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January, 1987 Asymptotic Expansions in Boundary Crossing Problems
Michael Woodroofe, Robert Keener
Ann. Probab. 15(1): 102-114 (January, 1987). DOI: 10.1214/aop/1176992258


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.


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Michael Woodroofe. Robert Keener. "Asymptotic Expansions in Boundary Crossing Problems." Ann. Probab. 15 (1) 102 - 114, January, 1987.


Published: January, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0619.60023
MathSciNet: MR877592
Digital Object Identifier: 10.1214/aop/1176992258

Primary: 60F05
Secondary: 60J15

Keywords: Edgeworth expansions , excess over the boundary , nonlinear renewal theory , Random walks

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 1 • January, 1987
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