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 $% $\mathcal{A}$_{i}(n)$betheeventthatallindividualsaliveattime$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 $n$).