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

TL;DR

This paper establishes limit theorems for critical branching processes in highly unfavorable random environments, focusing on the distribution of particles conditioned on survival and environmental constraints, under stable law attraction assumptions.

Contribution

It provides new conditional limit theorems for critical branching processes in random environments with stable law domain of attraction, extending understanding of their asymptotic behavior.

Findings

Conditional distribution of particles given survival and environment constraints

Limit theorems under stable law domain of attraction

Behavior of the process as time tends to infinity

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 $\alpha $-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 $S_{n}\leq const$.