## The Annals of Probability

### A Conditional Local Limit Theorem for Recurrent Random Walk

W. D. Kaigh

#### 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