Convergence of complex martingales in the branching random walk: the\n boundary

2016/11/16 by Konrad Kolesko, Matthias Meiners, Kolesko, Konrad +1
Mathematics · Physics and Astronomy · #60J80 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1611.05220

openalex publication_date 2016/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Biggins [Uniform convergence of martingales in the branching random walk.\n em Ann. Probab., 20(1):137--151, 1992] proved local uniform convergence of\nadditive martingales in d-dimensional supercritical branching random walks at\ncomplex parameters \λ from an open set \Λ \⊆ \ℂ^d.\nWe investigate the martingales corresponding to parameters from the boundary\n\∂ \Λ of \Λ. The boundary can be decomposed into several\nparts. There may be a part of the boundary, on which the martingales do not\nexist, on other parts it exists, but diverges or vanishes in the limit. In the\nremaining part, there is convergence to a non-degenerate limit. The arguments\nthat give this convergence also apply in \Λ and require weaker moment\nassumptions than the ones used by Biggins.\n

Citations

• Fixed points of multivariate smoothing transforms with scalar weights

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