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

Abstract

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 $\mm_n$.