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Abstract
Let (Z_{n},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, (Z_{n}/λ^{n}) 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 (Z_{n}(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 integervalued process which is assumed to be almost surely nondecreasing and such that E(X(λ))=λ>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)/λ^{n} 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.
Original language  English 

Pages (fromto)  339377 
Number of pages  39 
Journal  Stochastic Processes and their Applications 
Volume  152 
Early online date  16 Jul 2022 
DOIs  
Publication status  Published  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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 1 Finished

Fellowship  Random trees: analysis and applications
Engineering and Physical Sciences Research Council
1/06/18 → 31/05/22
Project: Research council