Divergence of a Random Walk Through Deterministic and Random Subsequences

Let {S_n}_n0 be a random walk on the line. We give criteria for the existence of a nonrandom sequence n_i for which respectively We thereby obtain conditions for to be a strong limit point of {S_n} or {S_n/n}. The first of these properties is shown to be equivalent to for some sequence a_i , where T(a) is the exit time from the interval [–a,a]. We also obtain a general equivalence between and for an increasing function fand suitable sequences n_i and a_i. These sorts of properties are of interest in sequential analysis. Known conditions for and (divergence through the whole sequence n) are also simplified.