it follows separable
chosen.
(cf Doob
and measurable
Then 1(w) is a.s.
that 1(w) is a.s.
(1953, p 61)) that there exists a
version
of t.
Assume t h a t
(~) B(R)-measurable.
(~) non-negative.
this
From
Thus the set W
version
is
(i) it follows = (wEW;
1(w) is
O
non-negative
and B(R)-measurable}
has ~-measure
one.
From ( i i i )
it
n
follows that the sets W n = ( w E W ~ ; measure
one for all n = I ,2, . . . .
l i m W = W' = { w ~ W ; t ( w ) E I } n
mapping
A : W § M
by 0
f 1(w,x)dx
< ~} also have E-
Since W n + 1 ~ W n and since
also
W' h a s B - m e a s u r e
one.
Define
the
14
if w6 w'
# o ~(w)
A<w) = if w ~ W - W' 0
where ~
for example is the measure with ~ {R) = O. 0
Thus ( w & W
0
; A(w) (B} x for all y ~ R we put ~ continuous nondecreasing
For any ~ E N
(x) = - ~. Thus p
function from (- ~, ~) into
-I
is a right-
~- ~, ~
we put t k = ~-1(k) and consider the infinite vector
t = (...,t_2,t_1,t0,tl,t2,...). have
-I
According to the properties
... ~ t_2 ~ t_1 ~ 0 < t O ~ t I ~ t 2 ~
of ~ we
... and further
lim t k = • ~. If ~ is considered as a realization k§177
of a point pro-
16
cess, the tk:s are the epoch of events of v provided the possible non-finite tk:s are properly
interpreted.
Let T be the set of all vectors t and let T be endowed with the oalgebra B(T) generated by { t ~ T ; t k ~ x}, k = 0,• As p o i n t e d out by e.g. Daley and Vere-Jones probability measure on (T, B(T)) generates
x6 ~
(1972, pp 308-309)
~, ~ . any
a p r o b a b i l i t y measure on
(N, B(N)), and conversely.
Let ~ N
and B E M be given and consider v = m v o ~. Put t k = v
and t k = v
-I
(k)
(k). Then we have
t k = sup(y : mv(~(y)) < k) < sup(y : ~(y) < m tk ) = B-I ( ~ k ) " On the other hand,
for every ~ > 0 we have
t k = sup(y
: v(~(y)) < k) > sup(y
: ~(y) < ~ t k - s) = ~ -1(t k - s)
-I m -I and thus t k = ~ (t k) p r o v i d e d ~ (x) is continuous
at x = t k.
Let N = N o A be a doubly stochastic Poisson process
as defined in
section
1.3.2.
L e t T and T be t h e random v e c t o r s
d e f i n e d by
%
Tk = N - l ( k ) pendent
and Tk = ~ - l ( k )
respectively.
S i n c e ~ and A a r e i n d e -
it follows that ~ and A -I almost surely have no common points
of discontinuity.
T=
Thus -I m
( .... A
(T_I),
-I m
A
(To),
A-I m
(~rl)...)
a.s.
and thus the two random vectors
are equally distributed.
This rela-
tion may serve as a definition,
based on interoccurrence
times, of
doubly stochastic Poisson processes.
Kingman
has used the above relation as definition by Serfozo
(1972:1, pp 290-291).
(1964),
see section 2.2,
and it has been discussed
17
1.4
Doubly stochastic Poisson sequences
Consider now the case X = Z, i.e. when the state space is the integers. A Borel measure on Z is a measure
assigning nonnegative
finite
mass to each integer and is completely determined b y these masses. Thus we may identify Borel measures
on Z and sequences of nonnegative
finite numbers.
By a point process or point sequence N with state space Z we m e a n a sequence of random variables Z+ = {0,1,2,...}. = {Uk ; k ~ Z }
A Poisson sequence with intensity measure
is then a sequence of independent
random variables all n ~ Z + .
{N k ; k @ Z} taking values in
such that
Poisson distributed
(~k)n -~k
Pr{N k = n} =
nl
e
for all k 6 Z and
By a random measure s with state space Z we mean a sequence
of random variables
{Zk ; k ~ Z }
taking values in R+.
The following definition is equivalent with definition 5.
Definition
5"
A point sequence N is called a doubly stochastic Poisson sequence if, for some random measure Z,
nk. m
Pr {n
m
{~k.
j=l
(Lk.)
= nk ) } = E { ~ j
j
j=1
J
'~
-~k e
J}
nk. J
for any positive integer m, any integers k I < k 2 < ... < k m and any nonnegative
integers
Parts of this paper Poisson sequences. applying methods
nkl,...,nkm. are devoted to the study of doubly stochastic
The main reason is that we are interested in
of time series analysis.
that in many cases observations
We will, however, point out
of a point process
are for measure-
ment reasons given in this form. There also exist cases where there is impossible to observe the exact ~time ~ of a point.
In e.g. sickness
18
statistics the number of people reported sick each day can be observed, but the exact time of the start of a disease is impossible to observe and even perhaps to define.
1.5
Some basic properties
We recall from section 1.2 that to each probability measure H on
(M, B(M))
the probability measure / H H(d~), which in this section M is denoted by PH' on (N, B(N)) is the distribution of a doubly stochastic Poisson process. In terms of Laplace transforms
(see defi-
nition A 1.2) we have the relation LpH(f) = LH(I - e -f) (cf B a r t l e t t % contribution to the discussion of Cox (1955, p 159) and Mecke (1968, P 75)). From this relation some theorems, most of them due to Krickeberg (1972) (cf also Kummer and Matthes (1970) and Kerstan, Matthes and Mecke (1974, pp 311-320)), follow as simple consequences.
Theorem
1
PHI = PH2
if and only if
H I = H 2.
Proof If H I = H 2 then PH
= PH2 follows from the definition. The converse I is proved by Krickeberg (1972, p 163) and will be reproduced here.
Assume that PHI = PH2 , which implies LPH I (f) = LPH2(f) and thus
LHI(I - e -f) = LH2(I - e -f) for all f~CK+.
Thus LH](g) = LH2(g) for
for all gE OK+ with sup g ~ I since to each such g there exists a f~ CK+ such that g = (I - e-f). To see this we just have to observe that f = - log(1 - g)~ CK+ for all g of the above kind. Consider now an arbitrary f~CK+.
Then LH1(sf) = LH2(sf) for all non-negative
s ~ (sup f)-1. Since f E C K +
it follows that sup f ~ ~ and thus
19
(sup f)-1 > 0. Since L(sf), as a function of s, is the Laplace transform of the random variable ; f(x)A(dx} where A is a random measure X with distribution H, it follows that L(sf) is determined by its values on ~O,a) for any a > 0. Thus L H (f) = LH2(f) for all f ~ C K + I and thus (see theorem A 1.3) H I = H 2.
Krickeberg (1972, p 165) notes that PHI~H 2 = PHIXPH2 for any H I and H 2 where ~ means convolution as defined in section A I.
Now we give a similar theorem about weak convergence, a concept which is discussed in section A I.
Theorem 2 Hn
_~w PH "
W,H if and only if PH n
Proof If Hn
W~ ~
then L H (f)--~ L~(f) and thus LPH (f)--+ Lp (f) which n H n implies (see theorem A 1.6) PH w~ PI[ " n If PH
n
w PH t h e n LH ( g ) - ~ LH(g) f o r a l l n
g~CK+ w i t h sup g < 1
and thus for an arbitrary f ~ CK4 it follows that L H (sf)-~ LH(sf) n for all n o n n e g a t i v e s < (sup f ) - I and t h u s LE (f)---* LH(f) (compare
n the proof of theorem I and the continuity theorem for Laplace transforms of random v a r i a b l e s )
Let
N o E B(N)
which i m p l i e s t h a t H ~
n
be the set of simple elements in N and let M
of n o n - a t o m i c e l e m e n t s i n M ( s e e d e f i n i t i o n
A 1.1).
theorem is due to Krickeberg (1972, p 164).
Theorem 3
Mo~ B(M)
~ .
and PH{No} = I if and only if H{M o} = I.
9 o
be the set
The f o l l o w i n g
20
Proof It is known that H {N ] = I if and only if bE M (cf e.g. Kerstan, o o Matthes and Mecke
(1974, p 31)) i.e. M
it follows from lemma I that PH{N~
= I H~{N~
o
= {~6M;
H IN } = I}. Thus ~ o
M ~ B(M) since N 6 B(N) and further o
o
= I if and only if ~ ( M ~ a.s.
(H).
9
Consider X = R and a random measure A with distribution H on
(M,B(M)). A (or H) is called strictly stationary if n Pr { ~ 2=1 n
=
{A{B. + y} < x . } } 1 -- 1 Bi ~ B ( R )
1,2,...,
is
independent
of y for
all
y ~ R,
a n d x i E R+ . (B + y = {x ; x - Y E B } ) .
Remark I This definition has an obvious extension to X = R k and may be further extended
(cf e.g. Mecke
(1967)) so that e.g. X = Z is
included.
We will sometimes consider strict stationarity when n X = R+. Then we mean that Pr {~] {A{B i + y] ~ xi}} is indepeni=I dent of y for all Y E R+, n = 1,2, .... Bi6 B(R+).
Theorem 4 PH is strictly stationary if and only if H is strictly stationary.
Proof It follows from theorems A 1.3 and A 1.4 that a random measure A is strictly stationary if and only if the distribution of f f(x - y)A{dx] is independent
of y for all f ~ C K + .
R
Define Ty : CK+-'* CK+ by T y f ( X )
= f(x
- y).
A is
stationary if and only if LH(Tyf) is independent f 6 CK+. S i n c e theorem
I.
Ty(1 - e - f )
= 1 - e-Ty f the
theorem
thus
strictly
of y for all
follows
from
21
Now we leave the stationary case and consequently X need not be the real line. L e t ~ d e n o t e The sets P g ~
the set of probability measures on
of all probability measures on
(N,B(N))
(M,B(M)).
and D ~ P
of all
distributions of doubly stochastic Poisson processes are of special interest to us.
Let D
: P § P for p ~ [~,I] denote the p-thinning operator, i.e. for P
any point process with distribution P6P the distribution of the point process obtained by independent selection of points with probability p is D P. The operator D is one to one (cf Kerstan, Matthes and Mecke P P (1974, p 311)). Mecke (1968) and (1972) has shown that D =
~ D P 0 ~ --
1.7
> 0 for all x ~ E - w,wJ
m
+ fZ(x). Thus
a fact that will be useful.
2w
A ch~acteriz~gion of ergodi~ity
The result to be given in this section will not be used in the sequel but the topic has some relevance to the questions treated in section 4.
Consider X = R and define for all Y 6 R the shift operator T : M + M Y by (Ty~){A} = ~{A + y} for all A ~ B ( R ) A + y = {x~R
; x - y ~A}.
and recall that
For any B ~ B(M) we put
TyB = {~6 M ; T_y~ 6 B}. For general properties of the shift operator we refer to Kerstan, Matthes and Mecke (1974, pp 133-140). A set
BEB(M) is called invariant if T B = B for all y ~ R. Let A be a Y
strictly stationary random measure with distribution H. Then H{TyB} is independent of y for all y ~ R and B ~ B ( M ) .
H (or A) is called
er~odic if H{B} = 0 or I for all invariant B 6 B ( M ) .
Let A(M) be the algebra
[~ B (M) where B (M) is the G-algebra gene, n= I n n rated by {~& M ; ~ { A N ~- n,n~} ~ x} for all A 6 B(R) and x ~ R + . From
theorem A 1.1 it follows that A(M) generates B(M).
The following lemma contains all ergodic theory to be used in this section.
Ler~ma 4
Let H be the distribution of a strictly stationary random measure. The following statements are equivalent:
28
(i) (ii)
is ergodic. For all B(M)-measurable I h(~)H{d~}
lim-~ t -~ (iii)
functions h : M § R+ with
< ~ we have
t f h(Ty~)dy = -t
For all B 1,B2ff
A(M)
h(~)~{d~}
a.s. (9).
we h a v e
lim ~-~ 9{B Ig]TyB2}dy = H{B I}9{B2}. t-~ -t
(iv)
Any representation H = ~91 + (I - ~)92
(0 < ~ < I)
of 9 as a mixture of stationary distributions
H I and 92
is a trivial, i.e. ~ = 0 or a = I or H 1 = H 2.
Proof (i) @
(ii) ~:~ (iv) follows from Kerstan, Matthes and Mecke
(1974, p 141).
(i)
0
( a , 8 > O)
=
0
if
x 0}
oo
Pr{N(t) = n) = f (xt)~n e -xt U{dx} 0 for all t > 0 and n = 0,1,2,...
and some distribution
point process need not be a weighted Poisson process. berg
(1969, p 123) gives an example.
