## The Annals of Probability

- Ann. Probab.
- Volume 5, Number 4 (1977), 577-581.

### Bonferroni Inequalities

#### Abstract

Let $A_1, A_2, \cdots, A_n$ be events on a probability space. Let $S_{k,n}$ be the $k$th binomial moment of the number $m_n$ of those $A$'s which occur. An estimate on the distribution $y_t = P(m_n \geqq t)$ by a linear combination of $S_{1,n}, S_{2,n}, \cdots, S_{n,n}$ is called a Bonferroni inequality. We present for proving Bonferroni inequalities a method which makes use of the following two facts: the sequence $y_t$ is decreasing and $S_{k,n}$ is a linear combination of the $y_t$. By this method, we significantly simplify a recent proof for the sharpest possible lower bound on $y_1$ in terms of $S_{1,n}$ and $S_{2,n}$. In addition, we obtain an extension of known bounds on $y_t$ in the spirit of a recent extension of the method of inclusion and exclusion.

#### Article information

**Source**

Ann. Probab., Volume 5, Number 4 (1977), 577-581.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995765

**Digital Object Identifier**

doi:10.1214/aop/1176995765

**Mathematical Reviews number (MathSciNet)**

MR448478

**Zentralblatt MATH identifier**

0369.60018

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 60E05: Distributions: general theory

**Keywords**

Events number of occurrences binomial moments Bonferroni inequalities best linear bounds method of inclusion and exclusion distribution of order statistics dependent samples

#### Citation

Galambos, Janos. Bonferroni Inequalities. Ann. Probab. 5 (1977), no. 4, 577--581. doi:10.1214/aop/1176995765. https://projecteuclid.org/euclid.aop/1176995765