## The Annals of Probability

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

### On the Law of Large Numbers

D. L. Hanson and Ralph P. Russo

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60F10: Large deviations

**Keywords**

Law of large numbers strong law of large numbers

#### Citation

Hanson, D. L.; Russo, Ralph P. On the Law of Large Numbers. Ann. Probab. 9 (1981), no. 3, 513--519. doi:10.1214/aop/1176994425. https://projecteuclid.org/euclid.aop/1176994425