If we, however,
function U this Jung and Lundassume that N is
33
a weakly
stationary
weighted
P o i s s o n process
(In section
doubly
1.6 stationarity
form of Pr(N(t)
rity it follows R(s,t)
U, and thus
where
2
'only if' direction
but the modifica-
1.1.) This
follows and
= ~2st.
from lemma
)%2 s
= (t -
Thus E(A(t) the
from
a random variable
it follows
- A(s))
since
Var N(t) =
= Var ~. From the assumption
that Var(A(t)
= Cov(A(s),A(t))
then N is a
= n) is as above.
for t ~ R
X here merely means
= o2(t2 + t 2 - 2t 2) = 0 w h i c h proves
2.2
process
= n) we get E N(t) = t E~
function
Var A(t) = t2~ 2
is defined
see remark
= t EX + t 2 Var ~, where distribution
Poisson
if and only if Pr(N(t)
tion to t > 0 is obvious, the
stochastic
with
1.3a that of stationa-
and thus
- tA(1)) 2 =
'if' direction.
The
is obvious.
Doubly stochastic Poisson proc~ses and r e n e w ~ p r o c ~ s e s
In this
section we will study the class
are both doubly
stochastic
Poisson
processes
Since both kinds of point processes the Poisson process,
interest,
is both a doubly
Kingman
in this
section
Kingman~s considered
In this
Poisson
may be helpful
(1964) has characterized
Poisson processes
for x < 0.
which
Such a study may also have
process
a process
a certain
out that
as a 'variation
which
and a renewal process,
in the analysis
which
of
common to the two
class
is somewhat
our p r e s e n t a t i o n
of the process.
of doubly
also are renewal processes.
give a discussion
we will point
section
generalizations
since if we are considering
stochastic
both representations
are natural
interest.
which
and renewal processes.
a study of the processes
classes m a y have a theoretical a practical
of point processes
stochastic
Although broader
we will than
may at most be
on a theme by Kingman'
all distribution
functions
are assumed
to be zero
34
We will consider tion
point processes
N = {N(x)
; x L @which,
1.3.4, may be defined by a random vector
To avoid
some minor
trouble
to zero with positive renewal process where
probability.
k = 1,2,...,
Since we only allow
finitely
< I. T O is allowed
tion H. The variables bility
and in that
renewal process distribution
have a common many events
function
intervals
+ ~ with positive transient.
if and only if at least
H and F are defective,
takes the value
i.e.
F.
we require
distribution
process
a
random variables
distribution
to have a different
is thus transient
T O to be equal
N is called
in finite
case we call the renewal
ding r a n d o m variable
section
are independent
T k may take the value
functions
T = (T0,TI,T2,...).
A point process
if T0,TI-T0,T2-TI,...
T k - Tk_1,
that F(0)
we allow in this
see sec-
funcprobaA
one of the
if the correspon-
+ ~ with positive
probability.
If
H(~)
= F(x)
the corresponding
I - e -x
if
x>
0
0
if
x
O) be a n o n d e c r e a s i n g
of section
1.3, such that A(O-)
rightcontinuous < 0 < A(O).
stochastic
For the same reason
as w h e n we allowed T O to be equal to zero with positive we allow Pr{A(O)
A-1(x)
> 0} > 0. The process
= sup
is called the inverse A-I(0)
(y
:
A(y)
; x > 0} defined by
of A. Due to the assumption
vector ~ = (~0,~i,~2 .... ) define
(~1) .... )
A(0-)
< 0 we have
= + ~} > 0. Let the random
a Poisson
1.3.4 it then follows
T = (A -I(TO),A
probability
< x)
> O. Further we allow Pr{A-1(x)
From section
{A-1(x)
pro-
process
with intensity
that the random vector
one.
35
defines a doubly stochastic Poisson process on R+.
Put oo
f(~)
=
S e-S~
F{dx}
0 and oo
~(s) =
S e-SX
H{dx}
0 where F and H are the distribution functions in the definition of a renewal process.
A point process N, with Pr{N(x) = 0 for all x > 0} = I, is both a doubly stochastic Poisson process and a renewal process. This uninteresting case will be left out of considerations.
Theorem I (i)
A doubly stochastic Poisson process corresponding to A is a
renewal process if and only if A -I has stationary and independent increments.
(ii) A renewal process is a doublx stochastic Poisson process if and only if
~(s) :
I I - log ~(s)
and
~(s) = ~o(S)~(s)
where g(s) = S e-SX G{dx} for some infinitely divisible distribution 0 function G with G(O) < I and go(S) = S e-SX Go{dX} for some distribu0 tion function G . O
(iii) The two representations are related through
E e -sA-1(~
= ~o(S)
36
and E e
-s(A-1(1) - A-I(0))
= g(s).
Proof (i) The 'only if' part, which is the difficult part is proved by Kingman (1964, pp 929-930) and will not be reproduced here. Con^
sider now the 'if' part. Let go and g be given by part (iii) of the theorem. For any n > 0 we have n
E exp{- s0T 0 - kZ=1 sk (Tk - Tk_1)} =
= E exp{- s0A-1(0) - So(A-I(T~ O ) - A-I(0)) n
-
Z sk
(i-1(~k)
- i -I (~T k _ 1 ) ) ) =
k=l
TO)
n
= ~o(So) E(~(s o)
Tk
n E(~(s k)
) =
k=l =
go(SO)
n
I
I - Zog ~(s o)
k:1
I - Zog ~(s~)
which proves part (i) of the theorem.
(iii) This follows from the proof of the 'if' part of (i).
(ii) To any G and Go, defective or not, satisfying the conditions in (ii) there exists a process A -I with stationary and independent increments such that g and go satisfy the relations in (iii). Conversely, for any process A
-I
with stationary and independent in-
crements g and go given by (iii) satisfy the conditions in (ii), since if G(O) = I then the corresponding doubly stochastic Poisson process will not necessarily have only finitely many events in finite intervals. Thus (ii) follows from (i) and (iii).
37
Now we will consider the class of point processes which are both doubly stochastic Poisson processes and renewal processes in more detail. In the analysis we will alternate freely between the two representations. We will follow Kingman and consider the stationary case. A renewal process is called stationary, provided F is not defective and has finite expectation p, if
1
(1
i
~(x) = ~ o
- F(y))dy
.
A stationary renewal process is a strictly stationary point process.
Corollary I A stationary renewal process is a doubly stochastic Poisson process if and only if
~<s) = [I + bs + f 0 and some measure B on (0,~) such that f x B{dx}
0 and some measure B on (0,~) with
f 7-Yqx x B{dx} < 0 For the distribution function 0 (and thus also F) is defective if an only if b
> 0 .
Thus in the stationary case b
= 0.
38
co
Kingman
(1964, p 925) showed that ~ = b + f x B{dx},
co
and thus
0
f x B{dx} = ~ - b < ~. Thus the 'only if' part follows from 0 theorem I (ii). The 'if' part also follows from theorem I (ii) ^
if a distribution exists.
Kingman
function G o such that h(s) = go(S)f(s)
always
(1964, p 925) has shown that X
co
I
7(b +# # ~{dz}dy)
if xLO
0 y
0o(X) = 0
if
x
0
if
x
_ 0 and some measure B on (0,co) such that co
x B{dx} _< 0 We may observe that since F(0) = lim f(s) we have F(0) > 0 if and oo
S-~co
only if b = 0 and 5 B{dx} < co. Since a stationary renewal process 0 is simple,
see definition A 1.1, if and only if F(0) = 0 it follows
from theorem
1.3 that A(t) is continuous
a.s. unless b = 0 and
39
f B{dx} < ~. 0 If b = 0 and S B{dx} = e < ~ we define the p r o b a b i l i t y measure C by 0 C{dx} = 2 B{dx}. Then C
9(s) = c f (I - e -sx) C{dx} = c(I - f e -sx C{dx}) 0 0 and thus A properties
-I
is compound Poisson process.
of A
-I
U s i n g the sample function
it is not difficult to see that A has the represen-
tation
9(x) k~1 ~
if
~(x) > 0
0
if
~(x) = 0
A(x) =
where N is a stationary renewal process with interoccurrence
distribution
C a n d {~k}k=l i s
a sequence of independent
variables all b e i n g exponentially
time
random 1
distributed with mean --. e
In the case b = 0 Kingman Pr{D+A(x) where D+A(x)
(1964, p 926) showed that
= 0 for almost all x ~ O} =
I
is the right-hand derivative.
Thus, if b = 0 and S B{dx} = *, almost all realizations of A are 0 continuous and, considered as measures, singular with respect to Lebesgue measure.
Kingman considered the important class of doubly stochastic Poisson processes,
discussed in section
1.3.3, where
X
A(x) = S ~(y)dy 0 for some stochastic process
{l(x)
; x > O} measurable
in the sense of
Doob and not identically equal to zero. He showed that a stationary
4o
renewal process can be expressed as such a doubly stochastic Poisson process if and only if b > 0. In this case ~(x) alternates between I the values 0 and ~ in such a way that ~(x) is proportional to a stationary regenerative phenomenon (cf Kingman 1972, p 48).
If f B(dx~ ~ ~ and if c and C are defined as above, it follows, see 0 I
Kingman (1964, p 9 2 8 ) , t h a t X(x) i s e q u a l t o 0 and ~ a l t e r n a t i v e l y on intervals whose lengths are independent random variables. The I lengths on the intervals where X(x) = ~ a r e
exponentially distributed
with mean ~ and the lengths where ~(x) = 0 have distribution function C. C
2.3
Some r e l i a b i l i t y models
Consider a doubly stochastic Poisson process (N(t)
; t ~ 0~. In this
section, with perhaps a somewhat misleading title, we will consider the distribution of the waiting time T for the first event. Since (T > t~ = (N(t) = 0~ this is the same problem as calculating the probability of no events in an interval.
2.3.1
An application on precipitation scavenging of an aerosol particle
In this section we will study a model, due to Rodhe and Grandell (1972), for precipitation scavenging of an aerosol particle from the atmosphere.
Information about the distribution of the waiting
time for the first event is of interest in connection with air pollution problems.
The intensity for the removal of a particle from the atmosphere is highly dependent on the weather. In the model we assume that the removal intensity only depends on whether it is raining or not. Let ~d denote the removal intensity during a dry period, i.e. during a dry period a particle has the probability ~d h + o(h) of getting
41
scavenged from the atmosphere in an interval of length h, and let
P
denote the removal intensity during a precipitation period. Let X(t) be a stochastic process defined by kd
if dry period at time t
kp
if precipitation period at time t
k(t) :
It is further assumed that k(t) is a continuous time Markov chain with stationary transition intensities qd and qp defined by I qd = lim ~ Pr(~(h) = ~pl~(O) = ~d } h+O I qp = lim ~ Pr{~(h) = XdlX(O) = Xp} , h+O and with initial distribution
Pd
: Pr{~(0)
: ~d }
pp = Pr{k(0) = k } . P For some discussion of the relevance of this model we refer to Rodhe and Grandell (1972).
Consider a particle which enters the atmosphere at time 0 and let T be the time for the removal of that particle from the atmosphere. Define G(t) by t G(t) = Pr t} = E(exp( - f k(s)ds}). 0 Put
Gd(t) = Pr{T > tl~(0) = kd } G(t)
: Pr t1~(0) : ~p}
and thus G(t) = PdGd(t) + ppGp(t) .
42
The chosen initial distribution describes the knowledge of the weather when the particle enters the atmosphere.
From the properties of k(t) it follows that
E(exp{ -
t+h 5 l(s)ds}ll(h)) h
is independent of h and by considering the possible changes of k(.) during (O,h) we get -Idh Gd(t + h) = (I - qdh)e
Gd(t) + q d h % ( t )
+ o(h)
and thus h + 0 gives
G~(t) = - (qd + Id ) Gd(t) + qdGp ( t ) and similarly G'(t)p = %
Gd(t) - (qp + Ip) Gp(t)
.
From the general theory of systems of linear differential equations it follows that -rlt Gd(t) = a d e
-r2t + Bd e
-rlt Gp(t) = ap e
-r2t + Bp e
where
rl
=
r2 =
1
-
I (qd+%+Xd+Xp) +
tl~ > t} B(t)
= Pr{T > tl~ = t} = E(e
-~dTd-Xp~ ~
PI~ = t) .
Note that it is irrelevant for B(t) if we start with a dry period or with a precipitation period. It can be shown that
Ad(t) (I - F(t)) = -~d t = e
t -XdT-~p(t-T) (I - F(t)) + 6 e (I - Fp(t-T))Fd{dT}
and that this function is non-increasing. q~
Separating the two cases T > t and ~ < t we get
52
t - F(t)) + f B(T) G d ( t 0
Gd(t) : Ad(t)(1
T)F{dT}.
