On largest offsprings in a critical branching process with finite variance

2012/09/18 by Jean Bertoin, Bertoin, Jean
Computer Science · Economics, Econometrics and Finance · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1209.3854

arxiv created 2012/09/18 · openalex publication_date 2012/09/18 · arxiv updated 2012/09/19 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

We continue our study of the distribution of the maximal number X^∗_k of offsprings amongst all individuals in a critical Galton-Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F with index -α for α>2 (and hence finite variance). We show that X^∗_k suitably normalized converges in distribution to a Frechet law with shape parameter α/2; this contrasts sharply with the case 1<α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in the decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

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