A self-normalized Erdo{combining double acute accent}s-Rényi type strong law of large numbers

Abstract

The original Erdo{combining double acute accent}s-Rényi theorem states that max0≤k≤n∑k+[clogn] i=k+1Xi/[clogn]→α(c),c>0, almost surely for i.i.d. random variables {Xn, n≥1} with mean zero and finite moment generating function in a neighbourhood of zero. The latter condition is also necessary for the Erdo{combining double acute accent}s-Rényi theorem, and the function α(c) uniquely determines the distribution function of X1. We prove that if the normalizing constant [c log n] is replaced by the random variable ∑k+[clogn] i=k+1(X2 i+1), then a corresponding result remains true under assuming only the exist first moment, or that the underlying distribution is symmetric. © 1994.