LAW OF THE ITERATED LOGARITHM FOR SUBSEQUENCES BY

Abstract. Let {S,},"=, denote the partial sums of i.i.d. random variables with mean 0. The present paper investigates the quantity lim sup snjJI1, log log n,, k-m where { n, ] z, is a strictly increasing subsequence of the positive integers. The first results are that if E X; < co, then the limit superior equals a & a.s. for subsequences which increase "at most geo~~etrically", and m*, where E * = inf { E> 0; (log nfi)-"/2 < GO}, k for subsequences which increase "at least geometrically". We also perform a refined analysis for the latter case and finally present criteria for the finiteness o f, in both cases. 1. Introduction Let (X,),"