Branching within branching II: Limit theorems

TL;DR

This paper establishes limit theorems for a complex branching-within-branching model in host-parasite co-evolution, focusing on asymptotic behaviors of contaminated cells and parasites using martingale techniques.

Contribution

It extends previous work by deriving Kesten-Stigum type limit theorems and norming results for the model, including cases where integrability conditions fail.

Findings

Limit theorems for contaminated cells and parasites are established.

Martingale and size-biasing techniques are used to analyze asymptotic behavior.

Conditions for positive martingale limits are characterized.

Abstract

This continues work started in part I on a general branching-within-branching model for host-parasite co-evolution. Here we focus on asymptotic results for relevant processes in the case when parasites survive. In particular, limit theorems for the processes of contaminated cells and of parasites are established by using martingale theory and the technique of size-biasing. The results for both processes are of Kesten-Stigum type by including equivalent integrability conditions for the martingale limits to be positive with positive probability. The case when these conditions fail is also studied. For the process of contaminated cells, we show that a proper Heyde-Seneta norming exists such that the limit is nondegenerate.