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

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 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 $n$ ( fixed, close to $n$, or going to infinity but far from $n$).

version, as published in the Proc. Steklov Inst. Math. arXiv admin note: text overlap with arXiv:1911.00316

References in corpus (4)

• Criticality for branching processes in random environment
• Limit theorems for weakly subcritical branching processes in random environment
• Branching processes in random environment with immigration stopped at zero
• Subcritical branching processes in random environment with immigration: survival of a single family