The Annals of Probability

A Conditional Local Limit Theorem for Recurrent Random Walk

W. D. Kaigh

Full-text: Open access

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.

Article information

Source
Ann. Probab., Volume 3, Number 5 (1975), 883-888.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996276

Digital Object Identifier
doi:10.1214/aop/1176996276

Mathematical Reviews number (MathSciNet)
MR388501

Zentralblatt MATH identifier
0322.60064

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J15 60F05: Central limit and other weak theorems

Keywords
Local limit theorem stopping time hitting time conditioned random walk random walk

Citation

Kaigh, W. D. A Conditional Local Limit Theorem for Recurrent Random Walk. Ann. Probab. 3 (1975), no. 5, 883--888. doi:10.1214/aop/1176996276. https://projecteuclid.org/euclid.aop/1176996276


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