Assume now, like in section 2.3.2, that there exists a K > 0 such that co
f e 0 the methods of calculating G(x,t) also applies to G (x,t) ~kWk(X ) x provided ~k(X) is replaced with m(x) . For the stationary case, i.e. Wk(X) = Wk' this result is obtained by Rudemo (1973:2, p 279).
It may be observed that the above reasoning holds also if the transition probabilities are not stationary, but then we have no methods for calculating G(x,t).
Example 4 Consider the case studied in section 2.3.2 where ~(x) = ~N(x) and assume that N(x) is a stationary renewal process. It was shown that under certain regularity assumptions
9
63
l i m e Kt G(t) = C. t~ Assume these regularity assumptions
and assume further that
0 < E kk < ~" Put m = E Xk'
Thus t
-
i fI ~
Gx(x't) --
n(x)
e
f n(y)dy x
o
and since N is stationary it follows that there exists a uniquely determined function G~
G~
such that (for almost every x)
= Gx(x,t+x)
For the general theory of Palm probabilities case we refer to Jagers
in the stationary
(1973, pp 25 - 26). Thus t
G~
i
=
f n(y)dy 0
e
n(o) m
~{dn}
o
and thus G~
is a monotone function.
We will now consider the behaviour of G~
for large values of t.
For arbitrary x E (0,t) we have x
x
f o~
=
0
f Oy(y,t)dy = 0 t
-• --
n(y) m
OI
e
f
n(z)dz
Y
0 t
t
- S n(z)dz
-•
(e x
-m
e
I o
= !
m
(o(t-x)
- f n(z)d~
- o(t))
0
) ~(dn}
=
62
which is a Palm-Khintchine
equation
(cf
e.g.
Daley and Vere-Jones
(1972, p 358)).
From the monotonicity
xG~
of G ~ it follows that
< A (G(t-x) - G(t)) < xG~ -- m
and thus
xe 0. Thus, see section measure.
Let ~ = {N(t)
of doubly
such that A(0) = 0 and A(t) < ].3.1, A corresponds to a random
; t 6 R+} be a Poisson process with inten-
sity one and independent
of A and let, see section
doubly stochastic Poisson process N = {N(t)
; t~R+}
%
to A be defined by N = NoA, i.e. N(t) = N(A(t)).
1.3.2, the corresponding
68
Theorem 3 d Let A,AI,A2,...
A if and only if
and A be as in theorem 2. Then A n
-i {A} n
(i)
E e
(ii)
E A {A} e
.~ E e
-A{A)
for all bounded A ~ A
-An{A}
-A{A} ," E A(A} e
for all b o u n d e d A s A
n
co
(iii)
{An) I is tight.
Proof Proceed as in the p r o o f of theorem 2 up to the construction of N n. From assumption
(iii) and lemma I it follows that {Nn} I is tight
and from assumptions
(i) and (ii) that Pr{N {A} = 0} § Pr{N{A} n
and that P r { N {A} = 1} § P r { N { A ) = 1} a n d t h u s
it
follows
= 0}
by theo-
n
d
rem A 1.8 that N ~
d
N which, by theorem
1.2, implies A --~ A .
n
n
m
4.
LIMIT THEOREMS
In this section we will consider asymptotic properties stochastic Poisson processes
Let A = {A(t) rightcontinuous
; t~R+}
on B+ = [0,~).
be a stochastic process with nondecreasing
sample functions,
for all t > 0. Thus, see section measure.
Let ~ = {N(t)
of doubly
such that A(0) = 0 and A(t) < ].3.1, A corresponds to a random
; t 6 R+} be a Poisson process with inten-
sity one and independent
of A and let, see section
doubly stochastic Poisson process N = {N(t)
; t~R+}
%
to A be defined by N = NoA, i.e. N(t) = N(A(t)).
1.3.2, the corresponding
69
One-dime~ional l i m i t theorems
4.1
We will n o w consider This question and
(1972:2)
the asymptotic
has been independently and Grandell
It is well-known
of N(t)
t r e a t e d by Serfozo
as t § ~. (1972:1)
(197]).
that
~(t) - t
d --§
where W is a normally
as
t §
distributed
d and Vat W = ] and where - - ~ means In many
distribution
cases there
exists
d
'convergence
constants
> 0 and a r a n d o m variable A(t) - Kt Y
r a n d o m variable
with E W = 0
in distribution'.
K~ y~ 8 with y > 6 > 0 and
S such that i
~S
as
t +~.
from Dobrushin
(1955)
t8 Then it follows
N(t)
-
Kt Y
that
S + /~KW if
V =
S
y < 26
26
d
as
t§
t 6
where
S and W are independent.
that the specific
if
It follows
form of the norming
constants
portant~
and we are led to the
(1972:1~
p 293 and ]972:2 pp 3]2-3]3).
Theorem
]
Suppose
that there
~t lim 8 t = ~ and lim ~ t-~o t§ 8t such that
A(t)
- ~t
= K, 0 _< K < ~
d ~S
Bt
following
exist nonnegative
as
t§
from Dobrushin~s
are not too im-
theorem
constants
proof
due to Serfozo
~t and Bt with
and a random
variable
S
7o
Then N(t) - ~t
d ~S+
~Was
t
§
oo
St where S and W are independent.
Proof We will give a proof slightly different from the one given by Serfozo.
Since N(t) : N(A(t)) one may suspect that N(t) behaves somewhat alike A(t) + N(st) - st. Put, in order to simplify the notations, No(t) = N(t) - t. Thus we have
N(t) - ~t
A(t) - ~t
~o(~t )
=
+
St
-
Bt
-
Bt
Due to the assumptions A(t) - ~t
d §
as
t §
St For the second term we have
N~
m ~ t~ Bt
If K > 0
then
=
N~
~-~
at + ~
~t _-~§ v/K'K as
~t
as
t
t § ~
-~
ce
~t and
~o(~t )
d
---~W
as
t §
and
No (A(t)) - ~o(~t ) +
Bt
7]
If K = 0 then No (mt ) mt Var (~-~j----)= -~ § 0
as
t §
~t Thus, if the last term is shown to tend to zero in probability
as
t § ~, the theorem is proved.
Since N and A are independent we have
d
No(A(t)) - N(mt)
No(I A(t) - mt I)
Bt
=~
Bt
IA(t) - mt[" INo(IA(t) - mtl) I Bt2
where = means
N iA(t) _ mt I
'equality in distribution'
preted as zero. From Chebyshev~s
and where o
gg
is inter-
inequality it follows that
(t) Pr {L ~
N
1 -
~
for
allt
> 0 a~d
(t)
{ o V~
all
~ > O,
-
; t ~ 0} is tight, and thus also {N~
IA(t) - mt I is tight. Since
i.e.
st
I), t > 0} ~IA(t) - ~tl --
d * IS1 as t § ~ it follows that
~t
i IA(t)
- ~tl
9 tends to zero in probability
as t § ~ and thus also
Bt No(IA(t ) - ~tl ) tends to zero in probability
as t § ~.
Bt
m
Consider now the case when E A2(t) < ~ for all t > O. Put M(t) = E A(t) and V(t) = Vat A(t). From lemma 1.3a it follows that E N(t) = M(t) and Vat N(t) = M(t) + V(t).
72
The
following
Grandell
(1971,
Corollary Suppose (i)
corollary
is a slight
reformulation
of results
due to
pp 207-213).
I
that
lim M(t) t-~
If k = ~
then,
N(t)
- M(t)
= ~
and
d ---+ W
as
M(t) l i m ~T-77~+ ~ =k,O 0.
B
Remark 2 If Pr (S(T) - S(T-) = 0) = I for all T ~ R+, then T S = R+ and the p r o o f of lemma I goes through if B is only assumed to be measurable
Lemma
(cf Feller
(1971, p 277)).
2
The function ~ : D x D and it is continuous given by r
o
+ D given by @(x,y) = x~y is measurable
for ( x , y ) 6 C
= x + y where
and it is continuous
x D . The function @ : D x D § D o
(x + y)(t) = x(t) + y(t) is m e a s u r a b l e
for (x,y)6 C x D.
Proof This lemma is a consequence of more general results given by Whitt (1972). A p r o o f will, however, be given. Consider first the function %. The m e a s u r a b i l i t y
follows from the p r o o f of lemma 1.2
slightly modified according to B i l l i n g s l e y observations xl,x2,... ~ D ,
about the Borel-algebras
(1968, p 232) and the
in remark
1. Let now
x 6 C and y,yl,Y2,... ~ D O be given. We will show that
if Xn + x and Yn § y than XnOY n + xoy. From the definitions
given
79
in s e c t i o n
A I it follows
that
Yn § y means
that
there
R~C
{Yn } =I' Yn 6 F '
such
that
ynOYn
exists
U
~ y
and Yn
~ e.
Since
U~C
x g C it follows tE
that
x
n
§ x means
that
x
) x. For
n
any
[0,~) we have
sup O<s 0 it follows from the elementary definition probabilities
that Q{ O'{]B~}
Q{Ba]0' }
=
Q{0' }
of conditional
89
where Q(BolO') is the conditional probability of B ~ @ B ( O )
We will now consider the case where X
given 0'.
is bounded. In theorem I it
O
will be shown that Q0' may be calculated as a limit of elementary de-fined conditional probabilities.
It will further be shown that
what in 'every day language' is meant with an observation of N on Xo, really is an observation in the sense of definition I.
Let X vo @ N
0
be bounded. Then v(X ) < ~ for all v ~ N . 0
the set O'(Vo) by O'(v o) = ( ~
For Vl,V 2 ~ N
Define for any
~ ; v(B) = Vo(B)for all B~B(Xo)).
the sets O'(v I) and O'(v 2) are either disjoint or equal
and further
~ O ' ( v ) = ~. Let d be a metric metrizing the topology vaN on X. Let (Bnl,...,Bnr) be a sequence of finer and finer partitions n
of X ~ (i.e. for each n and ~ = 1,...,r n the set Bnj. is a union of certain Bn+1,j, j = 1,...,rn+1, sets) such that B n j 6 B ( X o) and
lim n+~
max diam (Bnj) = 0. 1~_j~rn
Put 0 n ( v o) = ( ~ g 2
. ; V(Bnj)
=
Vo ( B nj.) for
I -~ j
~ r n ). --
0 !
For each v6 N thus O'(V)~n 0'. Define Qn (B~) : ~ § [~,I] by
Q ~B~t0~(~)~
if
~o~(~)
and
Q~O~(~)~ ~ 0
0
if
~0~(~)
and
Q~O~(~)~ = 0
QO'n (B2)(w) =
for every B ~ B ( ~ ) .
Theorem I For each ~ N
the set O'(v) is an observation,
Further for each B~E B(~) we have
i.e. O'(v)~O'.
O' lie Qn {B~} = QO'(B~} a.s. (Q). n-~oo
9o
Proof Consider any ~o 6 N. The set 0'(v o) is characterized by the vector (xl,nl,x2,n2,...,Xm,nm) where Xl,...,x m are the only points in X ~ with w ({x}) > 0 and where n. = v ({x.}}. Denote for each x 6 X by o j o J o Bn(X) the set among Bn],...,Bnr
which contains x. Then Bn(X) + {x}. n
Thus there exists n o such that for n > n o the sets Bn(Xl),...,Bn(Xm) are disjoint. Thus 0~(~ o) + O'(v o) for n > n o which implies that 0'(~o)6 0' since O~(Vo)6 0' for each n.
Let 0'n be the a-algebra generated by {wE 2 ; ~{Bnj} _< x}, x 6 R, j = 1,...,r n. Thus 0~(~)~ ~0'n and 0~g~ 0'n+1 " Define 0'~ to be the ~algebra generated by
[j 0'. If we can show that 0' = 0' then the n n=1 theorem follows from Doob (1953, pp 611-612).
Since 0 " ( 0 '
and since 0" is a c-algebra it is enough to show that
v{B)(~) : ~ § Z is 0"-measurable for each B E B(Xo). Put n
D = { D E B ( X o) ; ~{D}(~) is 0"-measurable}
[j B nj .and . Since X ~ = j=1
since v(.)(~) is a measure for each ~ it follows (cf the proof of lemma 1.1) thatP is a Dynkin system.
For any closed set F in X the set X o ~ F 6 lim n§
[~ X6Xo~F
Bn(X) = X o N F .
