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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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. The Annals of Probability 5 (1977), no. 4, 577--581. doi:10.1214/aop/1176995765.

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