Iterated Logarithm Inequalities

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, \ (x\) n = S n/n. For any sequence of positive constants a n, n ≥ 1, let P m = P (|\ (x\) n| ≥ a n for some n ≥ m).