Discrete Time Branching Processes in Random Environment

1Branching Processes in Varying Environment

1.1. Introduction

Branching processes are a fundamental object in probability theory. They serve as models for the reproduction of particles or individuals within a collective or a population. Here we act on the assumption that the population evolves within clearly distinguishable generations, which allows us to examine the population at the founding generation n = 0 and the subsequent generations n = 1, 2, … To begin with, we focus on the sequence of population sizes Z_n at generation n, n ≥ 0. Later, we shall study whole family trees.

Various kinds of randomness can be incorporated into such branching models. For this monograph, we have two such types in mind. On the one hand, we take randomness in reproduction into account. Here a main assumption is that different individuals give birth independently and that their offspring distributions coincide within each generation. On the other hand, we consider environmental stochasticity. This means that these offspring distributions may change at random from one generation to the next. A fundamental question concerns which one of the two random components will dominate and determine primarily the model’s long-term behavior. We shall get to know the considerable influence of environmental fluctuations.

This first chapter is of a preliminary nature. Here we look at branching models with reduced randomness. We allow that the offspring distributions vary among the generations but as a start ...