## The Annals of Probability

- Ann. Probab.
- Volume 15, Number 1 (1987), 102-114.

### Asymptotic Expansions in Boundary Crossing Problems

Michael Woodroofe and Robert Keener

#### Abstract

Let $S_n, n \geq 1$, be a random walk and $t = t_a = \inf\{n \geq 1: ng(S_n/n) > a\}$. The main results of this paper are two-term asymptotic expansions as $a \rightarrow \infty$ for the marginal distributions of $t_a$ and the normalized partial sum $S^\ast_t = (S_t - t\mu)/\sigma\sqrt t$. To leading order, $S^\ast_t$ has a standard normal distribution. The effect of the randomness in the sample size $t$ on the distribution of $S^\ast_t$ appears in the correction term of the expansion.

#### Article information

**Source**

Ann. Probab., Volume 15, Number 1 (1987), 102-114.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992258

**Digital Object Identifier**

doi:10.1214/aop/1176992258

**Mathematical Reviews number (MathSciNet)**

MR877592

**Zentralblatt MATH identifier**

0619.60023

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60J15

**Keywords**

Nonlinear renewal theory random walks excess over the boundary Edgeworth expansions

#### Citation

Woodroofe, Michael; Keener, Robert. Asymptotic Expansions in Boundary Crossing Problems. Ann. Probab. 15 (1987), no. 1, 102--114. doi:10.1214/aop/1176992258. https://projecteuclid.org/euclid.aop/1176992258