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Lecture Notes in Statistics 96
Edited by S. Fienberg, J. Gani, K. Krickeberg,
I. Olkin, and N. Wennuth
Ibrahim Rahimov
Random Sums and
Branching Stochastic
Processes
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Ibrahim Rahimov
The Institute of Mathematics
The Academy of Sciences of the Republic of Uzbekistan
Hodjaev Street. 29,
Tashkent. 700142, Uzbekistan
Department of Statistics
Middle East Technical University
06531, Ankara, Turkey
Library of Congress Cataloging-in-Publication Data Available
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to 1995 Springer-Verlag New York, Inc.
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Camera ready copy provided by the editor.
9 8 7 6 5 432 1
ISBN-13: 978-0-387-94446-3 e-ISBN-13: 978-1-4612-4216-1
001: 10.1007/978-1-4612-4216-1
TO TIlE MEMORY OF MY
FATIIER AND MY MOTIIER
Uzbekistan is one of the Central Asiatic Republics of the former Soviet
Union. The ancient Uzbek cities Samarkand, Bukhara, Khiva and Tashkent have
long been major attractions. They have over 4000 architectural monuments,
many of them under UNESCO protection. Uzbekistan is the mother country of M.
Khorezmi (Al-Goritm), Abu Raikhon Beruni and Mirzo Ulugbeg, famous
mathematicians and astronomers of the Middle Ages. Ibn Sina (Avitsenna) the
great medical scientist, philosopher and poet was born here.
CONTENTS
INTRODUCTION
CHAPTER I. SUMS OF A RANDOM NUMBER OF RANDOM VARIABLES. 5
§1.1. Sampling sums of dependent variables and mixtures
of infinitely divisible distributions. 5
§1a. Sums of a random number of random variables. 7
§1b. Multiple sums of dependent random variables. 9
§1c. Sampling sums from a finite population. 14
§1.2. Limit theorems for a sum of randomly indexed sequences. 18
§2a. Sufficient conditions. 18
§2b. Necessary and sufficient conditions. 21
§2c. An application. 25
§1.3. Necessary and sufficient conditions and limit
theorems for sampling sums. 27
§3a. Convergence theorems. 27
§3b. The rate of convergence. 35
CHAPTER II. BRANCHING PROCESSES WITH GENERALIZED IMMIGRATION. 44
§2.1.Classical models of branching processes. 44
§1a. Bellman-Harris processes. 45
§1b. Moments and extinction probabilities. 46
§1c. Asymptotics of non-extinction probability and exponential
limit distribution. 48
§1d. Branching processes with stationary immigration. 51
§1e. Continuous time branching processes with immigration. 54
§2.2.General branching processes with reproduction dependent
immigration. 58
§2a. The model. 59
§2b. The main theorem. 62
§2c. The proof of the main theorem. 64
§2d. Applications of the main theorem. 71
§2.3.Discrete time processes. 76
§3a. The model. 76
§3b. Limit theorems for discrete time processes. 78
§3c. Some examples. 84
§3d.Randomly stopped immigration. 87
§2.4.Convergence to Jirina processes and transfer theorems
for branching processes. 92
§4a. The model. 92
§4b. The main theorem and corollaries. 94
§4c. The proof of the main theorem. 97
CHAPTER III. BRANCHING PROCESSES vIm TIME-DEPENDENT IMMIGRATION. 105
§3.1.Decreasing immigration. 105
§1a. The main theorem. 106
§1b. The proof of the main theorem. 118
§1c. State-dependent immigration. 122
§3.2.Increasing immigration. 124
§2a. The process with infinite variance. 124
§2b. The process with finite variance. 132
§3.3.Local limit theorems. 136
§3a. Occupation of an increasing state. 136
§3b. Occupation of a fixed state. 154
CHAPTER IV. TIlE ASYMPTOTIC BEHAVIOR OF FAMILIES OF PARTICLES IN
BRANCHING PROCESSES. 156
§4.1. Sums of dependent indicators. 157
§1a. Sums of functions of independent random variables. 157
§1b. Sampling sums of dependent indicators. 163
§4.2.Family of particles in critical processes. 167
§2a. The model. 167
§2b. Limit theorems. 168
§4.3.Families of particles in supercritical and
subcritical processes. 177
§3a. Supercritical processes. 177
§3b. Subcritical processes. 184
REFERENCES. 184
INDEX. 194
INTRODUCTION
Let us consider the development of a population whose members live, give
birth to a finite number of new individuals, and die. Reproduction and death
are properties of all biological populations including demographic and cell
populations (see lagers, 1975, for example). Other examples are the
development of families of neutrons in atomic reactors (Vatutin et. aI.,
1985), the behavior of cosmic ray showers, and the growth of large organic
molecules (Uchaykin and Ryjov, 1988; Dorogov and Chistyakov, 1988). All of
these processes are characterized by the same property, namely, their
development has a branching form. A branching stochastic process is the
mathematical model of such kinds of empirical processes.
