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Discrete Time Branching Processes in Random Environment
book

Discrete Time Branching Processes in Random Environment

by Götz Kersting, Vladimir Vatutin
November 2017
Intermediate to advanced content levelIntermediate to advanced
306 pages
6h 24m
English
Wiley-ISTE
Content preview from 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

As xexx for all x Image, this assumption implies Image[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

Image

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

Image

which implies

Image

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

ASSUMPTION S2.– image

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

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Publisher Resources

ISBN: 9781786302526Purchase book