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

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 $n$).

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