Parametrised branching processes: a functional version of Kesten & Stigum theorem

2021/06/02 by Cécile Mailler, Mailler, Cécile, Jean‐François Marckert +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2106.01426

openalex publication_date 2021/06/02 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

Let (Z_n,n≥ 0) be a supercritical Galton-Watson process whose offspring distribution μ has mean λ>1 and is such that ∫ x(log(x))₊ dμ(x)1 for all λ∈ I. This allows us to define Z_n(λ) the number of elements in the nth generation at time λ. Set W_n(λ)= Z_n(λ)/λⁿ for all n≥ 0 and λ∈ I. We prove that, under some moment conditions on the process~X, the sequence of processes (W_n(λ), λ∈ I)_{n≥ 0} converges in probability as~n tends to infinity in the space of càdlàg processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.

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