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

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

2006-04-20

arxiv.org/abs/math/0604440v1

The original, Ukrainian variant of the paper was published in Ukrainian Mathematical Journal (2006), 58(3), 326-342

Mathematics

Probability

Scientific paper

Let $\mm_n, n=0,1,...$ 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 $\mm_n$ 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 $\sum_{n=1}^{\infty}a(n)(W-W_n)$. Also we establish a criterion of finiteness of $\me W\log^+ W a(\log^+W)$ and $\me \log^+|\zi| a(\log^+|\zi|)$, where $\zi:=Q_1+\sum_{n=2}^\infty M_1... M_n Q_{n+1}$, and $(M_n, Q_n)$ are independent identically distributed random vectors, not necessarily related to $\mm_n$.

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