D since
Since B(X o) is generated by (Xo~ F ;
F closed in X) and since for closed sets F I and F 2 also FI~]F 2 is closed and thus X o ~ F I ~ F 2 ~
~ it follows (of Bauer 1968, pp 17-18) that
p = B(Xo).
m Consider now the case where N and ~ are conditionally independent given A. To motivate a special study of this case, we just note that this is the case if ~ = A{B} for some B 6 B(X). In order to make our formulae somewhat more handsome, we denote the marginal distribution
of (A,~) by V. Thus V is a probability measure on (M • R, ~(M x R))
91
defined by V{B M x BE} = Q{N x BM x BR} . From the conditional independence we get Q(B N x BM x BR) = f
QB'(M)(B~) QB'(M)(B~} dQ =
= / n~{BN}QB'(M){B~}Q { N
• d~ • S} =
BM
BM
since
BM
i~ (W{Bn(Xi)} )niI e -w{X~ =I
Since H {O'(v)} = const. n
,
where Bn(X) and the vector (x],n 1,...,xm,n m) characterizing 0'(v) are defined in the proof of theorem I and where the constant depends on n and v but not on ~, it follows from theorem I that a.s. (Q)
(B(Bn(Xi)))niI e
~.!
Q~
=
lim
-~[x o )
V{d~ x BR }
I
n'+~
( ~ {Bn (x i ) } )
o
e
ff[d~)
1
for m~ O'(v) characterized by (xl,n I .... ,Xm,nm). Specializing further we consider X = R and X ~
(0,
and, see sec-
tion 1.3.3, the case where the model for the intensity is a stochastic process {l(x) ; x s
with distribution H on
(Io,B(Io))where
Io is a set of nonnegative Riemann integrable functions with finite integral over bounded sets. Let the space (~,B(~),Q) be modified in the obvious way. Then a.s. (Q)
92
t
m
QO'(BI)(~)
=
-S n(y)dy
I
( H n(y)dy) e o i=I B n x i)
Zim
o
V{dn x BR}
t
n-~co
-f ~(y)dy
m
/(~
n(y)ay)
I
e
0
H{dn}
I 0 i=l B n x i)
for ~O'(v)
by ( X l , 1 , . . . , X m , 1 ) .
characterized
(Multiple points do not occur.)
(t(j 2n
tj 2 Z]
=
n6 I
~(x) is continuous
O
such t h a t
2n
Jim n +~
l-
-
1)
Choose e.g. Bnj
f
Then for each x.E m (O,t] and each a t x = x. we h a v e 1
n(y)dy = n(x i) .
gn(X i )
Thus a.s. (Q), since a Riemann integrable
function is a.e. continuous,
t
-f
m
H (Bnlx i )
n(y)~y
n(y)dy) e 0
i=I lim
t_ (2 n)
n-~m
m
t = ( n n(xi))
e 0
i=I If e.g. t
-fn(yl~y supfl n
I 0
m 2n 0 H ((t--)B~n n(y)dy) e i=I (x i )
I+~ I
H{dn} < ~ a.s.(Q)
93
for some ~ > 0 it follows by uniform integrability
(cf Billingsley
(1968, p 32)) that
t
- f ~(y)dy
m
f ( H n(xi)) I Q0'{B{)(~ ) =
e
0
V{dn x BR}
i=I o
t
- f ~(y)dy
m
f ( ~ n(xi)) e I
a.s. (Q) .
0
~{an}
i=l
o
Remark I Two 'extreme'
cases where the condition of uniform integrability
holds are when
sup f O<xO
0
if
x 0.
Then
Q0'
{~; z < x} = f -
and thus for L(x,y) = (x
~+t
_
y
)2
e
-(~+t)y
the best estimate is
dy a.s. (q)
96
Instead of this criterion we may consider L(x,y) = Ix - YI" Then the best estimate In t ~ s
case ~
~
is a median of the conditional
can not be analytically
given.
It may be regarded as natural to choose ~ tional distribution.
distribution.
as the mode of the condi-
Since the density of that distribution is propor-
tional to
ym+B-1
e-(a+t)y
for y > 0 we get
~
= max(O,
B - I + m)
~§
'
This last estimate is not a best estimate in the sense of definition 3.
To compare these estimates we have for a = B = I and t = 5 computed ~x for m = 0,1,2,...,10.
In figure 2 these estimate are drawn. Though
the estimates only have m e a n i n g for m = 0,1,... m varies continuously.
they are drawn as if
97
L(x,y) mode
= (x - y)2
L(x,y)
Ix
yl
1.5
0.5
0
i
|
I
1
2
3
m
9
,
9
9
9
~
5
6
7
8
9
9
i
m
lO
Figure 2: lllustration of estimates in a P61ya process
Example 3 In this example some results derived by Rudemo, specialized to doubly stochastic Poisson processes, will be surveyed.
Consider the case indicated in section 2.3.1 where (~(t) ; t ~ 0} is a Markov chain with stationary transition probabilities and with distribution H on (Io,B(Io)). Here I
is the set of rightcontinuous O
piecewise constant functions R+ § R+ = ~,~) with only a finite number of jumps in every finite time interval and with range (~k ; k = 1,2,...K) where K may be finite or infinite. Put Hki(t) = Pr(~(s + t) = ~iI~(s) = ~k ) qki
ki (0)
(right-hand derivative)
wk(t) = Pr(~(t) = ~k ).
98
Consider X ~ = D , t ]
9
Let 0' be the c-algebra of observations t
and put
0 !
w~(slt) = Pr t { k ( s )
= kk }
and K
~(slt
) =
where thus ~ ( s l t )
Z ~k Wk( s l t ) k=1 is the best estimate of X(s) in terms of N on
~0,t] according to L(x,y) = (x - y)2. To simplify wk(t) = w k ( t l t ) a n d
~x(t)=
notations put
~x(tlt ).
This example turns out to be a special case of a partially observed Markov chain, since the vector process
(N(t),~(t)) is a Markov chain.
Consider first the case K < ~ , treated by Rudemo (1972). Rudemo (1972, p 323) shows that in intervals between events
K
Wk' (t) =
Z i=I
w~(t) qik + (k~(t)
- kk ) w~(t) a.s.
while if an event occurs at t
wk(t) = ~ k ( t -
O) ~(t
~k - O)
a.s.
Consider now the general case where K is not assumed to be finite.
Following Rudemo (1973:1) and (1973:2) we define for t > 0
Hki(t) = Pr{k(s + t) = k.m, N(s + t) - N(s) : OI~(s) = kk}.
These probabilities may be obtained from
Hki(t)
= ~. Hkj(t) qJi - ~i Hki (t) J
99
and I ~
i~
~ = i
Hki(O) = 6ki = I 0
if
k # i
Let H(t) be the matrix with elements Hki(t) and D the diagonal matrix with elements 6kik i and let w(t) and w~(t) be the row vectors with components wk(t) and w~(t) respectively.
Then
wa(t) = ~(0) H(t0) D H(t I - t0)D...D H(t - tv(t)_l ~ S
a.s. (Q) where v(t) = v { ~ , t ] } ,
tk = v
-I
(k) (cf section 1.3.4) and
S is the normalizing operator on row vectors, defined by
Pk
(PS)k = Z pj
for p satisfying p~ ~ 0 and 0 < Z. P~o < ~" From this it
9
j
J can be shown, see Rudemo (1973:2, 271), that Ik a.s.
wk(t) : wk(t - 0) ~(t
- 0)
at events, like in the case K < ~, while t
0 we have w (sit) = ~x(t) P(t,s) a.s.
As mentioned above the vector-matrix product representations be suited for computer calculations.
Rudemo
may not
(1975) shows that for
s > t s
=
+ I z t
(urt)
qik du
i
and for all s < t (i.e. also at events)
t
9{
~(slt) = ~(t) + f z (~(ult)qki ~k (u) s i
~. (u) l
x
- ~k(~It)
qik w~(u)
) du .
The above formulae seem suitable when t is fixed and s is varying. Rudemo
(1975) also gives recursive equations
for fixed s and varying
t and for both t and s varying but t-s constant.
101
With minor changes all the results given in this example hold true also when the intensity is a function of a Markov chain with stationary transition probabilities.
Problems
of this kind have also been studied by Snyder
(1972:1) and
(1972:2) when the intensity is a function of a vector M a r k o v process.
We will indicate how some of Rudemo's results may be proved. Assume that K < ~ since then all regularity assumptions are fulfilled. Define the random variables Z(t) and ~k(t) by
t Z(t) =
v(t)-1 H l(tk)) e k=O
S ~(u)d~ 0
and I
if
l(t) = kk
0
if
k(t) # kk
~k(t) =
From our general results it follows that
=~ (slt) = E ~kCS) Z(t) E zCt) and
=
E Z(t)
Consider s = t.
Assume that an event occurs at t, i.e. that ~(t) - ~(t - O) = I.
Then
~[t)
= E ~k(t) Z(t)
E ~k(t) k(t) Z(t - O)
E Z(t)
E k(t) Z(t - O)
k
~k E ~k(t) Z(t - O)
~k ~k(t - O)
=
E l(t).Z(t - O)
~(t
- O)
since k(t) = k(t - O) a . s . .
Assume that no event occurs at t.
Put wk(t) = E Ck(t) Z(t) and thus ~ ( t )
= (~(t)S) k. Note that
~k(t) = wk(t) E{Z(t)IX(t) = kk }.
For A > 0 such that . . . . . . t . . . . ur in the interval ~,t§
we have
102
t+A f x(u)du
t
- ~k(t)) Z(t)} =
- Ck(t))IA(t)
: ki}E(Z(~)Ik(t)
- ~k(t))IX(t)
= A i} ~i (t) .
wk(t+A) - wk(t ) = E{(~k(t+A)
e
t+A
S x(u)a~
-
= Z E{(~k(t+A) i
t
e
: Xi)~i(t)
t+A
S ~(u)au
-
= Z E{(~k(t+A) i
t
e
If i # k we have t+A
S
-
E {(s
+ ~) e
X(uldu
t
_ ~k(t))Ik(t ) = ki } =
t+A -
S
= E {e
x(u)du
t
IA(t + A) = kk' l(t) = k i} ~ik(A) =
= (I + O(A))(qikA
+ o(A)) = qik A + o(A)
and if i = k we have t+A -
E {(~k(t + A) e
S x(u)au $
- ~k(t)lil(t ) = ik } :
t+A = E
-
{(e
S xCu)du t
-
1)[x(t + a) = ~k' ~(~) = Xk } nkk(~) -
(I - nkk(n)) = - xka(1 + qkk A + o(A)) + qkk~ + o(~) =
= qkk A - ikA + o(A) 9
Thus we have
~k(t + A) - ~k(t) = A Z ~i(t ) - AlkWk(t ) + o(A) i qik and since a similar reasoning
goes through for A < 0 we have
~'wk(t) = iZ #i(t)qik - Ik#k(t)
.
:
103
Consider s > t. For A > 0 we have
E (~k(S + A) - ~k(s))Z(t) E Z(t)
= Z E {(~k(S + A) - ~k(S))lk(s) = ki} ,~(slt ) = i
iWk
= i~k (qikA + o(A)) ~ ( s l t )
= ~ i~ ~ ( s l t )
+ (qkk A + o(A)) ~ ( s l t )
=
qik + o(A) .
Since a similar reasoning goes through for A < 0 we have
@'~(slt) i
Consider s < t. For A > 0 such that no events OCCUr in the interval (s, s + A) we have, whether ~n event occurs at s or not,
"a(Sk + Aft) - ~ ( s l t ) k '
= E {(~k(S + A) - ~k(S)) Z(t)} E Z(t)
E {Z(s)(~k(S + A) - ~k(S))
Z(s + ~) Z(s)
z(t) } Z(s + A) =
E z(t)
= .E. (~i(s)~ij(A) E (Z(s)IA(s) = k i) "
E Z(t)
Since I + O(A) if i#k,j=k
E{(~(s+~)
- ~k(s)) ~IX(s)
= X i, ~(s*~) = Xj} =
-I + O(A) if i=k,j#k 0
otherwise
~o4
Ni~(4 ) = qij4 + o(4)
if
i # j
and
~i(s)E{Z<s)11(s) = ~i } S { ~ 0.
for example,
that
Pr{Jl(t
lim
f
-
)]
+ 4)
- t(t)
I L s} = 0
4§
for all E > 0 and almost
E(12(t))
for almost
< C
O. We will
sufficient. bility
all t > 0 and that
show that these
From the discussion
in the sense of Doob,
to use as a model
in section
it follows
u+4
1
~
1.3.3 about
that
{l(t)}
are
integra-
is possible
for the intensity.
Further
lim k+O
assumptlons
(e
U
- f ~(v)dv 0
- f l(v)~v 0 --
e
}
U
- f ~(v)dv e 0
= - t(u)
for all realizations
Since surely
for almost continuous
of derivation however,
are continuous
all u > 0 our assumptions in t = u it only remains
and integration
by u n i f o r m
p 32)) since
of l(t) which
imply that l(t)
(cf e.g.
is almost
to verify that the order
may be interchanged.
integrability
in t = u.
This
Billingsley
follows, (1968,
107
u+A
f
-
sup ~I~ (e A>-u
u
l(v)dv
o
-
-
e
f
)1
u+A
-u
A>-u
f
f
u
u
~,X(v)~(w)dvdw -~
Thus if lim ~{X n} = ~ and X ~ ~ X k then X n -~ k=1 does not have an i n t e g r a l r e p r e s e n t a t i o n .
