Let $S_n = X_1 + \cdots + X_n$ be partial sums of independent identically distributed random variables and let $a_n$ be an increasing sequence of positive constants tending to $\infty$. This paper concerns the almost sure lower limit of $\max_{1\leq j \leq n} S_j/a_n$. We prove that the lower limit is either 0 or $\infty$ under mild conditions and give integral tests to determine which is the case. Let $\tau = \inf\{n \geq 1: S_n > 0\}$ and $\tau_- = \inf\{n \geq 1: S_n \leq 0\}$. Several inequalities are given that determine up to scale constants various quantities involving truncated moments of the ladder variables $S_\tau$ and $\tau$ under three different conditions: $ES_\tau < \infty, E|S_{\tau-}| < \infty$ and $X$ symmetric. Moments of ladder variables are also discussed.
Publié le : 1994-10-14
Classification:
Random walk,
integral test,
rate of escape,
ladder variable,
truncated moment,
inequality,
60G50,
60J15,
60F15
@article{1176988487,
author = {Klass, Michael J. and Zhang, Cun-Hui},
title = {On the Almost Sure Minimal Growth Rate of Partial Sum Maxima},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 1857-1878},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988487}
}
Klass, Michael J.; Zhang, Cun-Hui. On the Almost Sure Minimal Growth Rate of Partial Sum Maxima. Ann. Probab., Tome 22 (1994) no. 4, pp. 1857-1878. http://gdmltest.u-ga.fr/item/1176988487/