The Annals of Probability

Asymptotic Expansions in Boundary Crossing Problems

Michael Woodroofe and Robert Keener

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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


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