## The Annals of Probability

### On the Law of Large Numbers

#### Abstract

Suppose $X_n$ is an i.i.d. sequence of random variables with mean $\mu$ and that $t_n$ is a nondecreasing sequence of positive integers such that $t_n \leq n$. Let $S_n = X_1 + \cdots + X_n$. We give conditions under which $\max_{t_n \leq k \leq n} \big|\frac{S_n - S_{n - k}}{k} - \mu \big| \rightarrow 0$ almost surely and we discuss sharpness.

#### Article information

Source
Ann. Probab., Volume 9, Number 3 (1981), 513-519.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994425

Digital Object Identifier
doi:10.1214/aop/1176994425

Mathematical Reviews number (MathSciNet)
MR614637

Zentralblatt MATH identifier
0466.60033

JSTOR