Critical Galton-Watson branching processes with countably infinitely many types and infinite second moments

TL;DR

This paper studies a critical Galton-Watson process with infinitely many types and infinite variance, analyzing its survival probability and establishing a limit theorem for the distribution of particles across types.

Contribution

It extends the theory of Galton-Watson processes to infinite types with infinite variance, providing new asymptotic results and a Yaglom-type limit theorem.

Findings

Derived asymptotic behavior of survival probability

Established a Yaglom-type limit theorem for the process

Analyzed infinite-dimensional distribution of particles

Abstract

We consider an indecomposable Galton-Watson branching process with countably infinitely many types. Assuming that the process is critical and allowing for infinite variance of the offspring sizes of some (or all) types of particles we describe the asymptotic behavior of the survival probability of the process and establish a Yaglom-type conditional limit theorem for the infinite-dimensional vector of the number of particles of all types.