Spinal decomposition, martingale convergence and the Seneta-Heyde scaling for matrix branching random walks

TL;DR

This paper extends martingale convergence and spinal decomposition results to matrix branching random walks, establishing a Seneta-Heyde scaling and providing duality results for renewal measures of Markov processes.

Contribution

It introduces an analogue of Biggins' martingale convergence theorem, a spinal decomposition, and Seneta-Heyde scaling for matrix branching random walks, along with explicit duality results for renewal measures.

Findings

Martingale convergence theorem for matrix branching random walks

Spinal decomposition theorem established

Seneta-Heyde scaling proven in the boundary case

Abstract

We consider a matrix branching random walk on the semi-group of nonnegative matrices, where we are able to derive, under general assumptions, an analogue of Biggins' martingale convergence theorem for the additive martingale $W_n$, a spinal decomposition theorem, convergence of the derivative martingale $D_n$, and finally, the Seneta-Heyde scaling stating that in the boundary case $c \sqrt{n} W_n \to D_\infty$ a.s., where $D_\infty$ is the limit of the derivative martingale and $c$ is a positive constant. As an important tool that is of interest in its own right, we provide explicit duality results for the renewal measure of centered Markov random walks, relating the renewal measure of the process, killed when the random walk component becomes negative, to the renewal measure of the ascending ladder process.