## The Annals of Probability

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

### 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.

#### 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