On largest offsprings in a critical branching process with finite variance

TL;DR

This paper investigates the distribution of the maximum number of offsprings in a critical Galton-Watson process with finite variance, revealing convergence to a Frechet law and contrasting behavior with infinite variance cases.

Contribution

It extends the understanding of offspring distribution extremes in critical branching processes with regularly varying tails, especially for finite variance cases.

Findings

Max offspring distribution converges to a Frechet law with shape α/2.

Contrasts with infinite variance case where variance is infinite.

Provides a weak limit theorem for ranked offspring sequences.

Abstract

We continue our study of the distribution of the maximal number $X^{\ast}_k$ 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 $\bar F$ with index $-\alpha$ for $\alpha>2$ (and hence finite variance). We show that $X^{\ast}_k$ suitably normalized converges in distribution to a Frechet law with shape parameter $\alpha/2$; this contrasts sharply with the case $1<\alpha <2$ 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.