文章基本信息

标题：Critical branching processes in random environment and Cauchy domain of attraction
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作者：C. Dong ; C. Smadi ; V. A. Vatutin 等
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2020
卷号：17
期号：2
页码：877
DOI：10.30757/ALEA.v17-34
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment and in the growth rate of the population given its survival up to a large time n. We assume that the random walk associated with the branching process is oscillating and satisfies a Doney-Spitzer condition P(Sn > 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 an asymmetric stable law with parameter 1, which implies that ρ = 0 or 1. We find the asymptotic behaviour of the survival probability of the population and prove a Yaglom-type conditional limit theorem for the population size in these two cases.
其他关键词：Branching process, random environment, random walk, conditioned random walk, Spitzer’s condition