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.
Publié le : 1990-10-14
Classification:
Probability estimates,
stochastic compactness,
lim inf,
integral test,
slowly varying tails,
60J15,
60F15
@article{1176990626,
author = {Pruitt, William E.},
title = {The Rate of Escape of Random Walk},
journal = {Ann. Probab.},
volume = {18},
number = {4},
year = {1990},
pages = { 1417-1461},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990626}
}
Pruitt, William E. The Rate of Escape of Random Walk. Ann. Probab., Tome 18 (1990) no. 4, pp. 1417-1461. http://gdmltest.u-ga.fr/item/1176990626/