Discrete Time Branching Processes in Random Environment

6Critical BPREs: the Quenched Approach

6.1. Introduction

In this chapter, we investigate the properties of critical BPREs under the quenched approach. Again, we rely on Assumption C. We first recall that when proving a Yaglom-type conditional limit theorem under the annealed approach in Chapter 5 we used a unique scaling e^S_m, 0 ≤ m ≤ n for the population size. The quenched approach gives more possibilities in the analogous situation. If a process survives up to a distant moment n, then the crucial role for the choice of scaling the population size at moment m ∈ [0, n] is played by the location of the point of observation with respect to the left-most point of the minimum of : τ(n) = min{0 ≤ k ≤ n : S_k = L_n}.

This is the case for the annealed approach as well. However, contrary to the annealed approach, where only trajectories with τ(n) = O(1) as n → ∞ provide survival, here the whole range of possible values for τ(n) is of importance. As shown in Theorem 6.3 below, to prove a Yaglom-type conditional limit theorem under the quenched approach it is necessary to scale the population size at moment t ∈ (0, 1] by . Moreover, according to Theorem 6.4, no scaling for the population size is needed at moment τ(n) → ∞ and the conditional limit distribution of the population size at this ...