Consider
now X = R, X ~ =
E0,t~
and ~ Lebesgue
m@L(Xo)
but
measure.
Then
~
m
8 + o2st { [o,t~ }
~=m
m+~t
2
or with m = ~/a and o 2 = B/a 2
~ + t
which,
as shown
L(x,y)
= (x - y)2 in the case of a P~lya process.
p 99) has the best implies is
this
shown, estimate
that
'better'
managed
in example
2, is the best estimate
see section coincide
for any other
2.1, that the best
distribution
than the best
is the worst
the same mean value
Lundberg
linear
to
(1940,
estimate
if and only if ~ is r-distributed.
linear
to give any plausible
sense
according
This
of ~ the best estimate
estimate.
explanation
distribution
and
We have,
however,
of not
why the r-distribution
among all distributions
in
with
and variance.
m Suppose
now that N is observed
be the random variable integral wellknown
equation
which
from e.g.
Assume
set X I and let ~(E~ = 0)
is to be estimated.
given in theorem
In most
4 is difficult
linear p r e d i c t i o n
set X o ~ X 1 is considered solution.
on a b o u n d e d
theory
it m a y be much
that R is absolutely
that
cases the
to solve.
It is
if some u n b o u n d e d
easier to get an explicit
dominated
by M on X ~ and define
131
gl
=.L (glL(xl))
go = ~ (~[L(Xo))
/
=
X o
f(x)N {dx} o
and g~
f(X)No{aX}.
= S
appr.
XI
gappr. ~ is a reasonable approximation of the best linear estimate gl
if
E (g:ppr. ~)2 E (g~ - g)2 is close to one. This quantity is mostly difficult to calculate, but since
E (gappr. -
~)2
.
E
E (g] - ~)2
(gappr.
E (~
E ( ~
~)2
-
_ ~)2.
~)2
~_
~appr. - go
+ E (go
E (g: - g)2 g:)2
E(g:ppr.= 1 +
(go
E
_
g)2
(1 . e)
S
< --
f2(x) M{~}
Xo~X I
0. Put ~ = ~(0) - I.
dx, X{B 1,~2 } =
B I
I
~dy
and
dx. Further we have
I
S 5 e-~Ix-Yl --~o
e -~lx-yl
I
p{B} = S e-~Ixl B co
S B 1•
dxdy = (2/~)M{B} and thus R is absolutely dominated by
B
M on the real line.
Assume that N is observed on Is,t]. If both -s and t are large, it seems natural to consider X ~ = R. Then ~o~ is determined by
f(x) +
f f(y) e -~Ix-yl
which has the solution
f<x)
and thus
=
dy = e -~Ixl
for x ~ R
133
E(I~(O)
_ t(0))2
=
c~
~/c~2 +
2~
where 1{t0)" " = 1.18 if a = I and for t > 0.101 if ~ = 10.
If only -s is large it seems natural to consider X ~ = (-~,t]. In 9{
this case 0
=
(+~~ - ~ 2 ~ -
~)e~t+(~
)(x-t)
ift
< 0
Let the function E{I~(0) - l(0)}2(t) denote the value of E(/~(0) - I(0)) 2 when N is observed on (-~,t~.
Instead of giving the rather complicated formula we have in figure 3 illustrated this function for some values of ~.
134
E
{A~(O)
-
A(O)} 2
(t)
1.0 '
~=i0
9
a = l
9
a=O.l
o.5
J ......
-I0
9
a
=
|
~
t
0.01
i0
Figure 3: Illustration of E{l~(O) - l(O)} 2 (t)
Consider t = 0, i.e. N is observed on Es,~
and X O
E{la(O)
- ~(0)} 2 = ~
--
Then
-
and
E( ~ p p r .
o
= e-2
27J
Ist
E(~o~_ g)2
2a
I
which is less than 0.01 for Is] > 22.9 if a = 0.01, for Isl > 5.66 if a = 0.1, for Is I > 1.04 if ~ = I and for isr > o.o713 if a = 1o.
For notational reasons we change the situation and consider the case when N is observed on ~ , T ] , estimated for t 6 E0,T~.
where T is large, and X(t) is to be
Since i(x) is stationary there is no real
change in the situation.
In this case we have ~ ( t l T ) = E(~(t)li~0,T]) where ft(x)__ is determined by
T = 1 + S ft(x)No{dX} 0
135
T ft(x) + ~ ft(y) e -~It-yl 0
~y = e -~It'x]
for x E EO,T]
which has the solution (ef van Trees (1968, pp 321-322))
ft(x ) : ~ {e-Slt-xl
A
+
[e-B(t+x)
+ e-B(2T-t-x)
+
1 - A2 e - 2 B T
+ A e-B(2T+t-x) + A e-B(2T-t+x)]}
where
B =
~2
+ 2a
and
A = ~ + 1 -
B.
For large T it seems reasonable to approximate this rather complicated T estimate with l~ (tiT) = I + f gt(x) N (dx} where appr. 0 o
gt(x)
=
~ {e -BIt-xl
+
A Ee - % ( t + x )
+ e-g(2T-t-xO
)
.
We have
E (X ~
appr.
(tiT)
- ~(tlT))
2
E (Xappr.(tlT)
- X~(tlT)) 2
5.7 if m = 0.I and for all T if ~ = I or 10.
Consider now l~(t) = E (l(t)Ii([O,t]). Then we have
t = I + f f(x) No{dX} 0
~(t)
where
e-~(t-x) + A e -(t+x) f(x)
=
(6
-
~) I - A 2 e-2~t
"
136
From this and the previous discussion in this example it follows t that I + f g(x) No(dX} with 0
g(x) = (B - e) e -B(t-x)
is a reasonable
approximation
of l~(t) p r o v i d e d t is large.
Consider as a further illustration
some random generations, of a
model within the class studied in section 2.3.2, which may be looked upon as continuous parameter analogues to the generations
described
in section 2.5.
Put in these generations T = 50 and ~(x) = I (N(x)
where
; x 6 [0~50]} is a Poisson process with parameter ~ and inde-
pendent of a sequence distribution
generations presented
{lk}k= 0 of independent
function U(x) = I - e
-X
random variables with
, x > O. In figures 4-6 these
together with some linear estimates of l(t) are
for e = 0.01, ~ = 0.1 and a = I. In the case ~ = 10 the
illustration value turned out to be very low, and this case is therefore omitted.
For e = I the curves representing E ( l ( t ) I L ( E O , t ] ) ) a n d
its approximation
coincide within the accuracy of the diagram.
137
I
% 5O
25 (a) The piecewise constant curve represents A(t). The continuous curve represents the approximation of E(
z(t) l[([0,50])).
5O
25 (b) The piecewise constant curve represents X(t). The picewise conti . . . . . . . . . . . . p ..... ts E(X(t)
i [([0,t])).
50
25 (C) The piecewise constant curve represents X(t). The p~ecewise conti ....... urve rep ..... ts the approximation of E(l(t)
IIIII II
i lJ
0
17( [0,t'])).
IN I
I
I
L 5O
25 (d) The spike train represents the location of the points of N.
Fi~ulre
4~
Illustration
of
linear
estimation
in
the
case
~ =
0.01.
t
138
!
!
25
50
(a) The piecewise constant curve represents ~(t). The continuous
curve represents the approximatlon of E( k(t) l ~ ( [0,50] )).
!
f
5O
25 (b) ~ e piecewise constant curve represents l(t). The piecewise cont~ ..... e~..... presonts ~(x(t)lY([o,t])).
50
25
(C) The piecewise constant curve represents A(t). The piecewise continuous curve represents the spgroximaton of E( t(t)[ [([0 t])).
i; IIHIli IIill l; l;IIrIIlllil]II IJiili~HlIIli ;lilil; o
50
25
(d) The spike train represents the location of the points of N.
Figure
5: I l l u s t r a t i o n
of linear
estimation
in the case
a = 0.1
139
0
!
9
25
50
(a) The piecewise constant curve represents l(t). The continuous curve represents the approximation of E(~(t) I T([0,50])).
i
25
0
(b) § (c)
5O
The piaeewise c o n s t ~ t curve represents ~ ( t ) . The pieeewise co~ti~uo~s ~ r v e ~epre~e~t~ ~( ~(t) l [ ( [ O , t ] ) ) .
rlllll lit I I il I III,I I ll!lllII o
11, 50
25 (d) The spike train represents the location of the points of N.
Figure
6:
Illustration
of
linear
estimation
in the
case
a =
I.
140
Example
9
We will now consider
a simple
generalization
example
8. Put X = R and let A have
chastic
process
of the case studied
density
l(x) where
with E l(x) = m and Cov(l(x),l(y))
= l(0) - m. If m = 62 = I we have the case
Suppose
that N is observed
in
X(x) is a sto-
= 62 e -~Ix-yl . Put
studied
in example
8.
on X . Then O
~
= ]~(r
S(x)
= /
(N{dx}
- m
dx)
X 0
where
f is the solution
m f(x)
+
O
= (-~,t]
{e-Blxl+
for x ~ X O.
~+~
e-B'2t-x'}(~
if
t
> 0
if
t < 0
=
(B-s) e
and if X
dy = 62 e -alxl
0
we have 62~
f(x)
e- l -Yl
62 J X
If X
of
= (s,0J we have
~t+6 (x-t)
(cf van Trees
(1968,
pp 321-322))
O
e
f(x)
=
-BIll 1 -
where
~T
eB(l l-21sl)
(B-h 2 ~
e
-2BIsl
in both cases
B =
Assume
+
(~B-~)
2 262e c~ + ~ m
that N is observed
on a b o u n d e d
set X
and let ~(E ~ = 0) be O
the random variable
which
is to be estimated.
We will n o w consider
141
a different kind of approximation
of [~ = ^'E([IL(Xo)) which may be
useful.
Let, like in theorem
finer partitions
lira n+~
I, (Bnl,...,Bnr} n
be a sequence
of finer and
of X ~ such that Bnj ~ B(X o) and
max I <j '~ ~
and that
143
Assume that R is absolutely dominated by M on X 9 Then it follows o from theorem 4 that
+
f
X
s
(N{~x} - M{dx})
o
where ~(x) = (f1(x),...,fn(X))
/ ~(x)mdx} + / s B
X
is determined by
R{B,dy} = ~{B}
, B6B(Xo),
o
where ~{B) = (Cov(~I~N{B}) ..... Cov(~n,N{B}) ).
Further it follows almost immediately
from the proof of theorem 4 that
X
(~' denotes the
5.3
transpose
o
of ~ and not the derivative.)
Some empirical comparisons be~een non-linea~ and linear estimation
The very restricted purpose of this section is to consider some random generations
illustrating a case where it seems reasonable
to believe that non-linear estimates are much
'better' than linear
ones.
Put X = R+ and X ~ = [0,t] and consider the process described in I example 3 for the special case K = 2, w1(0 ) = w2(0 ) = ~ q
if
k#i
-q
if
k = i
and
qki =
This means that {1(x)
; x ~ R+) is a Markov chain with stationary
transition probabilities,
alternating between the values 11 and 12
144
in such a way that Hki(Y) = Pr{1(x+y) = till(x) = Ik } = qy + o(y) if k # i, and hence
~I ( 1 Hki(Y
e-2qy
)
if"
k#i
if
k = i
= 1
7
(1 + e -2qy)-
Thus I
m ~(x
=~
r(x,y
= Cov(l(x),l(y))
( l 1 + ~2 )
and =
I
~ (11 - t2 )2 e - 2 q l x - y l
O' In this section we will use the notations XL(t) for ~(X(t)l[EO,q)
IB(t)' for E t(1(t)) and
i.e. IB(t) is the best estimate of ~(t) in
terms of N on EO,t] according to L(x,y) = (x-y) 2 and IL(t) is the corresponding best linear estimate.
Consider first the case q = O. Then N is a weighted Poisson process and it follows from example 2 that
~(t)
N(t)+1 e-11t ~N(t)+1 e-12t = 11 + A2 .N(t) e-11t .N(t) e-12t AI + A2
and from example 7 that
~(t) =
(I I + 12)2
+ (~I
- 12)2 N(t)
2(I I + 12 ) + (i I - 12 )2 t
In figures 7 and 8 these estimates are illustrated by random generations for t 6 [O.50~ and (11,12) = (0.5, 1.5).
145
We note that if ~I = 0 then -~2 t Z2 e if
~(t)
if
N(t) > 0
=
o
-~2 t I +e X~(t) = ~2
and
~(t)
X2(1 + N ( t ) ) 2 + ~2 t
and further
E(X~(t) -
~)2
_
2 -Z2 t ~2 e
2(1 - e-x2t) and
s(~(t)
2 12 _ ~)2 = 4 + 212t
where ~ = l(t).
