The Annals of Probability

Bonferroni Inequalities

Janos Galambos

Full-text: Open access

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

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176995765

Mathematical Reviews number (MathSciNet)
MR448478

Zentralblatt MATH identifier
0369.60018

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


Export citation