On Branching Processes in Random Environments

1969/06/01 by Walter L. Smith, William E. Wilkinson · 21 citations
Mathematics · Biochemistry, Genetics and Molecular Biology · #Stochastic processes and statistical mechanics #Diffusion and Search Dynamics #Mathematical Dynamics and Fractals

paper · doi:10.1214/aoms/1177697589

Abstract

\ζ_n\ is a sequence of iid "environmental" variables in an abstract space Θ. Each point ζ ε Θ is associated with a pgf φ_ζ(s). The branching process \Z_n\ is defined as a Markov chain such that Z₀ = k, a finite integer, and given Z_n and ζ_n, Z_n+1 is distributed as the sum of Z_n iid random variables, each with pgf φ_ζn(s). Set ξ(ζ_n) = φ'_ζn(1) and assume that E|logξ(ζ_n)| < ∞. Then: (i) P\Z_n = 0\ → 1 if E log ξ(ζ_n) \leqq 0; (ii) qk = _def lim P\Z_n = 0\ < 1 if E log ξ(ζ_n) > 0 and E|log (1 - φ_ζn(0))| < ∞. Furthermore \q_k\, k = 1, 2, ⋯, forms a moment sequence.

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• Multitype branching processes with immigration in random environment and\n polling systems
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• Perturbing transient Random Walk in a Random Environment with cookies of\n maximal strength
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