Let $S_n, n \geq 1$, be a random walk and $t = t_a = \inf\{n \geq 1: ng(S_n/n) > a\}$. The main results of this paper are two-term asymptotic expansions as $a \rightarrow \infty$ for the marginal distributions of $t_a$ and the normalized partial sum $S^\ast_t = (S_t - t\mu)/\sigma\sqrt t$. To leading order, $S^\ast_t$ has a standard normal distribution. The effect of the randomness in the sample size $t$ on the distribution of $S^\ast_t$ appears in the correction term of the expansion.
Publié le : 1987-01-14
Classification:
Nonlinear renewal theory,
random walks,
excess over the boundary,
Edgeworth expansions,
60F05,
60J15
@article{1176992258,
author = {Woodroofe, Michael and Keener, Robert},
title = {Asymptotic Expansions in Boundary Crossing Problems},
journal = {Ann. Probab.},
volume = {15},
number = {4},
year = {1987},
pages = { 102-114},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992258}
}
Woodroofe, Michael; Keener, Robert. Asymptotic Expansions in Boundary Crossing Problems. Ann. Probab., Tome 15 (1987) no. 4, pp. 102-114. http://gdmltest.u-ga.fr/item/1176992258/