Let $S_n, n = 1, 2, 3, \cdots$ denote the recurrent random walk formed by the partial sums of i.i.d. lattice random variables with mean zero and finite variance. Let $T_{\{x\}} = \min \lbrack n \geqq 1 \mid S_n = x \rbrack$ with $T \equiv T_{\{0\}}$. We obtain a local limit theorem for the random walk conditioned by the event $\lbrack T > n \rbrack$. This result is applied then to obtain an approximation for $P\lbrack T_{\{x\}} = n \rbrack$ and the asymptotic distribution of $T_{\{x\}}$ as $x$ approaches infinity.
Publié le : 1975-10-14
Classification:
Local limit theorem,
stopping time,
hitting time,
conditioned random walk,
random walk,
60F15,
60G40,
60J15,
60F05
@article{1176996276,
author = {Kaigh, W. D.},
title = {A Conditional Local Limit Theorem for Recurrent Random Walk},
journal = {Ann. Probab.},
volume = {3},
number = {6},
year = {1975},
pages = { 883-888},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996276}
}
Kaigh, W. D. A Conditional Local Limit Theorem for Recurrent Random Walk. Ann. Probab., Tome 3 (1975) no. 6, pp. 883-888. http://gdmltest.u-ga.fr/item/1176996276/