## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 6 (1972), 1874-1883.

### Inequalities for the Law of Large Numbers

#### Abstract

Let $X_1, X_2, X_3, \cdots$ be independent random variables and $a_1, a_2, a_3, \cdots$ positive real numbers. Define $F(t) = \sup_k P\{|X_k| > t\}$ and $S_m = \sum^m_{k=1} a_k X_k.$ Inequalities of the form $P\{\sup_m|S_m| > \delta\} \leqq C \sum_k \int^1_0 \varphi'(u)F(u/a_k) du$ are given for a large class of functions $\varphi$, as well as inequalities of a somewhat different form that are appropriate for considering exponential convergence rates. Examples of how the inequalities can be used to prove rate theorems are also given.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 6 (1972), 1874-1883.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177690858

**Digital Object Identifier**

doi:10.1214/aoms/1177690858

**Mathematical Reviews number (MathSciNet)**

MR378045

**Zentralblatt MATH identifier**

0251.60019

**JSTOR**

links.jstor.org

#### Citation

Kurtz, Thomas G. Inequalities for the Law of Large Numbers. Ann. Math. Statist. 43 (1972), no. 6, 1874--1883. doi:10.1214/aoms/1177690858. https://projecteuclid.org/euclid.aoms/1177690858