Sudden extinction of a critical branching process in random environment

Details Sudden extinction of a critical branching process in random environment Sudden extinction of a critical branching process in random environment

2008-09-05

arxiv.org/abs/0809.0986v1

Mathematics

Probability

Scientific paper

Let $T$ be the extinction moment of a critical branching process $Z=(Z_{n},n\geq 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 $n\to \infty$, 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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