Thus for large values of ~2 t the best estimate l~(t) is much 'better' than l~(t).
Consider now the more interesting case q > 0. From the results of Rudemo, described in example 3, it follows that
z~(t) = ~i~i(t) ~ + ~2~2(t) where w~(t) , k = 1,2, is determined by
146
o) = 2
~(~-o) at epochs of events and
~1 ' ( ~ )
= (~(=)
- Xl - q) ~I (T) + q ~2 (~)
~ 72~' (T) = q ~I(T) + ( k~(t)
- ~2 - q) w2(T)
in intervals between events.
Using the linear equations
for ~k(T) it follows that if no events
occur in (Sl,S2~ then for 9 ~ s 2 - s I
~+ eBT[Tr~(Sl)(J3+~)+qw2(sl)]+e ~TI(S 1 T ) =
-ST
x ~, [ITI(Sl)(B-(~)--qTr2~Sl) ]
eB~ [~+q+~ i ~1< s~ >-~I ~ I>] +e -~ [B-q-~ ] mFei X 2 n
2"~ Lg(elX)~ n
dx .
]68
Specializing to k = n and n+1 we get
E , n+1
- ~n+1
= ro~ - -wi
=
r@
=
" ei(n+l)x g(e-ZX)
- go ei(n+1)x 2 dx =
fN(x) + g 20 - go (g(e ix ) + g( e-iX) )
-
dx
=
-IT
2 2w go
R, =
r0
-~
(~
-
2 pred
~Fm +
--
r0
+
2-
4w
go'[~ :
2 2w go
-
m
=
m
and
E (s (n)~ _ s
s =
ro
=ro
-
-
i -w
='
einX
/ If
" g(e-ZX)
N(X)
+
m
I
2w
-7-
-w r0
s -
einX 2 go
- -m
2w go
dx
[m
+
r0
m
(g(e ix) + g ( e - i X ) ~
2
2 m
+
2
2m]
=
2w go
m
2
2w go
2 m -
m
2 pred
Because of stationarity we have
E (s
I )~
Zn+1
)2
Consider now the two estimates
We have
~-
go
L :
m 27
(n)~ = E (Ln
s )2 n
~(n)~ z(n+1)~ n+1 and n+1
dx =
169
s n+1
w I ~i(n+1)x fs = m + ~ . -w g(e-lx) L g (eix)
1
= m +
f I. -w g(e -Ix)
m e zN{ax} = 2w g(e Ix) I n
= m +
f -~
I. g(e -Ix)
i(n+1)x g(e-lX)
[el(n+1)x g(e-lX)
= ~ + f e i(n+1 )x
n
zN{dx} =
_ ei(n+1)x
~g(e-ix) - goI zN{dx}
L(o-ix)
and z(n+1)~ w [ei(n+1) n+1 = m + f I. x fZ(x zN{dx} = -w g(e -Ix) g(e Ix) ]n+1
= m +
= m +
= m +
lei(n+1)x g(e-lX) " f 1. -~ g(e -Ix)
~
I
-~
g(e -Ix)
i(n+1)x
ei(n+1)xl m
zN{ dx] =
2~ go
Ig (e-ix ) _ m 2g0] ZN{dx) = fw e i(n+1)x . -w g(e -mx) ~pred ] i(n+1)x
= m + f -~
:m+
g(e-~X)
m_ e__~.+ i(n 1)x __I zN{dx} = 2~ g(e Ix) ]n+1
(I
m
2 pred
e
g(e -Ix)
1
g(e-lX_ go 2m ) g(e-lX) + ~m ~pred ~pred
m ) - m) + m , (n)~ m) = 2 (Nn+1 7 - - - ~n+1 ~pred ~pred ~(n)~ + (I n+1
m ) 2 Nn+1 ~pred
zN{dx)=
170
We have
= s Un+1
s n+1
~ = -~
n+1
-e-(n+1)x ----:--. - ig g(e-lX) 0
m ~ 2w g
zN{dx}
and thus
EU 2 n*l
=
~
m = 2"rr
2 pred
+
0 -
2 m 2 epred
~
-
2
2Tr g'
2m
= 2~T
2+( 0
)
_
=
0
.
In Grandell (1972:1, pp 548-552) similar results were derived by use of Toeplitz forms. In these derivations the spectral distribution F Z was not assumed to be absolutely continuous, but since theorems A3.I and A3.2 can be generalized to not necessarily absolutely continuous spectral distributions, the results in Grandell (1972:1) can be derived by the method used in this example.
Let us go back to the application to insurance models.
The simple formula
z(n+1)~ _ n+1
m 2 pred
z(n)~ m ) n+1 + (I - - 7 - - Nn+ I pred
seems attractive, since it means that the policy holder has a possibility to understand how the number of claims year n+1, i.e. Nn+1, affect the bonus year n+1.
Example
3
Consider the case m = I and r~ = g
plJl, IpI< I , and
e x a m p l e may be r e g a r d e d as t h e d i s c r e t e
parameter
~ = Zk - m. This correspondence to
171
the
case
following
studied
in e x a m p l e
Hannan
(1970,
5.8.
pp
We w i l l
give
a direct
derivation
171-172).
We have
f~(x)
2 1 - p 2 1 + p - 2p cos x
= 1 27
and
fN(x ) = 1
2(1
27
In order
to derive
tion
g. We w i l l
g(z)
is a n a l y t i c
-
p cos x )
1 + p2 _ 2p
the
estimates
use the
facts
and without
cos
x
Zk we have
that
to c a l c u l a t e
g(eiX) 9 g(e -ix)
zeros
in
the
= fN(x)
and that
Izl < I.
Since
fN(x ) = I_ 27
.. 2 -
p e
2
l+p
ix
-p
- p e ix e -p
--ix e
-ix
we c o n s i d e r
I
I
2w
1
=_I_
-
+
p
p_
2~
b
pz
2
-
pz
pz -
-I
I
pz
-1
(z-h)(bz-
2~
I)=
(z - p ) ( p z
- I)
where
b =
1-
~/1-p
2"
P I
Thus I - bz
g(z) =
I - pz
+
(z - b ) ( z
(~
p 2wb
p)(z
-
I - bz I
-
pz
- b -I) = p-1 ) -
I - bz I
-
pz
func-
-I -I
172
Since
Ibl n. Consider first k < n. Since n - k > 0 we have
h(x)
=
#1-p 2 I+-~7
e
ikx(1_pe-iX) l-be -ix
eikX(1_pe -ix)
1+~-p
l-be
--iX
(1-bp)e ikx -
be
n-k 9 .. I {( Z bJe IJx) + "} = j=1 1-pe -ix
ixn-k ei(n-k)x) I1-b 1_be ix
1_pe-iX)b n-k+1 ei(n+1)x
(1-beZX)(1-be -zx)
+
-
I
-
1_oe -ix
~n-k
"
174
oo
= ,, ~
{(1_bp)eikX
I z b Ij leijX 1-b 2 j=-.
_ (1_pe-iX)bn-k+lei(n+1)x)
Thus
hj = --~{(~ bp) blkJl
bn-k+1
bln+l-Jl + phi-k+1 bln-Jl}
or
{blk-Jt
2 h.
+ b2n-k-J +2}
if
j n
=
J 0
and thus
X /- 1p2
E(Z k - s ) 2 -
Consider
2
{1 + b 2(n-k+1)}
.
k > n. Then
{I h(x)
1 +
eikx(
-ix 1 - pe ) -ix I - b e
- p2
=
2 1 - p ~ _ p
P
k-n
I -be
e
1 - p e -i
inx -ix
Thus
~1-p h,
J
2
pk-n
bn-D
if
j n
~-
0
n-k
175
Further
1T
E(,~-
.~k)2 = 1 -
]" Ih(x)l 2 f"(x)
d~ =
--'IT
=
1 -
(1
-
s
p2(k-n)
~
2
1 -
=
dx
-~
I1 - p e
ix
I
2
p2(k-n) 1 +
ql
-
p2
Put k = 0 and consider E(s
"
- ~)2
as a function
of n. To summarize
we have
21nl I
P
ifn 0
of n.
of 0. In
176
1.0
0.5
0,25
0.5O
0.75
Figure 17: Illustration of E(L~ - ~0 )2
Consider now Zn = {1,2,...,n). Then a natural approximation of Lk' when ~ = Zk - I for k ~ Z n ,
~
2
p2 ~Z
n
=~
I~
{i~p2
is
n b2n_k_j+2) Z (b Ik-jl + (Nj-I) j=l
if k equal to or near n
n j=1
h Ik-jl
n (blk-J
Z j=1
(Nj-
I +
I)
b~+j ) (Nj-I
if k not near n or I
)
if k equal to or near I
As illustration we have applied this approximation to the random generation G7 described in section 2.5. Thus we use 1 + nZ5 0 as approximation of Lk. In figure 18 which is equal to figure I with the approximations of Zk added, this is illustrated. Figure 18 may be compared with figures 4(a), 5(a) and 6(a).
177
50
25 In e&ch point k the height of the spike represents N k , the value of the plecewise constant curve represents ~k smd the v~lue of the continuous curve represents the approximation of Z k.
Fisure
18: l l l u s t r a t i o n
A natural
question
of e s t i m a t i o n
on g e n e r a t i o n
is a r e a s o n a b l e
is now if qZ
G7.
approximation
M
of ~Z n
n in the sense that E(q Z
_ ~)2 ~
E(~
n this we have c o n s i d e r e d means
integer part.
_ ~)2
To get some idea of
n ~ = ~ n+1 - I and ~ = ~ - I, where En~] n
In table
2 and 3 we have
calculated
E
E(q Z
] _ ~)2
n and E ( ~
_ ~)2 for some values
of n and some values
of ~. The tables
n illustrate
both h o w
convergence
'good'
of E(q Z
the a p p r o x i m a t i o n s
_ ~)2 and E ( ~ n
where
~ = Z
are and the rate of the
_ ~)2 to their
limits.
In table
2,
n
- I is c o n s i d e r e d ,
the a p p r o x i m a t i o n
n
n
~I-P2 : ----Z--
~Z
(I + b 2)
n
is used. In table
z j=1
3~where
Ln ] -
~ = ~ n+1
b n-J (~
- m) J
I is c o n s i d e r e d ~ t h e
approximation
178
1_~,2
nzn is
=
2
n+1 l IL~]-Dl
n Z
j=1
b
(N'-S -
m)
used.
~ =s
p=o.25
n
1
0=0.5o
p=o.75
E(~Zn-~)2
E(nZn-~)2
E(~Zn-~)2
E(nZn-~)2
E(~Zn-$)2
E(nzn-$)2
i
0.50000
O.5OO13
o.50o00
o.5o258
0.50000
0.52076
2
I0.492o6
0.49207
0.46667
0.46686
0.41818
0.42311
3
0.49194
0.49194
0.46429
0.46430
0.40217
0.40321
4
0.49193
0.49193
0.46411
0.46412
0.39894
0.39915
0.46410
0.46410
0.39828
0.39832
I
0.39815
0.39816
7
0.39812
0.39812
8
0.39811
0.39811
0.39811
0.39811
5 6
|
0.49193
0.49193
o.4641o
Table 2: Illustration of the rate of convergence.
o.4641o
'goodness'
of approximations
and the
179
~ =s
~=o~5
n
1
-i
~:o5o
~~
E(~n-g)2
E(RZn-~)2
E(~n-~)2
E(.Zn-~)2
E(~n-~)2
E(~Zn-g)2
0.50000
0.50050
O.5O0O0
O.50898
0.50000
0.55731
2
0.49206
0.49221
0.46667
0.47011
0.41818
0.45201
3
10.48438
0.48438
0.43750
0.43798
0.35938
0.37178
4
0.48425
0.48425
0.43541
O.43561
0.34749
0.35371
5
0.48413
0.48413
0.43333
0.43336
0.33636
0.33850
6
0.48412
0.48413
0.43318
0.43320
0.33410
0.33521
0.48412
0.43304
0.43304
0.33186
0.33224
8
0.43303
0.43303
0.33141
0.33161
9
0.43301
0.43301
0.33095
0.33102
i0
0.33086
0.33090
25
O.33072
O.33072
O. 33072
O. 33072
7
0.&8412
Table
3:
0.48412
lllustration
0.43301
of the
0.43301
'goodness ' of approximations
and the
rate o f c o n v e r g e n c e .
Now we will give
any help.
First the A3.2
consider
we
case
some
For the
consider
cases
rest
linear
estimation
compute
the
case w h e r e
the
not
absolutely
continuous.
l.i.m. n-~o
of this
~k = m + ~ + Sk w h e r e
in o r d e r t o
where
~Z
spectral
theorems
section
We will then which
of the get
~Z
- A3.3 Z
n
i.e.
