Limit distributions of branching Markov chains

TL;DR

This paper investigates the long-term behavior of branching Markov chains on countable spaces, establishing conditions for convergence of population distributions and linking them to stationary spaces of the underlying Markov chain.

Contribution

It introduces new conditions for the uniform integrability of the population martingale and connects the convergence of population averages to stationary spaces of the associated Markov chain.

Findings

Population martingale is uniformly integrable.

Convergence of population averages linked to stationary spaces.

Applied to boundaries of compactifications of the state space.

Abstract

We study branching Markov chains on a countable state space (space of types) $\mathscr{X}$, with the focus on the qualitative aspects of the limit behaviour of the evolving empirical population distributions. No conditions are imposed on the multitype offspring distributions at the points of $\mathscr{X}$ other than to have the same average and to satisfy a uniform $L \log L$ moment condition. We show that the arising population martingale is uniformly integrable. Convergence of population averages of the branching chain is then put in connection with stationary spaces of the associated ordinary Markov chain on $\mathscr{X}$ (assumed to be irreducible and transient). This is applied, in particular, to the boundaries of appropriate compactifications of $\mathscr{X}$. Final considerations consider the general interplay between the measure theoretic boundaries of the branching chain and the associated ordinary chain.