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

摘要

Let $\mm_n, n=0,1,...$ be the supercritical branching random walk, in whichthe number of direct descendants of one individual may be infinite withpositive 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 anonnegative function which regularly varies at infinity, with exponent greaterthan -1. The paper presents sufficient conditions of the almost sureconvergence of the series $\sum_{n=1}^{\infty}a(n)(W-W_n)$. Also we establish acriterion 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, notnecessarily related to $\mm_n$.