Subcritical branching processes in random environment with immigration: survival of a single family

2020/09/05 by E. E. Dyakonova, Vladimir Vatutin, Dyakonova, E. E. +1
Economics, Econometrics and Finance · Mathematics · Medicine · #60G50 (Secondary) Primary 60J80 #60J80 (Primary) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #Secondary 60G50 #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2009.03672

openalex publication_date 2020/09/05 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We consider a subcritical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We consider the event % A_i(n) that all individuals alive at time n are offspring of the immigrant which joined the population at time i and investigate the asymptotic probability of this extreme event when n→∞ and i is either fixed, or the difference n-i is fixed, or min(i,n-i)→∞. To deduce the desired asymptotics we establish some limit theorems for random walks conditioned to be nonnegative or negative.

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