Subcritical branching processes in random environment with immigration: survival of a single family

TL;DR

This paper analyzes the asymptotic probability of all individuals at a large time being descendants of a single immigrant in a subcritical branching process within a random environment, using limit theorems for conditioned random walks.

Contribution

It introduces new asymptotic results for the probability that all individuals originate from a single immigrant in a subcritical process with immigration, extending understanding of such processes.

Findings

Asymptotic probabilities are derived for various regimes of the immigration time.

Limit theorems for conditioned random walks are established.

Results provide insights into the genealogy of populations in random environments.

Abstract

We consider a subcritical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We consider the event $% \mathcal{A}_{i}(n)$ that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$ and investigate the asymptotic probability of this extreme event when $n\to\infty$ and $i$ is either fixed, or the difference $n-i$ is fixed, or $\min(i,n-i)\to\infty.$ To deduce the desired asymptotics we establish some limit theorems for random walks conditioned to be nonnegative or negative.