Stochastic Processes

135 Pages·1966·3.305 MB·English

by  M. Girault (auth.)

Okonometrie
und Unternehmensforschung
Econometrics
and Operations Research
III
Herausgegeben von / Edited by
M. Beckmann, Bonn· R. Henn, Gottingen . A. Jaeger, Cincinnati
W. Krelle, Bonn· H. P. Kunzi, Zurich
K. Wenke, Ludwigshafen . Ph. Wolfe, Santa Monica (Cal.)
Geschajt.rJiihrende Herausgeber / Managing Editors
W. Krelle . H. P. Kunzi
Stochastic Processes
M.Girault
Professor at the University of Paris
With 35 figures
Springer-Verlag Berlin Heidelberg New York 1966
ISBN-13: 978-3-642-88271-5 .,.ISBN-13: 978-3-642-88269-2
DOl: 10.10071978-3-642-88269-2
All rights reserved, especially that of translation into foreign languages
It is also not pennitted to re produce this book, either whole or in part, by photomechanical means
(photostat, microfilm and/or microcard) or any other means
without written penuission from the publishers
© by Springer-Verlag, Berlin· Heidelberg 1966
Softcover reprint of the hardcover 1st edition 1966
Library of Congress Catalog Card Number 66-22463
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Preface
Existing works on stochastic processes belong to a field of abstract
mathematics which puts them beyond the scope of the non-specialist.
The preoccupations of research mathematicians being more often than
not distant from the practical problems of experimental methodology,
the needs of practical workers, though real, are not met by the
majority of works that. deal with processes. By "practical workers",
we mean research scientists in all the different disciplines: Physics,
Chemistry, Biology, Medicine, Population, Economics, Organisation,
Operational Research etc.
Indeed, all scientific research today touches upon complex fields
in which deterministic models can be useful for no more than an element
ary and simple approximation. The Calculus of Probability although
offering some interesting models is still inadequate in many instances,
particularly in the study of evolving systems.
The practical worker must therefore have at his disposal a set of
original and varied stochastic models.
These models must not be too general, for in that case not only
would their theoretical study prove difficult, but above all the adaptation
of such models to an observed system would lead to an estimation of
a great number of parameters on the basis of a necessarily restricted
sample. This would constitute an insuperable difficulty for the practical
scientist. It is therefore essential for him to have at his disposal a
varied range of very characteristic models.
These considerations have determined the orientation of the present
work, which is intended for the research scientist who has acquired the
basic mathematical training and is familiar with Probability-Calculus.
The models presented here are all likely to be of interest to the prac
tical scientist.
The theory of processes is in itself a difficult one; I have therefore
tried to present each question as simply as possible, where necessary
by adopting several methods of approach which, taken together throw
light on the subject as a whole. Each class of models is presented in
the first instance by its simplest and most characteristic case (uniform
process of Poisson or Laplace for example); after which a generalisation
is made by disposing of certain hypotheses (uniformity, for example).
VI Preface
In my opinion, the systematic use of variance-distributions makes
it easier to understand and explain the processes with independent
increments.
The Markov processes are studied by taking full advantage of the
properties of convex-linear operations so that most of the properties
are established in an elementary and natural manner.
The study of reach-delays is set forth with the help of matrix
calculus which provides a very convenient method both for calculating
and expressing results.
The study of Markov processes is restricted to the finite case;
however, the last chapter is devoted to a study of some simple and
characteristic permanent processes.
Paris, April 1966
MAURICE GIRAULT
Contents
Chapter I
Introduction
Page
1. Examples ............... . 2
2. Definitions . . . . . . . . . . . . . . . 3
3. Classification according to the nature of E 4
4. Classification according to the nature of the set T of instants where
the system is observed . . . . . . . . . . . . . . . • . . . 4
Chapter II
Poisson processes
A. Generalities - Point processes 8
1. Definition . . . . . . . . 8
2. Point processes . . . . . 9
3. Definition of a point process 9
B. The uniform Poisson process . . 10
4. Definition of the Poisson point process. 10
5. Distribution of the random interval between two consecutive events 11
6. Conclusion. . . . . . . . . . . . . . . . . . . . . 12
7. The Gamma probability distributions. . . . . . . . . . . . . 12
8. Distribution of "survival" of an interval that obeys 1'1 . . . . 13
+
9. Distribution of the interval embracing (k 1) consecutive events 14
10. Probability distribution of the number of events occurring in a given in-
terval of time . . . . . . . . . . . . . . . . . . 15
11. Second stochastical definition of the Poisson process. . 16
12. Conditional distributions . . . . . . . . . . . . . . 16
13. Number of occurrences in an interval of random length 17
C. Probability distributions associated to Poisson processes 17
14. Poisson distribution 18
