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Abstract: Let $T$ be the extinction moment of a critical branching process$Z=Z {n},n\geq 0 $ in a random environment specified by iid probabilitygenerating functions. We study the asymptotic behavior of the probability ofextinction of the process $Z$ at moment $n\to \infty$, and show that if thelogarithm of the random expectation of the offspring number belongs to thedomain of attraction of a non-gaussian stable law then the extinction occursowing to very unfavorable environment forcing the process, having at moment$T-1$ exponentially large population, to die out. We also give aninterpretation of the obtained results in terms of random walks in randomenvironment.



Author: V.A. Vatutin V. Wachtel

Source: https://arxiv.org/







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