The Annals of Probability

Asymptotic Expansions for the Distributions of Stopped Random Walks and First Passage Times

Tze Leung Lai and Julia Qizhi Wang

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Abstract

Let $S_n = X_1 + \cdots + X_n, n \geq 1$, be a $d$-dimensional random walk and let $T_a = \inf\{n \geq n_a: ng(S_n/n) \geq a\}$, where $n_a = o(a)$. Let $\theta = g(EX_1), \hat{\theta}_n = g(S_n/n)$ and $\Delta_a = T_a\hat{\theta}_{T_a} - a$. Edgeworth-type expansions are developed for $P\{T_a = n, y_1 \leq \Delta_a \leq y_2\}$ and for the distribution functions of $T_a$ and of $\sqrt T_a(h(\hat{\theta}_{T_a}) - h(\theta))$, where $h$ is a real-valued function such that $h'(\theta) \neq 0$.

Article information

Source
Ann. Probab., Volume 22, Number 4 (1994), 1957-1992.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988491

Digital Object Identifier
doi:10.1214/aop/1176988491

Mathematical Reviews number (MathSciNet)
MR1331212

Zentralblatt MATH identifier
0843.60046

JSTOR
links.jstor.org

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60F05: Central limit and other weak theorems 60J15 62L12: Sequential estimation

Keywords
Random walks nonlinear renewal theory boundary crossing probabilities bootstrap Edgeworth expansions

Citation

Lai, Tze Leung; Wang, Julia Qizhi. Asymptotic Expansions for the Distributions of Stopped Random Walks and First Passage Times. Ann. Probab. 22 (1994), no. 4, 1957--1992. doi:10.1214/aop/1176988491. https://projecteuclid.org/euclid.aop/1176988491


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