More on the asymptotic behaviour of moments of branching Markov processes

TL;DR

This paper investigates the long-term behavior of moments in branching Markov processes with non-local mechanisms, extending previous results to more complex eigenvalues and higher order fluctuations, aiding in limit theorem proofs.

Contribution

It generalizes existing asymptotic moment results to include non-simple eigenvalues and analyzes higher order fluctuations of the process.

Findings

Extended asymptotic moment analysis to non-simple eigenvalues.

Studied higher order fluctuations of the moments.

Provided tools for proving central limit theorems and LLN extensions.

Abstract

Consider a branching Markov process, $X = (X(t), t \ge 0)$, with non-local branching mechanism. Studying the asymptotic behaviour of the moments of X has recently received attention in the literature [6, 7] due to the importance of these results in understanding the underlying genealogical structure of $X$. In this article, we generalise the results of [7] to allow for a non-simple leading eigenvalue and to also study the higher order fluctuations of the moments of $X$. These results will be useful for proving central limit theorems and extending well-known LLN results.