Strong and Weak Limit Points of a Normalized Random Walk

Abstract

Let $S_n = \sum^n_1 X_i$ be a random walk. A point $b$ is called a strong limit point of $n^{-\alpha}S_n$ if there exists a nonrandom sequence $n_k\rightarrow\infty$ such that $n_k^{-\alpha}S_{n_k}\rightarrow b$ w.p. 1. The possible structures for the set of strong limit points of $n^{-\alpha}S_n$ are determined. We also give a sufficient condition for $n^{-1}S_n$ to be dense in $\mathbb{R}$. In particular $n^{-1}S_n$ is dense in $\mathbb{R}$ when $E|X_1| = \infty$ and $n^{-1}S_n$ has a finite strong limit point.