Sharp L log L condition for supercritical Galton-Watson processes with countable types

2025/03/07 by André, Mathilde, Duchamps, Jean-Jil · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.05575

Abstract

We investigate Kesten-Stigum-like results for multi-type Galton-Watson processes with a countable number of types in a general setting, allowing us in particular to consider processes with an infinite total population at each generation. Specifically, a sharp Llog L condition is found under the only assumption that the mean reproduction matrix is positive recurrent in the sense of Vere-Jones (1967). The type distribution is shown to always converge in probability in the recurrent case, and under conditions covering many cases it is shown to converge almost surely.

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