The Annals of Probability

First Exit Times from Moving Boundaries for Sums of Independent Random Variables

Tze Leung Lai

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Abstract

Let $X_1, X_2, \cdots$ be independent random variables such that $EX_n = 0, EX_n^2 = 1, n = 1,2, \cdots$ and the uniform Lindeberg condition is satisfied. Let $S_n = X_1 + \cdots + X_n$. In this paper, we study the first exit time $N_c = \inf \{n \geqq m: |S_n| \geqq cb(n)\}$ for general lower-class boundaries $b(n)$. Our results extend the theorems of Breiman, Brown, Chow, Robbins and Teicher, Gundy and Siegmund who studied the case $b(n) = n^{\frac{1}{2}}$. We also obtain the limiting moments of $N_c$ in the case $b(n) = n^\alpha (0 < \alpha < \frac{1}{2})$ as analogues of recent results in extended renewal theory.

Article information

Source
Ann. Probab., Volume 5, Number 2 (1977), 210-221.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995846

Digital Object Identifier
doi:10.1214/aop/1176995846

Mathematical Reviews number (MathSciNet)
MR433590

Zentralblatt MATH identifier
0379.60026

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60K05: Renewal theory

Keywords
First exit times lower-class boundaries Lindeberg condition uniform invariance principle delayed sums extended renewal theory without drift

Citation

Lai, Tze Leung. First Exit Times from Moving Boundaries for Sums of Independent Random Variables. Ann. Probab. 5 (1977), no. 2, 210--221. doi:10.1214/aop/1176995846. https://projecteuclid.org/euclid.aop/1176995846


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