Parameterised branching processes: A functional version of Kesten & Stigum theorem

Cécile Mailler, Jean François Marckert

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Abstract

Let (Zn,n≥0) be a supercritical Galton–Watson process whose offspring distribution μ has mean λ>1 and is such that ∫xlog+(x)dμ(x)<+∞. According to the famous Kesten & Stigum theorem, (Znn) converges almost surely, as n→+∞. The limiting random variable has mean 1, and its distribution is characterised as the solution of a fixed point equation. In this paper, we consider a family of Galton–Watson processes (Zn(λ),n≥0) defined for λ ranging in an interval I⊂(1,∞), and where we interpret λ as the time (when n is the generation). The number of children of an individual at time λ is given by X(λ), where (X(λ))λ∈I is a càdlàg integer-valued process which is assumed to be almost surely non-decreasing and such that E(X(λ))=λ>1 for all λ∈I. This allows us to define Zn(λ) the number of elements in the nth generation at time λ. Set Wn(λ)=Zn(λ)/λn for all n≥0 and λ∈I. We prove that, under some moment conditions on the process X, the sequence of processes (Wn(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.

Original languageEnglish
Pages (from-to)339-377
Number of pages39
JournalStochastic Processes and their Applications
Volume152
Early online date16 Jul 2022
DOIs
Publication statusPublished - 31 Oct 2022

Bibliographical note

Funding Information:
CM is grateful to EPSRC, UK for support through the fellowship EP/R022186/1.

Keywords

  • Functional limit theorems
  • Galton–Watson trees
  • Martingale limit

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics

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