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

We consider a subcritical branching process in an independent and identically distributed (i.i.d.) random environment, where one immigrant arrives at each generation. We consider the event $\mathcal{A}_{i}(n)$ in which all individuals alive at time $n$ are descendants of the immigrant, who joined the population at time $i$, and investigate the asymptotic probability of this extreme event for $n\to \infty$ when $i$ is fixed, 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 on $[0,n]$.