Critical branching processes in random environment and Cauchy domain of attraction

2019/10/29 by Dong, Congzao, Smadi, Charline, Vatutin, Vladimir A.
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1910.13190

Abstract

We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment. We assume that the random walk associated with the branching process is oscillating and satisfies a Spitzer condition P(S_n>0)→ ρ, n→ ∞ , which is a standard condition in fluctuation theory of random walks. Unlike the previously studied case ρ∈ (0,1), we investigate the case where the offspring distribution is in the domain of attraction of a stable law with parameter 1, which implies that ρ=0 or 1. We find the asymptotic behaviour of the survival probability of the population in these two cases.

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