Open Access
August, 1977 Bonferroni Inequalities
Janos Galambos
Ann. Probab. 5(4): 577-581 (August, 1977). DOI: 10.1214/aop/1176995765


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.


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Janos Galambos. "Bonferroni Inequalities." Ann. Probab. 5 (4) 577 - 581, August, 1977.


Published: August, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0369.60018
MathSciNet: MR448478
Digital Object Identifier: 10.1214/aop/1176995765

Primary: 60C05
Secondary: 60E05

Keywords: best linear bounds , binomial moments , Bonferroni inequalities , Dependent samples , distribution of order statistics , events , method of inclusion and exclusion , number of occurrences

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 4 • August, 1977
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