Divergence of a Random Walk Through Deterministic and Random Subsequences

Abstract

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.