The Annals of Probability

Bonferroni Inequalities

Janos Galambos

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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

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

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60C05: Combinatorial probability
Secondary: 60E05: Distributions: general theory

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


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

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