Let $S_n = X_1 + \cdots + X_n, n \geq 1$, be a $d$-dimensional random walk and let $T_a = \inf\{n \geq n_a: ng(S_n/n) \geq a\}$, where $n_a = o(a)$. Let $\theta = g(EX_1), \hat{\theta}_n = g(S_n/n)$ and $\Delta_a = T_a\hat{\theta}_{T_a} - a$. Edgeworth-type expansions are developed for $P\{T_a = n, y_1 \leq \Delta_a \leq y_2\}$ and for the distribution functions of $T_a$ and of $\sqrt T_a(h(\hat{\theta}_{T_a}) - h(\theta))$, where $h$ is a real-valued function such that $h'(\theta) \neq 0$.
Publié le : 1994-10-14
Classification:
Random walks,
nonlinear renewal theory,
boundary crossing probabilities,
bootstrap,
Edgeworth expansions,
60G40,
60F05,
60J15,
62L12
@article{1176988491,
author = {Lai, Tze Leung and Wang, Julia Qizhi},
title = {Asymptotic Expansions for the Distributions of Stopped Random Walks and First Passage Times},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 1957-1992},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988491}
}
Lai, Tze Leung; Wang, Julia Qizhi. Asymptotic Expansions for the Distributions of Stopped Random Walks and First Passage Times. Ann. Probab., Tome 22 (1994) no. 4, pp. 1957-1992. http://gdmltest.u-ga.fr/item/1176988491/