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# Bonferroni Inequalities

Janos Galambos
The Annals of Probability
Vol. 5, No. 4 (Aug., 1977), pp. 577-581
Stable URL: http://www.jstor.org/stable/2243081
Page Count: 5
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## 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.

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