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

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.