Discrete Time Branching Processes in Random Environment

9Strongly Subcritical BPREs

9.1. Introduction

In this chapter, we investigate the strongly subcritical case in more detail by applying the annealed approach. We make the following two assumptions.

ASSUMPTION S1.– Assume

[9.1]

As xe^x ≥ x for all x ∈ , this assumption implies [X] < 0. Hence, [9.1] is indeed the condition of a strong subcriticality, as introduced in Chapter 2.

This has an important consequence. We may rewrite formula [9.1] as

Since the function x log x is convex in (0, ∞), an application of Jensen’s inequality yields

which implies

Our second assumption is the so-called x log x condition for ordinary branching processes.

ASSUMPTION S2.–

It will be shown that for a strongly subcritical BPRE {Z_n, n ≥ 0} meeting conditions S1 and S2 the non-extinction probability at generation ...