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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

5Critical BPREs: the Annealed Approach

5.1. Introduction

In this chapter, we consider under the annealed approach critical branching processes in random environment. Thus, we assume that the associated random walk S = {Sn, n ≥ 0} satisfies the condition

with probability 1. In this case, the population becomes extinct with probability 1 as already stated in Chapter 2. Indeed, the estimate

Image

implies P(Zn > 0 | Z0) 0 a.s. as n → ∞ in the critical case. Consequently,

Image

Thus, it is of interest to investigate the rate of decay of the survival probability images(Zn > 0) as n → ∞ as well as the asymptotic properties of the conditional law images(Zn dx | Zn > 0).

To achieve this goal, we first construct, using the functions U(x) and V (x), some martingales in section 5.2 and introduce a change of measures using these martingales. In conclusion, we investigate the properties of the prospective minimum value of in section 5.3.

These preliminary results will enable us to describe, in section 5.5

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ISBN: 9781786302526Purchase book