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

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.