The theory of branching random processes is a rapidly developing part of
the general theory of random processes. A great number of papers dealing
wi th investigations of different models of branching processes have been
published. The books of Harris (1963), Sevast'yanov (1971), Mode (1971),
Athreya and Ney (1972), lagers (1975), Asmussen and Hering (1983) and the
review articles of Sevast'yanov (1951), Kendall (1966), Vatutin and Zubkov
(1985) are some of the results of this growth and they show the development
of the theory.
First of all, the interest in this theory is connected with its
applications to a wide spectrum of practical problems. They include the
description of various biological populations, the investigation of
the transformation processes of particles in nuclear reactors, the
investigation of cascade processes, of chemical processes, problems of
queueing theory, the theory of graphs and other problems.
We will use the following intuitive description of branching processes
instead of the formal definition of the process. We consider the scheme of
evolution and reproduction of some particles. Each of these particles lives
independently of the others in a random time L and generates a random number
v of new particles. These new particles undergo analogous transformation.
Models of branching processes differ from each other by additional
assumptions on the reproduction process of new particles and on the
2 Introduction
distribution of the vector (L,v). The random process ~(t) (the size of the
population at the time t e [0,00)) is the main object of the investigation
in the theory of branching processes.
In branching processes with immigration new particles emerge not only
after reproduction, but may also emerge at the moments of jumps of some
integer-valued random process X(t) with non-decreasing trajectories. The
number of immigrants is equal to the size of the jump of the process X(t).
The number Z(t) of particles at the time t is termed a branching process
with immigration; X(t) is called the immigration process.
In the first papers devoted to branching processes with immigration it
was assumed that the immigration process X(t) is either a homogeneous
Poisson process or a partial sum of independent and identically distributed
random variables. It was established that the asymptotic behavior of the
process does not alter greatly in the presence of stationary immigration of
particles.
However, in real processes jumps of the immigration process can depend on
each other. The distribution of the number of immigrants can vary in time
and the immigration process can depend on reproduction in a specific way.
For example, if we consider the process of urban population growth, the
number of immigrants at present depends on the lives of past immigrants and
their descendants. Another example is the neutron multiplication process
with an external neutron source (Dorogov and Chistyakov, 1988). If we want
to support a process by immigration, it is apparent that the immigration
process depends on reproduction.
More general models of branching processes with immigration were
considered in the 1970's by Foster and Williamson (1971), Durham (1971) and
S. Nagayev (1975). Immigration processes in these papers were more general
than processes with independent increments. Later on, various proofs of
I imi t theorems for branching processes wi th immigration were suggested by
Athreya, Parthasarathy and Sankaranarayanan (1974), Shurenkov (1976),
Asmussen and Hering (1976), Badalbayev and Zubkov (1983), S. Nagayevand
Asadullin (1985) and others. However, an assumption of the independence of
immigration and reproduction processes still characterizes these efforts. In
a sense, immigration processes so far considered have been similar to
stationary processes.
The proofs of limi t theorems for branching processes are based on a
representation of the process in the form of a sum of the random number of
branching processes "shifted" over time. Since these processes are
