Critical branching processes in random environment with immigration:\n survival of a single family

2019/11/01 by Charline Smadi, Vladimir Vatutin, Smadi, Charline +1
Mathematics · Medicine · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1911.00316

openalex publication_date 2019/11/01 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We consider a critical branching process in an i.i.d. random environment, in\nwhich one immigrant arrives at each generation. We are interested in the event\n\A_i(n) that all individuals alive at time n are offspring of the\nimmigrant which joined the population at time i. We study the asymptotic\nprobability of this event when n is large and i follows different\nasymptotics which may be related to n (i fixed, close to n, or going to\ninfinity but far from n). In order to do so, we establish some conditional\nlimit theorems for random walks, which are of independent interest.\n

Related

• Subcritical branching processes in random environment with immigration: survival of a single family
• A critical branching process with immigration in random environment
• Subcritical branching processes in random environment with immigration stopped at zero
• Branching processes in random environment with immigration stopped at zero
• Asymptotic Behaviors for Critical Branching Processes with Immigration