Divergence of a Random Walk Through Deterministic and Random Subsequences

Let {S n} n≥0 be a random walk on the line. We give criteria for the existence of a nonrandom sequence n i → ∞ for which $$S_{n_i } \xrightarrow{P}\infty $$ respectively $$S_{n_i } /n_i \xrightarrow{P}\infty $$ 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 $$S_{T(a_i )} \xrightarrow{P}\infty $$ for some sequence a i→ ∞, where T(a) is the exit time from the interval [−a,a]. We also obtain a general equivalence between $$S_{n_i } /f(n_i )\xrightarrow{P}\infty $$ and $$S_{T(a_i )} /f(T(a_i ))\xrightarrow{P}\infty $$ 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 $$S_n \xrightarrow{P}\infty $$ and $$S_n /n\xrightarrow{P}\infty $$ (divergence through the whole sequence n) are also simplified.