Discrete Time Branching Processes in Random Environment

4Properties of Random Walks

4.1. Introduction

The study of BPREs essentially uses properties of the ordinary random walk S : = {S_n, n ≥ 0} with S₀ := 0. This chapter contains some basic results needed in the sequel.

To begin with, we introduce four sequences of random variables related to S. Let

and

be the sequences of strict descending ladder epochs of the random walk and let

be the sequences of weak descending ladder epochs.

In a similar way, we specify the sequences of strict ascending ladder epochs

and weak ascending ladder epochs

The random variables and are called ladder heights. Ladder epochs and ladder heights will repeatedly appear in our subsequent arguments. Rather often they will occur when using the so-called duality principle, which we formulate as a theorem.

THEOREM ...