Path to survival for the critical branching processes in a random\n environment

2016/03/10 by Vladimir Vatutin, Vatutin, Vladimir, Elena Dyakonova +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #60F17 #60G50 #Diffusion and Search Dynamics #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Primary 60J80 #Probability (math.PR) #Stochastic processes and statistical mechanics #secondary 60K37

paper · pdf · doi:10.48550/arxiv.1603.03199

openalex publication_date 2016/03/10 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

A critical branching process \ Z_k,k=0,1,2,... \ in a random\nenvironment is considered. A conditional functional limit theorem for the\nproperly scaled process \ \log Z_pu,0\≤ u<\∞ \ is\nestablished under the assumptions Z_n>0 and p\≪ n. It is shown that the\nlimiting process is a Levy process conditioned to stay nonnegative. The proof\nof this result is based on a limit theorem describing the distribution of the\ninitial part of the trajectories of a driftless random walk conditioned to stay\nnonnegative.\n

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