Reinforced Galton--Watson processes III: Empirical offspring distributions

TL;DR

This paper investigates the long-term behavior of reinforced Galton--Watson processes, focusing on empirical offspring distributions, and employs large deviations theory to analyze properties like concentration, evanescence, and persistence.

Contribution

It extends previous work by applying large deviations techniques to analyze the asymptotic properties of empirical offspring distributions in reinforced Galton--Watson processes.

Findings

Analysis of concentration properties of empirical distributions

Conditions for evanescence and persistence identified

Application of large deviations to reinforced branching processes

Abstract

Reinforced Galton--Watson processes describe the dynamics of a population where reproduction events are reinforced, in the sense that offspring numbers of forebears can be repeated randomly by descendants. More specifically, the evolution depends on the empirical offspring distribution of each individual along its ancestral lineage. We are interested here in asymptotic properties of the empirical distributions observed in the population, such as concentration, evanescence and persistence. For this, we incorporate tools from the theory of large deviations to our preceding analysis [arXiv:2306.02476,arXiv:2310.19030].