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

2025/07/13 by Ion Grama, Grama, Ion, Sebastian Mentemeier +3
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2507.09737

openalex publication_date 2025/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 √(n) W_n → D_∞ a.s., where D_∞ 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.

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