Formally
the theorem
distribution
-- Z N. = m + ~, a r e s u l t n j=1 J
we p u t
of the level,
~ = Z{{0}}.
' since
A3.1
we
can b e
= {1,...,n}.
we
consider
can u s e t h e o r e m generalized
observed = ~ since
is o f no help.
do not
process
to is
18o
We use the n o t a t i o n
Rs = n
{r[
~-0
.}
=
{Cov(si,E
and further we denote by I vector
(I,...,I)
n
J
)}
the n•
with n components.
1 < i --'
'
j
< n --
identity matrix
'
and by ~
the
It shall be r e m e m b e r e d that
and s k are u n c o r r e l a t e d
for all k.
The f o l l o w i n g t e c h n i c a l
lemma will be used in order to i n v e s t i g a t e
the asymptotic
properties
of ~Z
" n
Lemma
I
For all n > I we have
1 =
(mI
1 --I1
+ o 2 1' n
-I1
1
+ RE ) - I
~
2'1
0
2
1 +
1'
1
-"n
-"rl
(ml
+ RS) -I I' n
n
--n
Proof
In Grandell given 9
(1972, pp
103-104)
a probabilistic
The p r o o f to be given here was
Put R = ml n + R ne
and -a = oI -n'
p r o o f of this
lemma is
s u g g e s t e d by B. yon Bahr.
Thus we shall prove that
(~ (R + a'_a)-I _a')-1 = I + (a_ R -I _a')-1
Let B be a symmetric positive put ~ = ~ B -I
definite m a t r i x
Let C be an o r t o n o r m a l m a t r i x
such that B 2 = R and
I~I
such that b C =
i
I n
where Ikl = (~ _~ b2) 2
and ! = (1,0 ..... 0). Then we have
a = bB =
I and thus
(~ (R + a'~) -I S ) -I =
= 0, then
2 m n
m 2 n
I
Fs
- Fs
-) + o(1)
Thus
lim n E (s~ - s )2 = m
and since
-n n E (N
~n )2 =
m,
n
where N
"~ Z Nj, it follows, with a slight m o d i f i c a t i o n n n j=l
definition
1, t h a t
{N } i s n
asymptotically
More interesting is the case Fs Fs
is absolutely
continuous
that some version fs
of
efficient.
- Fs
-) = 0. We assume that
in a n e i g h b o u r h o o d
dFs dx
of
is continuous
of x = 0 and in a n e i g h b o u r h o o d
of x = 0. Then it follows from t h e o r e m A3.4 that
E
(#~ - s )2
m
m
n
2 2
I
m + 2w fs
+ o(I)
Thus
2 lim n E (~n ~ _ ~ )2 = m n n+~
Obviously however,
m
. . =
m + 2w fs
{N } is in this case not asymptotically n consider
aN
n
+ b.
T h e n we h a v e
2w mfs m + 2w fs
efficient.
Let
US
187
-
2
n E "--,(aNn + b - ,%n ).
=
= n
(1
E
~a(N n
-
~n ) -
= n ~a._~ + ( 1 - a )
2
-
a)(~ n
- m) + b -
2w f ' % ( O ) +
o(1)+
(1 -
(b-
a)m] 2 =
(1-a)m)21
=
n = a2m + (1 - a) 2 2w f ~ ( O )
+ o(1)
+ n(b
-
(1 - a)m) 2 .
Thus we must have b = (I - a)m. To get the asymptotically best choice of a we minimize a2m + (I - a) 2 2w fZ(0) and thus we get
a ---
2w f Z ( O ) m + 2~
f~(0)
and m
b =
2
m + 2~ f~(0)
For this choice o2 a and b we have
lim n§
n E (aN
+ b - ~ )2 = n
n
2w mfs m + 2w fZ (0)
and thus, p r o v i d e d fZ(0) > 0, it follows that
m2 + 2w f~(O)
n
m + 27 f'%(O)
is asymptotically
Example
efficient.
5 (Continuation of example 3)
We have m = I and 2w fZ(0) =
1-p (1
-
2 p)2
= l+p 1 -
p
188
Thus
lim n E (s ~ - s )2 = n-~co
I + P 2
and
1 - p + (1 + p) N n ]
I
2
ia asymptotically
In figure 1 -
efficient.
22 we have p +
(1
+ p)
n E (
N
n E ( ~~ - ~n )2
for some values
0.50 and 0.75 respectively
1 -
p +
(1
'good'
rate of convergence
and
2
n _ ~ ) n
2 p = 0.25,
drawn
+ p)
N
n
of n and for
in order to illustrate
is as an a p p r o x i m a t i o n
of the drawn quantities
of ~
n
how
and the
to their limits.
189
0.9 --
0 ~ 0 . 7 5
--
p=0.50
__
p:0.25
0.8
=
0.7
-0.6
0.5
I
I
I
i
2
3
I
J4
Figure 22: lllustration
I
I
I
I
I
5
6
?
8
9
of the
'goodness'
rate of convergence.
I
!
io
of approximations
1-p+(I+p)Nn n E (
and the
)2 ~
and n
2 -
I
For each value of p the curves
represent from above
nE(~ - ~n)
.
25
2
Consider now the r ~ n d o m generations
GI-G7 described in section 2.6.
In table 4 we give the values of ~
taken from table
n
mative estimates
I
~ 1 +p p + 2(I + p) Nn -
approximation
of
I, the approxi-
p + (1
E ( 2
+ o)
and N
2n 2
n _ ~ ) n
w h i c h is an
190
I-0+( 1+p )N n
Name of n
p
generation
n
GI
500
0.0
0.993
0.991
0.032
G2
500
0.0
1.025
I .026
0.032
G3
500
0.0
1.018
I .023
0.032
G4
500
0.75
0.929
0.899
0.042
G5
500
0.75
0.878
0.815
O.042
G6
500
0.75
0.933
0.979
0.042
G7
50
0.75
0.860
0.876
0. 132
Table 4: lllustration
7.
of estimates on random generations.
ESTIMATION OF SECOND ORDER PROPERTIES
OF STATIONARY DOUBLY
STOCHASTIC P01SSON SEQUENCES
Consider~ like in section 6, a stationary doubly stochastic Poisson sequence N = {N k ; k E Z} together with its underlying random measure = {Zk ; k s
In section 6, where linear estimation of random
variables was treated, we assumed m = E ~k to be known.
In general these quantities
and
rk = C o v
are unknown,
(~j ,~j+k )
and therefore
have to be estimated.
In this section we will study estimates of the
covariance structure.
We will, however,
assume m to be known.
If it was possible to observe Z the problem to find the estimates were
'standard'
time series analysis.
observed and we have to find estimates
In general ~ can not be in terms of an observation
Since also N is a stationary time series, we do really never leave
of N.
191
'standard'
time series analysis.
We will in this section assume that we have an observation NI,...,N n of N and we will compare natural estimates natural estimates
in terms of N with the
if we had an observation s
of Z.
In section 6.1, where linear estimates was studied for finite observations, the results were b a s e d on the covariances
r~ while in sec-
tion 6.2 the results were b a s e d on the spectral density fs therefore
study estimates
We will
of r k when n is 'small' and estimates of
fZ when n is 'large'. This division will also from the point of v i e w of estimation be natural.
As will be seen in example
I, the word
'small' has to be liberally interpreted.
We will always assume that E s
4
< ~ and that s is stationary up to
the 4th order. Thus the quantities
m
=Es
rk
= E (s
rk, ~
= E (.~) - m ) ( ~ . v + k - m)(.~ +~
-m)
r k , j , .i
= E (~)
-
D
exist and are independent
- m) ( ~ u + k
-m)
- m) (g~)+k - m ) ( g v + j
m)(s
-
m)
of ~. Observe that m and r k are defined
as before and, more important, that rk, j must not be confused with Cov (Zk,Zj) for a non-stationary
stochastic
sequence.
We note that, contrary to the situation when linear estimation is studied
(cf remark 6.3), N k - m can not here be considered as an
observation of a 'signal'
~k - m with an independent
Nk - Zk added. The reason is that here properties
'noise'
up to the 4th
order are needed, while for linear estimation only properties to the 2nd order are needed
(cf t h e o r e m
1.6).
up
]92
The quantities
n lkl Ck =
(~. J
Z j=1
-
)
m (s
I
-
m)
and
n-lkl CkN =
j=IZ
will be important
7. I
(Nj - m)(Nj+ k - m)
for the construction
of the estimates.
Esgimation of t h e cova~iances
Suppose
that ~]'''''~n
rk = ~
is a natural
and N I,...,N n are observed.
Ck
estimate
of r k in terms of s
Vat rk = 0( 1 ) under rather general see section
and it is known that
conditions.
Since r N = 6km + r k,
1.6,
=
is a natural
We observe
Then
ckN
I
estimate
_ 6k m
of r kZ in terms
of N.
that E r k = E r k = r~ and will compare Var r k with
Var r k. After some calculations,
cf Grandell
(1971, pp 227-229),
we get n
n Vat r 0 = n Var r O + Var
+ m + 2 In2 + ( 6 - m ) r ~
and for k r 0 and 2k < n
n
(1 E ~.) + 2 E (n-j) r~ n j=1 J n j=-n 'J
+ 2r~,~
193
(n-k) Var rk = (n-k) Vat r k + (n-k)
[m2 + r k~ + 2mr 0 +
~ + rk, k + r0,k] + 2(n-2k)
[Zr~2k + r kZ ,2k]
For large values of n these formulae do not give much information on the behaviour of Var r k. If {~k - m} is a linear process,
closed
forms for lim Var r k exist, but unfortunately these formulae can not n§ be applied to Var r k since {N k - m} is not a linear process. An unpleasant property of estimates of r k is that in general lira n Coy (rk, r~) is equal to a non-zero n-~ A good discussion of estimation Hannan
Example
constant
of covariances
also when k ~ j.
is found in e.g.
(1960, pp 34-45).
I
In order to get some idea of the relation between Var r k and Var r k we consider the case described in section 2.5. These random generations were used by Grandell tion of the estimates.
In spite of what is said above
of r k for large values of n, we have in figure 23
drawn Var
lim--
109-113) as illustra-
It shall be observed that in this case
{~k - m} is not a linear process. using estimates
(1972:2,
~k
n+~ Var r k
as a function of p for s'ome values of k.
194
lira
n~
Vat
~k
Vat
r~ k
I
0.25
Figure
23:
Illustration a n d Var r_ K
Consider
the
estimate
of the
I
1
I
0,50
0.75
I
asymptotic
relation
between
V a r rk
.
r0"
F o r this
estimate
we h a v e
(cf G r a n d e l l
(1972:2, p 110))
l i m n V a r ro =
and thus must
to f u l f i l
the r a t h e r
modest
requirement
V a ~ r r0 ~ 0.1 we
have
n
Thus,
13 1 + p + 21 I - p
~
100 {13
as e x a m p l e s ,
p = 0.25,
n
~
6000
1 + p + 21} l - p
we m u s t if
have
n
p = 0.50
.
~
3400 and
if n
~
p = 0, n 11200
if
~
4267
if
p = 0.75.
195
7.2
Estimation of the spectral density
Assume that F ~ is absolutely continuous and consider estimates of the spectral density f~ (see section 1.6). In section A3 a short discussion of spectral estimation is given. Suppose, like in section 7.1, that Z1,...,~n and NI,...,N n are observed. Since
f~(x) : 2~m + f~(x) (see section 1.6) it is natural to compare the estimates
~(x)
:
I
n-1
z
2~n
(n)(x)
wk
z
Ck e
-ikx
k=-n+1 and
Rx) :
I 2~n
I
2wn
of fs
n-1 (n) (x) N e-ikx _ m__ = k=_~n+1 Wk Ck 2w
n-1 E
(n)(x) ( N nm6k ) e-ikx wk Ck -
k=-n+ I
The coefficients w~n)(x) correspond to the chosen weight
function Wn(Y:X ). Since E f(x) = E f~(x) we do not consider the bias of the estimates.
If s is Quasi-normal, see section A3, we have good knowledge of the asymptotic behaviour of Var f~(x), see theorem A3.5. It is thus natural to investigate under which conditions on Z also N is quasi-normal, since then we also have good knowledge of the asymptotic behaviour of
Va~ }(~).
Put
. . - r.r. Pk,j,i = rk,j,i - rk r l-j J 1-k - r.r. i j-k"
is quasi-normal if, in addition to the general assumptions given in the beginning of this section
196
oo
z
I < Iik
I - ~ for all n. n A sequence {~n)1 is called tight if the corresponding of distributions
sequence {H n)
is tight.
A sequence of probability measures on (S,B(S)) is called relatively compact if each subsequence
of it contains a further subsequence which
is weakly convergent.
For Polish spaces Prohorov~s theorem (cf Billingsley states the equivalence between tightness
(1968, pp 35-40))
and relative compactness,
and this fact explains the importance of tightness.
