Sudden extinction of a critical branching process in random environment

TL;DR

This paper investigates the asymptotic probability of extinction in a critical branching process within a random environment, revealing that unfavorable environmental conditions can cause sudden extinction even when the population is large.

Contribution

It characterizes the asymptotic extinction probability when the environment's expectation follows a non-Gaussian stable law, linking extinction to extreme environmental conditions.

Findings

Extinction probability asymptotics depend on the stable law domain of the environment's expectation.

Unfavorable environments can cause sudden extinction despite large populations.

Results interpreted via random walks in random environments.

Abstract

Let $T$ be the extinction moment of a critical branching process $Z=(Z_{n},n\geq 0) $ in a random environment specified by iid probability generating functions. We study the asymptotic behavior of the probability of extinction of the process $Z$ at moment $n\to \infty$, and show that if the logarithm of the (random) expectation of the offspring number belongs to the domain of attraction of a non-gaussian stable law then the extinction occurs owing to very unfavorable environment forcing the process, having at moment $T-1$ exponentially large population, to die out. We also give an interpretation of the obtained results in terms of random walks in random environment.