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

We consider a critical branching process in an i.i.d. random environment, which one immigrant arrives at each generation. are interested the event $\mathcal {A}_{i}(n)$ that all individuals alive time n offsprings of joined population i. study asymptotic probability this extreme when is large and i follows different asymptotics may be related to (i fixed, close n, or going infinity but far from n). In order do so, we establish some limit theorems for walks conditioned stay positive nonnegative, independent interest.