Limit theorems for critical branching processes in an extremely unfavorable random environment

TL;DR

This work analyzes a critical branching process in a random environment, focusing on extremely unfavorable settings where the associated random walk remains below a fixed threshold. By leveraging a renewal-based change of measure and stable-law attraction for the environmental increments, the authors derive conditional limit theorems for the population size $Z_n$ given $Z_n>0$ and $S_n\le K$, and they characterize the limiting shape of the entire trajectory via a rescaled process $\mathcal{Y}^{\theta n,n}$. The main contributions are (i) showing that the conditional law of $Z_n$ converges to a proper discrete distribution, and (ii) proving that the rescaled trajectory converges to a continuous-limit process $\mathcal{W}$, with the limit law decomposed into a left and a right part expressed as explicit functionals of underlying random walks and renewal structures. These results provide a precise description of population dynamics in hostile environments and furnish explicit limiting objects for further study of BPREs under extreme environmental stress.

Abstract

Let $\{Z_{m},m\geq 0\}$ be a critical branching process in random environment and $\{S_{m},m\geq 0\}$ be its associated random walk. Assuming that the increments distribution of the associated random walk belongs without centering to the domain of attraction of an $α$-stable law we prove conditional limit theorems describing, as $n\rightarrow \infty $, the distribution the number of particles in the process $\{Z_{m},0\leq m\leq n\}$ given $Z_{n}>0$ and $S_{n}\leq const$.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Corollary 7
• Lemma 8
• Corollary 9