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A self-normalized Erdos--Rényi type strong law of large numbers

Author

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  • Csörgo, Miklós
  • Shao, Qi-Man

Abstract

The original Erdos--Rényi theorem states that max0[less-than-or-equals, slant]k[less-than-or-equals, slant]n[summation operator]k+[clogn]i=k+1Xi/[clogn]-->[alpha](c),c>0, almost surely for i.i.d. random variables {Xn, n[greater-or-equal, slanted]1} with mean zero and finite moment generating function in a neighbourhood of zero. The latter condition is also necessary for the Erdos--Rényi theorem, and the function [alpha](c) uniquely determines the distribution function of X1. We prove that if the normalizing constant [c log n] is replaced by the random variable [summation operator]k+[clogn]i=k+1(X2i+1), then a corresponding result remains true under assuming only the exist first moment, or that the underlying distribution is symmetric.

Suggested Citation

  • Csörgo, Miklós & Shao, Qi-Man, 1994. "A self-normalized Erdos--Rényi type strong law of large numbers," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 187-196, April.
    • Handle: RePEc:eee:spapps:v:50:y:1994:i:2:p:187-196
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    Cited by:

    1. Xia Chen, 1999. "The Law of the Iterated Logarithm for Functionals of Harris Recurrent Markov Chains: Self Normalization," Journal of Theoretical Probability, Springer, vol. 12(2), pages 421-445, April.

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