A special family of Galton-Watson processes with explosions

2015/02/26 by Serik Sagitov, Sagitov, Serik, Alexey Lindo +1 · 2 citations
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1502.07538

openalex publication_date 2015/02/26 · arxiv created 2015/12/10 · arxiv updated 2015/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The linear-fractional Galton-Watson processes is a well known case when many characteristics of a branching process can be computed explicitly. In this paper we extend the two-parameter linear-fractional family to a much richer four-parameter family of reproduction laws. The corresponding Galton-Watson processes also allow for explicit calculations, now with possibility for infinite mean, or even infinite number of offspring. We study the properties of this special family of branching processes, and show, in particular, that in some explosive cases the time to explosion can be approximated by the Gumbel distribution.

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