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October, 1975 A Conditional Local Limit Theorem for Recurrent Random Walk
W. D. Kaigh
Ann. Probab. 3(5): 883-888 (October, 1975). DOI: 10.1214/aop/1176996276

Abstract

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.

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W. D. Kaigh. "A Conditional Local Limit Theorem for Recurrent Random Walk." Ann. Probab. 3 (5) 883 - 888, October, 1975. https://doi.org/10.1214/aop/1176996276

Information

Published: October, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0322.60064
MathSciNet: MR388501
Digital Object Identifier: 10.1214/aop/1176996276

Subjects:
Primary: 60F15
Secondary: 60F05 , 60G40 , 60J15

Keywords: Conditioned random walk , hitting time , local limit theorem , Random walk , stopping time

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 5 • October, 1975
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