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, in which one immigrant arrives at each generation. We are interested in the event A(i) (n) that all individuals alive at time n are offsprings of the immigrant which joined the population at time i. We study the asymptotic probability of this extreme event 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). In order to do so, we establish some limit theorems for random walks conditioned to stay positive or nonnegative, which are of independent interest.