Open Access
January, 1977 Some Extensions of a Theorem of Stein on Cumulative Sums
S.-S. Perng
Ann. Statist. 5(1): 98-109 (January, 1977). DOI: 10.1214/aos/1176343743


Let $u, u_1, u_2, \cdots$ be a sequence of i.i.d. random $k$-vectors and $a_1, a_2, \cdots$ be a sequence of $k$-vectors. Let $S_n = \sum^n_1 a_i'u_i$. For any positive $L$, let $N = \min \{n \geqq 1: |S_n| \geqq L\}$. In case $k = 1$ and all $a_n$'s are equal and nonzero, Stein [5] showed that $N$ is exponentially bounded provided that $u$ is nondegenerate at 0. In this paper, conditions on the $a_n$'s and on $u$ which guarantee the exponential boundedness of $N$ defined above are obtained. The exponential boundedness of $N' = \min \{n \geqq 1: |S_n + C_n| \geqq L\}$, where $C_1, C_2, \cdots$ is an arbitrary sequence of real numbers, is also considered. Some applications are given.


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S.-S. Perng. "Some Extensions of a Theorem of Stein on Cumulative Sums." Ann. Statist. 5 (1) 98 - 109, January, 1977.


Published: January, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0358.62056
MathSciNet: MR426327
Digital Object Identifier: 10.1214/aos/1176343743

Primary: 62L99
Secondary: 60G40

Keywords: Exponential boundedness , stopping time

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 1 • January, 1977
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