Functional law of large numbers and central limit theorem for Crump-Mode-Jagers branching processes

TL;DR

This paper proves a Law of Large Numbers and a Central Limit Theorem for age-dependent, random Crump-Mode-Jagers branching processes, extending understanding of their asymptotic behavior in large populations.

Contribution

It establishes the CLT for Crump-Mode-Jagers processes with age-dependent, random birth rates, addressing tightness in Skorohod space, which was previously unresolved.

Findings

Law of Large Numbers proven for the process

Central Limit Theorem established with tightness in Skorohod space

Methodology based on Hahn's criterion for CLT in D

Abstract

We establish a Law of Large Numbers and a Central Limit Theorem for a class of Crump Mode Jagers continuous time branching processes, where the birth rate is age dependent, and also random (different from one individual to the next), in the limit of a large number of ancestors. The only difficulty concerns the tightness in the Skorohod space $D$ for the central limit theorem. We exploit a criterion for the CLT in $D$ due to M. Hahn \cite{MH}.