A SELF-NORMALIZED ERDOS-RENYI TYPE STRONG LAW OF LARGE NUMBERS

The original Erdos-Renyi theorem states that max(0 less-than-or-equal- to k less-than-or-equal-to n)SIGMA(i=k+1)k+[c log n] X(i)/[c log n] -- > alpha(c), c > 0, almost surely for i.i.d. random variables {X(n), n greater-than-or-equal-to 1) with mean zero and finite moment generatin g function in a neighbourhood of zero. The latter condition is also ne cessary for the Erdos-Renyi theorem, and the function alpha(c) uniquel y determines the distribution function of X1. We prove that if the nor malizing constant [c log n] is replaced by the random variable SIGMA(i =k+1)k+[c log n] (X1(2) + 1), then a corresponding result remains true under assuming only the existence of the first moment, or that the un derlying distribution is symmetric.