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Generalized Bonferroni inequalities

Part of: Combinatorial probability Limit theorems

Published online by Cambridge University Press:  14 July 2016

Thomas H. Spencer
Thomas H. Spencer*
Affiliation:
University of Nebraska at Omaha
*
∗ Postal address: Department of Mathematics and Computer Science, University of Nebraska at Omaha, Omaha, NE 68182, USA. Email address: [email protected]
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Abstract

Consider a number of events in a probability space. Let X be a random variable that is the number of events that occur. Given some of the moments of the distribution of X, it is possible to obtain bounds on the probability that at least one event occurs. The best possible bounds are given here. If there are many equiprobable events that are d- wise independent, and d is even, then the probability that at least one event happens is at least 1 — O(µ–d/2), where μ = E(X).

Keywords

BOUNDSOCCURRENCE OF INDEPENDENT EVENTSESCAPE PROBABILITY

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 409 - 417
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Supported in part by the National Science Foundation under grants CCR-8810609 and CDA-8805910 and by the University Committee on Research, University of Nebraska at Omaha.

Much of the work for this paper was done while the author was at the Computer Science Department, Rensselaer Polytechnic Institute, Troy, NY 12180.

References

[1] Dawson, D. A. and Sankoff, D. (1967) An inequality for probabilities. Proc. Amer. Math. Soc. 18, 453–461.CrossRefGoogle Scholar
[2] Galambos, J. (1975) Methods for proving Bonferroni type inequalities. J. London Math. Soc. 9, 472–479.Google Scholar
[3] Galambos, J. (1977) Bonferroni inequalities. Ann. Prob. 5, 577–581.CrossRefGoogle Scholar
[4] Galambos, J. and Xu, Y. (1990) A new method for generating Bonferroni-type inequalities by iteration. Math. Proc. Camb. Phil. Soc. 107, 601–607.CrossRefGoogle Scholar
[5] Knutr, D. E. (1980) The Art of Computer Programming. Volume 1: Fundamental Algorithms. Addison-Wesley, Reading, MA.Google Scholar
[6] Margaritescu, E. (1987) On some Bonferroni inequalities. Stud. Cercet. Mat. 246–251.Google Scholar
[7] Móri, T. and Székely, G. (1985) A note on the background of several Bonferroni-Galambos-type inequalities. J. Appl. Prob. 22, 836–843.CrossRefGoogle Scholar
[8] Prékopa, A. (1988) Boole-Bonferroni inequalities and linear programming. Operat. Res. 36, 145–162.CrossRefGoogle Scholar
[9] Recsei, E. and Seneta, E. (1987) Bonferroni-type inequalities. Adv. Appl. Prob. 18, 508–511.CrossRefGoogle Scholar
[10] Sobel, ?. and Uppuluri, P. (1972) On Bonferroni-type inequalities of the same degree for the probability on unions and intersections. Ann. Math. Statist. 43, 1549–1558.CrossRefGoogle Scholar
[11] SpençEr, T. (1993) Provably good pattern generators for random pattern test. Algorithmica.CrossRefGoogle Scholar
[12] Tan, X. and Xu, Y. (1989) Some inequalities of the Bonferroni-Galambos type. Statist. Prob. Lett. 8, 17–20.CrossRefGoogle Scholar
[13] Tomescu, I. (1986) Hypertrees and Bonferroni inequalities. J. Combinatorial Theory B 41, 209–217.CrossRefGoogle Scholar
[14] Walker, A. (1981) On the classical Bonferroni inequalities and the corresponding Galambos inequalities. J. Appl. Prob. 13, 757–763.CrossRefGoogle Scholar