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

Abstract

Let M n , n = 1, 2, ..., be a supercritical branching random walk in which the 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 regularly varying at infinity with index greater than −1. We present sufficient conditions for the almost-sure convergence of the series $$\sum\nolimits_{n = 1}^\infty {a(n)(W - W_n )} $$ . We also establish criteria for the finiteness of EW ln+ Wa(ln+ W) and E ln+|Z ∞|a(ln+|Z ∞|), where $$Z_\infty : = Q_1 + \sum\nolimits_{n = 2}^\infty {M_1 \ldots M_n Q_{n + 1} } $$ and (M n , Q n ) are independent identically distributed random vectors not necessarily related to M n .