Population size of critical Galton-Watson processes under small deviations and infinite variance

TL;DR

This paper investigates the behavior of critical Galton-Watson processes with infinite variance, focusing on the probability and characteristics of the population being unusually small at large time scales.

Contribution

It provides new insights into the distribution of population sizes under small deviations in processes with infinite variance offspring distributions.

Findings

Characterizes the asymptotic behavior of small deviations

Derives probability estimates for small population sizes

Extends understanding of critical processes with infinite variance

Abstract

We study the evolution of the population size distribution of a critical Galton-Watson process with infinite variance of the offspring size of particles assuming that the population size is unusually small at the distant moment $n$ of observation.