Critical branching processes in random environment with immigration: the size of the only surviving family

TL;DR

This paper investigates the size distribution of the surviving family in a critical branching process within a random environment with immigration, focusing on the conditional distribution given the family origin over large time scales.

Contribution

It provides a detailed analysis of the conditional distribution of the process given the family origin event, considering various asymptotic regimes for the family’s origin time.

Findings

Characterizes the distribution of the process conditioned on family origin.

Analyzes different asymptotic regimes for the family’s origin time.

Provides insights into the survival structure of the process.

Abstract

We consider a critical branching process $Y_{n}$ in an i.i.d. random environment, in which one immigrant arrives at each generation. Let $% \mathcal{A}_{i}(n)$ be the event that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$. We study the conditional distribution of $Y_{n}$ given $\mathcal{A}_{i}(n)$ 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 $n$).