Let $\{S_n\}$ be a random walk with underlying distribution function $F(x)$ and $\{\gamma_n\}$ be a sequence of constants such that $\gamma_n/n$ is nondecreasing. A universal integral test is given which determines the lower limit of $S_n/\gamma_n$ up to a constant scale for $\lim \sup \gamma_{2n}/\gamma_n < \infty$. The generalized LIL is obtained which contains the main result of Fristedt-Pruitt (1971). The rapidly growing random walks and the limit points of $\{S_n/\gamma_n\}$ are also studied.
Publié le : 1986-04-14
Classification:
Normalized random walks,
lower limits,
generalized law of the iterated logarithm,
exponential bounds,
truncated moments,
60G50,
60J15,
60F15,
60F20
@article{1176992531,
author = {Zhang, Cun-Hui},
title = {The Lower Limit of a Normalized Random Walk},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 560-581},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992531}
}
Zhang, Cun-Hui. The Lower Limit of a Normalized Random Walk. Ann. Probab., Tome 14 (1986) no. 4, pp. 560-581. http://gdmltest.u-ga.fr/item/1176992531/