Discrete Time Branching Processes in Random Environment

Preface

Branching processes constitute a fundamental part in the theory of stochastic processes. Very roughly speaking, the theory of branching processes deals with the issue of exponential growth or decay of random sequences or processes. Its central concept consists of a system or a population made up of particles or individuals which independently produce descendants. This is an extensive topic dating back to a publication of F. Galton and H. W. Watson in 1874 on the extinction of family names and afterwards dividing into many subareas. Correspondingly, it is treated in a number of monographs starting in 1963 with T. E. Harris’ seminal The Theory of Branching Processes and supported in the middle of the 1970s by Sevastyanov’s Verzweigungsprozesse, Athreya and Ney’s Branching Processes, Jagers’ Branching Processes with Biological Applications and others. The models considered in these books mainly concern branching processes evolving in a constant environment. Important tools in proving limit theorems for such processes are generating functions, renewal type equations and functional limit theorems.

However, these monographs rarely touch the matter of branching processes in a random environment (BPREs). These objects form not so much a subclass but rather an extension of the area of branching processes. In such models, two types of stochasticity are incorporated: on the one hand, demographic stochasticity resulting from the reproduction of individuals and, on the other hand, ...