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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

7Weakly Subcritical BPREs

7.1. Introduction

In this chapter, we consider one more important class of BPREs, namely, weakly subcritical BPREs. We show that the methods, developed in Chapter 5 for criticality, are also available for weak subcriticality.

We recall the definition of the standardized truncated second moment of F :

Image

and assume throughout this chapter the validity of the following condition:

ASSUMPTION W1.– The process Z is weakly subcritical, that is, there is a number 0 < β < 1 such that

and also

Image

As we have x xeβx with strict inequality for x ≠ 0, Assumption [7.1] implies

Image

Thus, the associated random walk image has a negative drift with respect to images. Also, we have, for λβ, the inequality eλxeβx +βxeβx(λβ), due to the convexity of the exponential function and with strict inequality for x ≠ 0. It follows [eλX] > [eβX]. Thus, letting

and choosing λ = 0 and 1, we obtain ...

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ISBN: 9781786302526Purchase book