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Convergence rates in the law of large numbers for martingales

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  • Alsmeyer, Gerold

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

Suggested Citation

  • Alsmeyer, Gerold, 1990. "Convergence rates in the law of large numbers for martingales," Stochastic Processes and their Applications, Elsevier, vol. 36(2), pages 181-194, December.
    • Handle: RePEc:eee:spapps:v:36:y:1990:i:2:p:181-194
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    Cited by:

    1. Rüdiger Kiesel, 1998. "Strong Laws and Summability for φ-Mixing Sequences of Random Variables," Journal of Theoretical Probability, Springer, vol. 11(1), pages 209-224, January.
    2. Fuh, Cheng-Der & Zhang, Cun-Hui, 2000. "Poisson equation, moment inequalities and quick convergence for Markov random walks," Stochastic Processes and their Applications, Elsevier, vol. 87(1), pages 53-67, May.

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