PhD Thesis on Mathematics :

Discrete Self-Decomposable Distributions

by Karina Schreiber

Otto-von-Guericke-Universität Magdeburg, Germany
Fakultät für Mathematik

Date: 30.08.1999

60E05 Distribution theory,
60E07 Infinitely divisible distributions; stable distributions, See also {65Exx}
Keywords: branching processes (with immigration), F-self-decomposability, discrete self-decomposable, heavy tails, discrete stable, discrete Linnik distribution, Sibuya distribution, convergence to stable law, convergence rate, asymptotic behaviour, survival distribution

Abstract: The elementary concept of probability and its first applications arose in the 17th century when studying chances in games. Applications to scientific and technological problems, as the description of motion of molecules, confined in main in the last hundred years. Here, the theory of stochastic processes provides a powerful theoretical background for describing these observations and similar phenomena.

In particular, branching processes can describe the behaviour of particles, which are able to die, survive or produce more descendants. This branching behaviour is determined by a probability generating function which may be particles, which are able to die, survive or produce more descendants. This branching behaviour is determined by a probability generating function which may be identified as a composition semigroup. For the special case of pure death processes, each particle will either die or survive. In this connection an interesting class of distributions is obtained: discrete self-decomposable distributions. They result as stationary distributions of pure death processes, where at random times a random number of particles immigrate and they will also either die or survive. Discrete self-decomposable distributions can also be obtained as the distribution of a random stopped pure death process with a random number of particles at the beginning. This class of discrete self-decomposable distributions may also be defined as a discrete analogue to the class of self-decomposable distributions in the sense of a transformation. It is well-known that nondegenerated self-decomposable distributions are continuous, only. Both versions, the discrete and continuous one, of self-decomposable distributions are infinitely divisible, a property introduced by de Finetti in 1929. In the thirties, many results for this class were derived for instance by Kolmogorov, Levy and Khintchine. Infinitely divisible random variablesjoined large attention because these are precisely the limits of sums of random variablesin a sequence of series. Here, the additional restriction of infinite smallness is impossed, i.e., the influence of any individual random variable becomes infinitely small when the number of summands grows. In the absence of any such restriction, any distribution function can serve a limit of this kind.

In the sequel, we concentrate on discrete self-decomposable distributions. We consider discrete stable and discrete Linnik distributed random variables, which are discrete self-decomposable. They may be represented as a random sum of Sibuya distributed ones. The Sibuya distribution arises in analysing one-dimensional symmetric random walks. It is shown that a random sum of Sibuya distributed random variables with discrete self-decomposable summation index is again discrete self-decomposable.

The present work is devided in three chapters. In Chapter 1 we firstly give a brief review of some basic terms related to stochastic processes, and secondly we introduce the Sibuya distribution. Chapter 2 is mainly about branching processes and F-self-decomposability introduced by van Harn, Steutel and Vervaat in 1982, as a more general class containing the discrete self-decomposability. The main results are presented in Chapter 3. There, we consider the special case of discrete self-decomposability, and some distributions with this property. We will investigate whether or not the Sibuya distribution is discrete self-decomposable. Surprisingly, it will turn out that the discrete self-decomposablity of this distribution depends on the involved parameters. Investigations of the asymptotic behaviour of discrete Linnik distributions and their relatives, and characterizations of discrete self-decomposable distributions in terms of survival distributions will complete our analysis.

Last, but not least, I would like to take this opportunity to express my thanks to many people. It is a special pleasure for me to thank Professor Gerd Christoph for supporting my dissertation. He encouraged me to take up this project, and then he applied the appropriate mix of cheerfulness and tolerance with each new mathematical crisis.

I also owe Professor Radu Theodorescu a debt of gratitude for his great support during my research stay in Quebec. His lectures given in Magdeburg drawn our attention to the discrete stable and discrete Linnik distribution.

I am indebted to my colleagues of our Institute of Mathematical Stochastics. They were closely involved with many aspects of this project.

I would also like to thank the Friedrich-Naumann-Stiftung for liberal support and financial assistance of my research subsidized from public funds of the Bundesministerium für Bildung und Forschung. In particular, I would like to thank Professor Karl-Heinz Paque for intensive support and interesting talks about daily and topical questions of life and politics as a welcomed change of mathematical everydays life.

Mainly I thank my parents. Without their understanding and support this dissertation would probably never has been finished. I dedicate this dissertation to my parents as an expression, although insufficient, of my appreciation.

Magdeburg, August 30, 1999

Language: written in ENG