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 = {S_n, n ≥ 0} satisfies the condition

[5.1]

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

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

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