On the rate of convergence of a regular martingale related to a branching random walk

DOI10.1007/S11253-006-0072-YzbMATH Open1097.60013arXivmath/0604440OpenAlexW2964227622MaRDI QIDQ5487605

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Publication date: 19 September 2006

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Abstract: Let $m{m}_n,n=0,1,\mathrm{...}$ be the supercritical branching random walk, in which the number of direct descendants of one individual may be infinite with positive probability. Assume that the standard martingale $W_n$ related to $m{m}_n$ is regular, and $W$ is a limit random variable. Let $a(x)$ be a nonnegative function which regularly varies at infinity, with exponent greater than -1. The paper presents sufficient conditions of the almost sure convergence of the series $su{m}_{n=1}^{infty}a(n)(W-W_n)$. Also we establish a criterion of finiteness of $meWlo{g}^{+}Wa(lo{g}^{+}W)$ and $melo{g}^{+}|zi|a(lo{g}^{+}|zi|)$, where $zi:=Q_1+su{m}_{n=2}^{i}nfty{M}_1\mathrm{...}{M}_n{Q}_{n+1}$, and $(M_n,Q_n)$ are independent identically distributed random vectors, not necessarily related to $m{m}_n$.

Full work available at URL: https://arxiv.org/abs/math/0604440

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