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

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 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.

20 pages, 14 references. To be published in Theory of Probability and its Applications, 65(4), 2020. arXiv admin note: text overlap with arXiv:1911.00316