Surviving particles for subcritical branching processes in random environment |
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Authors: | Vincent Bansaye |
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Institution: | Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie et C.N.R.S. UMR 7599, 175, rue du Chevaleret, 75 013 Paris, France |
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Abstract: | The asymptotic behavior of a subcritical Branching Process in Random Environment (BPRE) starting with several particles depends on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS). 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 new results on random walks with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition on 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. |
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Keywords: | 60J80 60J85 60K37 |
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