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
and also
As we have x ≤ xeβx 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 eλx ≥ eβx +βxeβx(λ−β), 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 ...