Sudden extinction of a critical branching process in random environment

2008/09/05 by Wachtel, V. A. Vatutin V.
#60G50 #60J80 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0809.0986

Abstract

Let T be the extinction moment of a critical branching process Z=(Z_n,n≥ 0) in a random environment specified by iid probability generating functions. We study the asymptotic behavior of the probability of extinction of the process Z at moment n→ ∞, and show that if the logarithm of the (random) expectation of the offspring number belongs to the domain of attraction of a non-gaussian stable law then the extinction occurs owing to very unfavorable environment forcing the process, having at moment T-1 exponentially large population, to die out. We also give an interpretation of the obtained results in terms of random walks in random environment.

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