Convergence of complex martingales in supercritical multi-type general branching processes in L^q for 1 < q ≤ 2

2025/07/29 by Kolesko, Konrad, Meiners, Matthias, Tomic, Ivana
#60J80 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2507.21887

Abstract

Nerman's martingale plays a central role in the law of large numbers for both, single- and multi-type, supercritical general branching processes. There are further, complex-valued Nerman-type martingales in the single-type process that figure in the finer fluctuations of these processes. We construct the analogous martingales for the process with finitely many types and give sufficient conditions for these martingales to converge in L^q for q ∈ (1,2].

Citations

• Asymptotic expansions of solutions to Markov renewal equations and their application to general branching processes
• Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes
• Fluctuations of Biggins' martingales at complex parameters

Related

• Gaussian fluctuations and a law of the iterated logarithm for Nerman's martingale in the supercritical general branching process
• Fluctuations of Biggins' martingales at complex parameters
• On L^p-convergence of the Biggins martingale with complex parameter
• Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes
• Complete left tail asymptotic for supercritical multitype branching processes