Limit theorems for subcritical Branching Process in Random Environment depending on the initial number of particles

Asymptotic behaviors for subcritical Branching Processes in Random Environment (BPRE) starting with several particles depend on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS) (see [12]). Descendances of particles for BPRE are not independent. In the (SS+IS) case, the asymptotic probability of survival is proportional to the initial number of particles. And conditionally on the survival of the population, only one initial particle survives a.s. These two properties do not hold in the (WS) case and different asymptotics are established, which require to prove new results on random walk with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition by the survival of particles. This also raises the problem of the dependence of the Yaglom quasistationary distributions on the initial number of particles and the asymptotic behavior of the Q-process associated with a subcritical BPRE.