Discrete Time Branching Processes in Random Environment

8Intermediate Subcritical BPREs

8.1. Introduction

In this chapter, we investigate the intermediate subcritical case under the annealed approach. It is located on the borderline between weakly and strongly subcritical BPREs. The passage corresponds to a phase transition in the model, thus a particularly rich behavior can be expected from the intermediate case. This is reflected in our below results. In particular, we will observe a kind of bottleneck phenomenon, which does not occur elsewhere under the annealed approach. Similar behavior has been noted under the quenched approach in the critical regime, as we explained in Chapter 6.

Thus, we assume:

ASSUMPTION I1.– Assume that X is non-degenerate and that

Again, since xe^x > x for all x ≠ 0, this assumption entails, as in Chapter 7, that

On the other hand, by the convexity of the function y log y, y > 0, and by Jensen’s inequality, we have [e^X] log [e^X] < [e^XX] = 0, and consequently

Hence, [Z_n] = [Z₀] [e^X]ⁿ is decreasing at an exponential ...