Critical Branching Processes in a Random Environment with Immigration: The Size of the Only Surviving Family

We consider a critical branching process $\{Y_n,\,n\geq 0\}$ 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 $n$).