On the rate of convergence of a regular martingale related to a branching random walk - Iksanov, O. M. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
Let M_n,n = 1, 2, ..., be a supercritical branching random walk in which a number of direct descendants of an individual may be infinite with positive probability. Assume that the standard martingale W_n related to 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. We present sufficient conditions of almost sure convergence of the series ^∞_n=1a(n)(W - W_n). We also establish a criteria of finiteness of EW ^+Wa(ln+W) and EW ^+|Z_∞|a(ln+|Z_∞|), where Z_∞ = Q_1 + ^∞_n=2M_1 ... M_nQ_n+1 and (M_n, Q_n) are independent identically distributed random vectors, not necessarily related to M_n.