Some Extensions of a Theorem of Stein on Cumulative Sums

Abstract

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.