Convergence rates in the law of large numbers for martingales

Gerold Alsmeyer

Stochastic Processes and their Applications, 1990, vol. 36, issue 2, 181-194

Abstract: In this paper we extend well-known results by Baum and Katz (1965) and others on the rate of convergence in the law of large numbers for sums of i.i.d. random variables to general zero-mean martingales S. For , p>1/[alpha] and f(x) = x (two-sided case) OR = x+ or x- (one-sided case), it is e.g. shown that if, for some [gamma] [epsilon] (1/[alpha], 2] and q>(p[alpha] - 1)/([gamma][alpha] - 1), and an additional mixing condition holds in the one-sided case, then holds iff , X1, X2, ... being the increments of S. The latter condition reduces to the well-known moment condition Ef(X1)p

Date: 1990
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