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

TL;DR

This paper analyzes the survival probability of a single immigrant in a critical branching process within a random environment, establishing asymptotic behaviors and new limit theorems for related random walks.

Contribution

It introduces new asymptotic results for the probability that all individuals at a large time descend from a single immigrant, with novel limit theorems for associated random walks.

Findings

Asymptotic probabilities depend on the timing of immigrant arrivals.

Conditional limit theorems for random walks are established.

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

Abstract

We consider a critical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We are interested in the event $\mathcal{A}_i(n)$ that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$. We study the asymptotic probability of this event when $n$ is large and $i$ follows different asymptotics which may be related to $n$ ($i$ fixed, close to $n$, or going to infinity but far from $n$). In order to do so, we establish some conditional limit theorems for random walks, which are of independent interest.