## The Annals of Probability

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

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

#### 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