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April, 1977 First Exit Times from Moving Boundaries for Sums of Independent Random Variables
Tze Leung Lai
Ann. Probab. 5(2): 210-221 (April, 1977). DOI: 10.1214/aop/1176995846


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.


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Tze Leung Lai. "First Exit Times from Moving Boundaries for Sums of Independent Random Variables." Ann. Probab. 5 (2) 210 - 221, April, 1977.


Published: April, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0379.60026
MathSciNet: MR433590
Digital Object Identifier: 10.1214/aop/1176995846

Primary: 60F05
Secondary: 60K05

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

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 2 • April, 1977
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