The Rate of Escape of Random Walk
William E. Pruitt · The Annals of Probability · 1990
Let $\{X_k\}$ be an i.i.d. sequence and define $S_n = X_1 + \cdots + X_n$. The problem is to determine for a given sequence $\{\beta_n\}$ whether $P\{|S_n| \leq \beta_n \mathrm{i.o.}\}$ is 0 or 1. A history of the problem is given along with two new results for the case when $P\{X_1 \geq 0\} = 1$: (a) An integral test that solves the problem in case the summands satisfy Feller's condition for stochastic compactness of the appropriately normalized sums and (b) necessary and sufficient conditions for a sequence $\{\beta_n\}$ to exist such that $\lim \inf S_n/\beta_n = 1$ a.s.