Divergence of a random walk through deterministic and random subsequences

Let {S-n}(n greater than or equal to 0) be a random walk on the line. We give criteria for the existence of a nonrandom sequence n(i) --> infinity for which [GRAPHICS] respectively [GRAPHICS] We thereby obtain conditions for infinity to be a strong limit point of {S,} or {S-n/n}. The first of these properties is shown to be equivalent to [GRAPHICS] for some sequence a(i) --> infinity, where T(a) is the exit time from the interval [-a, a]. We also obtain a general equivalence between [GRAPHICS] and [GRAPHICS] for an increasing function f and suitable sequences n(i) and a(i). These sorts of properties are of interest in sequential analysis. Known conditions for [GRAPHICS] and [GRAPHICS] (divergence through the whole sequence n) are also simplified.