Limit theorems for discrete multitype branching processes counted with a characteristic

Konrad Kolesko and Ecaterina Sava-Huss

Stochastic Processes and their Applications, 2023, vol. 162, issue C, 49-75

Abstract: For a discrete time multitype supercritical Galton–Watson process (Zn)n∈N and corresponding genealogical tree T, we associate a new discrete time process (ZnΦ)n∈N such that, for each n∈N, the contribution of each individual u∈T to ZnΦ is determined by a (random) characteristic Φ evaluated at the age of u at time n. In other words, ZnΦ is obtained by summing over all u∈T the corresponding contributions Φu, where (Φu)u∈T are i.i.d. copies of Φ. Such processes are known in the literature under the name of Crump–Mode–Jagers (CMJ) processes counted with characteristicΦ. We derive a LLN and a CLT for the process (ZnΦ)n∈N in the discrete time setting, and in particular, we show a dichotomy in its limit behavior. By applying our main result, we also obtain a generalization of the results in Kesten and Stigum (1966).

Keywords: Branching process; Characteristics; Fluctuations; Martingale; Kesten–Stigum; Spectrum (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2023.04.009

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