The main motivation for the study of weak convergence
is that if h
209
is a measurable mapping from S into another metric space S' and if d Sn---* < then also h(6 n)
d h(~5) provided P r { ~
the set of dis-
continuity points of h} = 0. Thus the finer the topology the stronger a weak convergence result.
Consider now convergence in distribution
of random measures.
Theorem 6 (continuity) Let A,A I ,A2, ... be random measures with distributions H ,H I ,H 2, . . . . Then An
d .... A if and only if ~
(f) § ~ ( f )
This result is due to v. Waldenfels
(1968). His proposition is
stronger and formulated for characteristic p 13) gives a similar strengthening
for all f 6 C K +
functionals.
for Laplace transforms.
The following two theorems are weaker formulations Kallenberg
Mecke (1972,
of results due to
(1973, pp 10-11). For any subset A of X we denote its
boundary by 8A.
Theorem 7 Let A,AI,A2,...
be random measures and let A o C B ( X )
~-system containing
a basis
be a
on X s u c h t h a t d Pr(A{~A} = 0} = I for all bounded A ~ A . Then A ---~ A if and only o n if (An{A I} .... ,An{Aj)) all bounded A 1 , . . . , A j ~
d
for the topology
(A{AI],...,A{Aj})
for all j = 1,2 .... and
Ao.
Theorem 8 Let NI,~2,... be point processes, and let A C B ( X ) b e an a l g e b r a
let N be a simple point process
containing
a basis
for the topology d on X such that Pr{N{~A} = 0} = I for all bounded A ~ A . Then N : N n
210
if and only if (i)
Pr{N {A) = 0)
§
Pr{N{A}
= 0)
for all bounded
A
~
Pr{N{A)
> I}
for all bounded A
n
(ii)
Pr{N (A) > I} n
(iii) {Nn) I
is tight.
This is the first time in our discussion tion of random measures needed.
The explicit
where
a tightness
condition
by the following weaker
of convergence condition
is, however,
formulation
in distribu-
is explicitly
easy to remove
of theorem
as seen
8.
Theorem 9 Let N,NI,N2,... if N {A)
d~
and A be as in theorem
N{A)
for all bounded A ~ A
d 8. Then N n ---* N if and only .
n
Proof We have to show that N {A} n Tightness
of {N n}
compact K C X
d
N(A) implies
is equivalent
that {Nn}l is tight.
to tightness
of (Nn(K)} I for all
and thus we only have to show that tightness
of
oo
{Nn{A}) 1 implies tightness
of (Nn{K}} I
Take a compact K C X. Since X can be covered by countably many bounded basis many.
sets it follows that K can be covered by finitely
Thus there exists
> 0 there exists
for all n. Since
space
on s and A, such that
Pr{N {K) < k) > Pr{N {A} < k) n -n --
with left hand limits
DF0,A ~ , A ~ ~, of all rightcontinuous defined
dowed with the Skorohod J1 topology. properties
For every
{Nn{K)) T is tight.
Consider now the function functions
such that A ~ K .
a real number k, depending
Pr{N {A) ~ k} > I - s n it follows that
a bounded A ~ A
as DEO,I],
for which
on E0,A]. Let DE0,A ~ be en-
The space DE0,A ~ has the same
Billingsley
(1968)
is the standard
211
reference. on E0,|
In many situations
it is natural to consider
Let D be the set of all rightcontinuous
functions
functions with
left ha~d limits defined on E0,~). The following topology on D is studied by Lindvall
(1973:1) and (1973:2) who develops
Stone (1963) and Whitt
(1970).
Let F be the set of strictly increa-
sing, continuous mappings of F0, ~) onto itself. identity element of F. Take X,Xl,X2,. .. ~ D .
Let e denote the
Let x n + x mean that
U,C
there exist u where
U
U
F such that Xn~ Yn
~ stands for u n i f o r m convergence
form convergence
ideas due to
x and y n and
on compact subsets of D , ~ ) .
U~C
~
~ e
ands for uni-
With thirstsdefinition
of convergence D is Polish.
Let for A ~ ~ , ~ )
the function r A : D § DE0,A] be the restriction
operator to ~,AI,_ i.e. rA(x)(t) theorem given by Lindvall
= x(t)
, t 6 E0,A]. The following
(1973:2, p 21) and (1973:1, p 120) brings
the question about weak convergence of stochastic processes
in D
back to the finite interval case.
T h e o r e m 10 Let X,X I,X2,... be stochastic processes
in D. Suppose there exists
co
a sequence
{Ai)i= I , A.l > 0 and A.1 § ~ as i § ~
d
rA. (Xn) --~ rA. (X) 1
1
for i = 1,2, . . . .
X
d n
~ X
Then
as
n§
as
n
-~ oo
, such that
212
A2.
HILBERT SPACE AND RANDOM VARIABLES
The reader is assumed to be acquainted with the formal definition of a Hilbert space. A good introduction well suited for our purposes is, however,
given by Cram6r and Leadbetter
(1967, pp 96-104).
Let H be a Hilhert s~ace. Let h,hl,h2~ H. In general
(hl,h 2) denotes
the inner ~roduct between h I and h 2 and Ilhll = (h/~-~,h)denotes the norm of h. Let h,hl,h 2 .... { H .
Convergence h
§ h means that
n
H is complete in its norm. The operations
llhn - hll + O.
of addition and multiplica-
tions with real or complex numbers are defined for the elements in H. If (hl,h 2) is real for all hl,h 2 6 H , space.
then H is called a real Hilbert
Let {hi ; j E J } be a family of elements in H. Let H(J) be the
collection of all finite linear combinations
of elements in {hi
or limits of sequences of such combinations.
H(J) is Hilbert subspace
of H and is called the Hilbert space spanned by {hi denoted by S({hj
; j EJ]).
; jEJ}
; j ~J}
and often
It is obvious that if Jo is a subset of J
then H(J o) is a Hilbert subspace of H(J). For our applications
of
Hilbert space geometry the following theorem is of great importance.
Theorem I. Let H h
o
6H
The projection theorem
be a Hilbert subspace of H and let h 6 H .
o
called the projection of h on H
o
two equivalent
0 for all x E [ - ~ , ~ restriction.
We assume that
since for our purposes this is no
The time series itself has the spectral representation
xk =
f ei~z{d~} --IT
215
where,
in differential
notations,
E(Z(dx} Z(dy})
the process
F{dx}
if
x = y
0
if
x r y
Z(x) fulfils
=
(The reader is assumed to be acquainted
with the formal defini-
tion of this kind of representations.)
Define the Hilbert
space L
= S(X. ; j < n}, L = S(X. ; j 6 Z} J ~ O
n
with inner product E hlh 2 and with inner product
L = S{e iJx
~ hl(X) h2(x)
F{dx}.
; j < n}
L = S{emJX;
j ~Z}
For all n (including ~)
L
--IT
and L j
inf -- ~ n E A n
n ( Z j=1
Var
inf
Var
(a
proves
We w i l l
now
and thus
estimation
studied
for n o r m a l
problems
of the e s t i m a t e s ,
up to the
4th order
such
{X k
; k6Z}
that
= E
We w i l l
spectral
series only
and t h e r e f Q r e
k =-~E Irkl
assume
which
analysis, consider
only
series
< ~ and E
with
known
This
and was the
is one first
(asymptotic)
assumptions
of v for all k , j , i 6 Z .
for a n o r m a l
mean
(Xv - m ) ( X v + k
on m o m e n t s
process
value
E X k = m,
- m)(Xv+ j - m)(X
Put
(X v - m ) ( X v + k - m ) ( X v + j - m ) ( X v + i - m)
a quantity
density.
are required.
be a time
and is i n d e p e n d e n t
of the
of time
processes.
variances
Let
= 2~ f(0)
theorem.
consider
of the most studied
the
--
J
lim inf n Var m~n -> 2w f1(0)
which
(a I ..... a n )
Y' ) >
--n
-%hEAn
X!I) )
a. J
=
-lq
Pk,j,i
~i-m)
=
- rkr.l_j. - rjri_ k - r.r.1j-k'
is equal
to zero.
Let us
further
that
z
Ipk,j,iL
O.
Let XI,...,X n be observed
and put
n-l~l Ck =
E j=1
(xj - m)(X.~+
k
- m)
.
Then the periodogram
1 n-1 Z In(X) = 2~n k=-n+1
might
-ikx Ck
e
seem to be a good estimate
some unpleasant
properties
I
k~1= (X k - m) e -ikx 2
= 2~---~
of f(x). This estimate
has, however,
and we are led to consider weighted
esti-
mates of the form
ii
fn~(X) =
/ Wn(Y:X)
In(Y) dy
where
fw n
(y:x) ~=
and where the weight
I
functions
Wn(Y:X)
for all x accumulates
mass in
224
the neighbourhood
of y = x at a 'suitable'
rate as n § ~.
Put
w(kn)(x) = ~ e ik(x-y) Wn(Y:X)dy --7[
and thus we get
f~(x) = 1 n 2wn
Usually only estimates wk(n)(x)
n-1 ~n) e-ikx Z w (x) Ck k=-n+1
f (x) where w ~n)(x)
= 0 for Ikl < m m
(m
--
is independent
of x and where
much smaller than n) are considered. n
The simplest such estimate is the Grenander and Rosenblatt
'truncated'
estimate
(cf e.g.
(1956, p 148)) given by
m
fn~(x)
=
n
I 2~n
-ikx
E
Ck
e
k=-m n m
where m
n
§ ~ as n § ~ in such way that
n n
§ 0.
The following theorem is taken from Roseablatt where also the required conditions tions Wn(Y:X)
(1959, pp 253-255)
on the sequence of weight
func-
are given.
Theorem 5 Let
{Xk
; k~Z}
be a quasi-normal time series and let Wn(Y:X) be
a sequence of 'suitable'
weight
functions.
w n
Var
Then
(y:x) dy i f
x r 0,'~
-Tr
f~(x) 4w nf2(x)
if f(x) > 0. Further,
J w2(y'x)dy
if
x=0
if 0 0 and
225
f(x 2) ~ O, the estimates
f~(x I) and f~(x 2) are asymptotically
un-
correlated.
It may be observed that Var f~(x)~~ tends to zero slower than I_ since n n f w~ (y:x) dy § ~ as n ~ ~. -7 For
the 'truncated' estimate we have
f w 2 (y:xl n
dy
m ~--~
.
226
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232
INDEX
Absolutely dominated
121
Additive see completely random Asymptotically
efficient
Average intensity
162
185
Best estimate 88 Best linear estimate
116, 142, 213
Borel
-
algebra 3, 205 measure 3
Bounded set 5 Completely random 5 Completion
13
Convergence in distribution 69, 74, 208 - vague 205 - weak 19, 208 Convolution 207 Covariance 23 Cox process see doubly stochastic Poisson process Cross spectral density
163, 216
Diffuse see non-atomic Doubly stochastic Poisson process 7 alternative definitions of a - 10, 12, 16 Doubly stochastic Poisson sequence Dynkin system 7 Ergodic 27 Estimate 88 best - 88 best linear - 116, 142, 213 best linear unbiased - 220 linear - 116 Functional limit t h e o r e m 76 Hilbert
space
Instantaneous
115~ 212 intensity
14, 15
Intensity average - 185 instantaneous - 14, 15 - function 12 measure 5 -
Laplace-transform Leading function
18, 206 10
17
233
Level
160
Linear
estimate
116
Local convergence see vague convergence Loss
function
88
Mean 23 Measurable
process
13
M i x e d Poisson process see w e i g h t e d Poisson Non-atomic - measure 8, 206 random measure Observation
process
19, 206
87
Operator c-amplifying - 21 p-thinning - 21 shift - 27 P a l m measure w-system
54
205
Point process 4 simple - 19, 206 Poisson process with intensity measure 5 intensity one 11 - leading function 10 Polish
space
3, 205
P61ya process Quasi-normal
32 195, 222
Radon measure see Borel measure R a n d o m measure 4 distribution of a - 4 non-atomic - 19, 206 Regular
variation
Relatively
76
compact
208
Renewal process 34 alternating - 50 arithmetic - 44 non-arithmetic - 44 ordinary - 44 stationary - 37 transient - 34 Simple point Skorohod
process
topology
19, 206
74, 210,
211
Spectral density 27, 214 - distribution 27, 214 -
Standard
Poisson
see Poisson
process
process
with intensity
one
234 State space 3 Stationary strictly ~ 20 (weakly) - 26 Thinning 21 Tight 208 Topology Skorohod - 74, 210, 211 vague - 3, 205 "Truncated"
estimate
198, 224
Vague - convergence 205 - topology 3, 205 Version
13
Weak convergence
19, 208
Weighted Poisson process Without after-effects see completely random Without multiple points see simple
31