Open Access
August, 1974 Strong and Weak Limit Points of a Normalized Random Walk
K. Bruce Erickson, Harry Kesten
Ann. Probab. 2(4): 553-579 (August, 1974). DOI: 10.1214/aop/1176996604


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.


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K. Bruce Erickson. Harry Kesten. "Strong and Weak Limit Points of a Normalized Random Walk." Ann. Probab. 2 (4) 553 - 579, August, 1974.


Published: August, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0291.60031
MathSciNet: MR359003
Digital Object Identifier: 10.1214/aop/1176996604

Primary: 60G50
Secondary: 60F05 , 60F15 , 60J15

Keywords: accumulation points , denseness of averages in the reals , limit points , Normed sums of independent random variables , Random walk , truncated moments

Rights: Copyright © 1974 Institute of Mathematical Statistics

Vol.2 • No. 4 • August, 1974
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