Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case

TL;DR

This paper investigates the tail behavior of the limit and scaled versions of a supercritical Galton-Watson process with heavy-tailed offspring distribution, revealing how different distributions influence asymptotic tail behavior.

Contribution

It provides a detailed analysis of tail asymptotics for the process in the heavy-tailed case, highlighting the impact of offspring distribution types on tail behavior.

Findings

Different offspring distributions lead to distinct tail asymptotics.

The paper characterizes the most probable scenarios for large deviations.

It extends understanding of supercritical processes with heavy-tailed offspring.

Abstract

As well known, for a supercritical Galton-Watson process $Z_n$ whose offspring distribution has mean $m>1$, the ratio $W_n:=Z_n/m^n$ has a.s. limit, say $W$. We study tail behaviour of the distributions of $W_n$ and $W$ in the case where $Z_1$ has heavy-tailed distribution, that is, $\E e^{\lambda Z_1}=\infty$ for every $\lambda>0$. We show how different types of distributions of $Z_1$ lead to different asymptotic behaviour of the tail of $W_n$ and $W$. We describe the most likely way how large values of the process occur.