Central limit theorem for supercritical binary homogeneous Crump-Mode-Jagers processes

TL;DR

This paper establishes a central limit theorem for supercritical binary homogeneous Crump-Mode-Jagers processes, analyzing the convergence error using renewal theory and Lévy processes, and providing precise asymptotic behavior of moments.

Contribution

It introduces a new decomposition of splitting trees and applies Lévy process theory to derive a CLT for the convergence error in these branching processes.

Findings

Convergence error moments are explicitly characterized.

A new decomposition of splitting trees is proposed.

A central limit theorem for the process is proved.

Abstract

We consider a supercritical general branching population where the lifetimes of individuals are i.i.d. with arbitrary distribution and each individual gives birth to new individuals at Poisson times independently from each others. The population counting process of such population is a known as binary homogeneous Crump-Jargers-Mode process. It is known that such processes converges almost surely when correctly renormalized. In this paper, we study the error of this convergence. To this end, we use classical renewal theory and recent works [17, 6, 5] on this model to obtain the moments of the error. Then, we can precisely study the asymptotic behaviour of these moments thanks to L{\'e}vy processes theory. These results in conjunction with a new decomposition of the splitting trees allow us to obtain a central limit theorem. MSC 2000 subject classifications: Primary 60J80; secondary 92D10, 60J85, 60G51, 60K15, 60F05.