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A limit theorem with applications in order statistics

Published online by Cambridge University Press:  14 July 2016

János Galambos
János Galambos*
Affiliation:
Temple University, Philadelphia
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Abstract

Let A1, A2, ···, An be events on a given probability space and let Br, n be the event that exactly r of the A's occur. Let further Sk (n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n), as n→ + ∞, in terms of limits of the Sk(n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X1, X2, ···, Xn by choosing Aj = {Xj ≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.

Keywords

SIEVE THEOREMBINOMIAL MOMENTSLIMIT DISTRIBUTIONGENERATING FUNCTIONCONTINUITY THEOREMEXTREME VALUE DISTRIBUTION
Type
Short Communications
Information
Journal of Applied Probability , Volume 11 , Issue 1 , March 1974 , pp. 219 - 222
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1] Berman, S. M. (1962) Limiting distribution of the maximum term in sequences of dependent random variables Ann. Math. Statist. 33, 894–908.Google Scholar
[2] Feller, W. (1957) An Introduction to Probability Theory and its Applications. 2nd ed. Wiley, New York.Google Scholar
[3] Jordán, Ch. (1934) Le théorème de probabilité de Poincaré, généralisé au cas de plusieurs variables indépendantes Acta Sci. Math. (Szeged) 7, 103–111.Google Scholar
[4] Takács, L. (1967) On the method of inclusion and exclusion J. Amer. Statist. Assoc. 62, 102–113.Google Scholar