15. Gamma distribution 18
16. Beta distributions. . 20
D. Extensions: Poisson-type-processes 23
17. Continuous, non-uniform Poisson processes 23
18. Poisson-type processes . . . . . . . . . 24
19. Parameter-distribution . . . . . . . . . 25
20. Decomposition ot the parameter distribution 26
VIII Contents
Page
21. Interval distribution and conditional distributions . 27
22. Poisson cluster process . . . . . . . . . . . . . 28
Problems for solution . . . . . . . . . . . . . . 29
Random functions associated to a Poisson process. 30
Chapter III
Numerical processes with independent random increments
1. Definition . . . . . . . . . . . . 31
2. Mean values - zero mean process. . . . . . 32
3. Variance distribution . . . . . . . . . . . . 32
4. Cumulants and second characteristic functions 33
5. Indicator function of a Poisson process 34
6. The Wiener.Levy process . . . . . . . . . . 35
7. Laplace processes with independent increments 40
8. General form of the random functions X (t) with independent increments 41
9. Infinitely divisible distributions 45
10. Stable distributions. . 49
Problems for solution. . . . . 50
Chapter IV
Dlarkov processes
A. Generalities and definitions 50
1. Generalities . . . . . . . . . . . . . . 50
2. Notation . . . . . . . . . . . . . . . 51
3. Representation by temporal development . 52
4. Markov sequences . . . . . . . 53
5. Probability law of states . . . . 54
6. Homogeneous Markov sequences . 56
B. Study by means of convexity . . . 57
7. Limit for k = 00 of Mk when all the terms of M are positive. 57
8. Regular positive case. . . . . . 60
C. Study by means of spectral analysis 62
9. Definitions ......... . 62
10. Eigenvalues and eigenvectors . . 64
11. Properties of the eigenvalues of a Markov matrix. 65
12. Canonical representation. 68
D. Direct algebraic study ............. . 69
13. Classification of states . . . . . . . . . . . . 70
14. Order on classes: transient classes and final classes 71
15. Examples . . . . . . . . . . . . . . . . 71
16. Summary . . . . . . . . . . . . . . . • 72
E. Reaching delays and sojourn duration problems 73
17. Introduction. . 73
18. Example 73
19. General method 76
Contents IX
Page
F. Miscellaneous ....... . 82
20. Ergodicity. . . . . . . . 82
21. Inversion of a homogeneous Markov sequence 87
Problems for solution 87
Chapter V
Laplace processlls and second order processes
I. Introduction . . . . . 89
A. S('cond order properties . 89
2. Second order processes 89
a. Stationary second order processes 92
B. Laplace processcs . . . . . . . . 97
4. Definition . . . . . . . . . . . 97
5. Remindcr of a few definitions and properties concerning Laplace
random sets (or Laplace random vectors) of k dimensions 97
6. Covariance Function of a Laplace process 102
7. Discrete process - Levy canonical decomposition 103
Problems for solution ............ 104
Chapter VI
Some Markov processes on continuous-time
1. Homogeneous Laplace.Markov sequences 105
2. Stationary Laplace-Markov ;;equence 107
a. Estimation problems . . . . . . . . . 108
4. Interpolation - permanent process . . 109
no
5. Non homogeneous standardized Laplace-Markov processes
Ii. General form of Laplace-Markov processes III
7. \\Tiener-Levy processes . 113
8. Poisson-Markov processes 113
Problems for solution 115
Answers to problcms 116
Index ...... . 126
Girault, Stochastic Processes a
Stochastic Processes
Chapter I
Introduction
·We denote by the name "Random ~-1odel" any abstract scheme
of a probabilistic nature, which can represent certain real phenomena. As
long as these models provide adequate representation of the phenomena
under observation (in the same way as any mathematical model used
by the engineer) they allow us to forecast the consequences of certain
situations. Thus, these models afford the possibility of making calculated
choices. For this reason, they play an important part in the science
of decision, more so as their field of application is very wide, since
they occur frequently.
Let us describe briefly a random model:
A trial is made, defined by conditions H. The result is not deter
mined by H. However, H determines a set .f; of possible elementary
events, and a probability distribution on .f;.
This scheme is too general to provide the basis for interesting
computations. For this, we should give a structure to the set .f;.
The simplest and most widespread method consists in t,aking for .f;
the real numbers, by associating to each trial a number (integer,
or rational or real). This number is named a random variable (or a
numerical alea).
More general models are obtained by considering elementary events
consisting of two or several ordered numbers; we then have a multi
dimensional random vector. The classical probability theory (in the
narrow sense of this terml) deals with such random events.
For applications, it is m0st important to consider more general
models. Let us consider a few examples.
1 The classical Probability Theory is limited to the study of the three following
cases:
.£ includes a finite number of states .
.£ is R (real numbers). The model is called random variable .
.£ is the set Rk of vectors with k real components or random vectors.
Of course, in order to study the law of large numbers, infinite sequences of
random variables are considered but nevertheless the systematic study of Random
infinite successions, is not undertaken.
Girauit, Stochastic Processes 1