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

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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 .

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 326–342, March, 2006.

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Iksanov, O.M. On the rate of convergence of a regular martingale related to a branching random walk. Ukr Math J 58, 368–387 (2006). https://doi.org/10.1007/s11253-006-0072-y

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  • DOI: https://doi.org/10.1007/s11253-006-0072-y

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