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An Erdös-Rényi strong law for sample quantiles

Published online by Cambridge University Press:  14 July 2016

Stephen A. Book  and
Donald R. Truax
Stephen A. Book
Affiliation:
California State College, Dominguez Hills
Donald R. Truax
Affiliation:
California State College, Dominguez Hills
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Abstract

From a random sample X1, X2, …, XN there can be constructed N – K + 1 successive sample means of the form for 0 ≦ n ≦ N − K, where Erdös and Rényi (1970) studied the maximum Σ(N, K) of these N – K + 1 sample means. Under appropriate conditions, they showed that for a wide interval of λ's there exist constants C(λ), depending only on λ and the distribution from which the sample was selected, such that Σ(N, [C(λ) log N])→ λ a.s. as N→∞. In the present article, analogous results are developed for the maximum of the N – K + 1 successive sample medians and, more generally, for all sample quantiles.

Keywords

ORDER STATISTICSLARGE DEVIATIONSLAWS OF LARGE NUMBERS
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 3 , September 1976 , pp. 578 - 583
Copyright
Copyright © Applied Probability Trust 1976 

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References

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[3] Book, S. A. (1975) Large deviation probabilities and the Erdö s-Rényi law of large numbers. Canad. J. Statist. To appear.Google Scholar
[4] