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 :

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

[7.1]

and also

As we have x ≤ xe^βx with strict inequality for x ≠ 0, Assumption [7.1] implies

Thus, the associated random walk has a negative drift with respect to . Also, we have, for λ ≠ β, the inequality e^λx ≥ e^βx +βxe^βx(λ−β), due to the convexity of the exponential function and with strict inequality for x ≠ 0. It follows [e^λX] > [e^βX]. Thus, letting

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