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.
Publié le : 1974-08-14
Classification:
Random walk,
limit points,
accumulation points,
denseness of averages in the reals,
normed sums of independent random variables,
truncated moments,
60G50,
60J15,
60F05,
60F15
@article{1176996604,
author = {Erickson, K. Bruce and Kesten, Harry},
title = {Strong and Weak Limit Points of a Normalized Random Walk},
journal = {Ann. Probab.},
volume = {2},
number = {6},
year = {1974},
pages = { 553-579},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996604}
}
Erickson, K. Bruce; Kesten, Harry. Strong and Weak Limit Points of a Normalized Random Walk. Ann. Probab., Tome 2 (1974) no. 6, pp. 553-579. http://gdmltest.u-ga.fr/item/1176996604/