Iterated logarithm inequalities.

1. Introduction—Let x,x 1, x 2 … be a sequence of independent, identically distributed random variables with mean 0, variance 1, and moment generating function ϕ(t) = E(etz) finite in some neighborhood of t= 0, and put S n = x 1, + … + x n, \(\bar x\) n = S n/n. For any sequence of positive constants a n, n ≥ 1, let P m = P(|\(\bar x\) n| ≥ a n for some n ≥ m).