Some results on two-sided LIL behavior

Abstract

Let {X,X_n;n≥1} be a sequence of i.i.d. mean-zero random variables, and let S_n=∑_i=1ⁿX_i,n≥1. We establish necessary and sufficient conditions for having with probability 1, 0<lim sup _n→∞|S_n|/$\sqrt{nh(n)}$<∞, where h is from a suitable subclass of the positive, nondecreasing slowly varying functions. Specializing our result to h(n)=(loglogn)^p, where p>1 and to h(n)=(logn)^r, r>0, we obtain analogues of the Hartman–Wintner LIL in the infinite variance case. Our proof is based on a general result dealing with LIL behavior of the normalized sums {S_n/c_n;n≥1}, where c_n is a sufficiently regular normalizing sequence.