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

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.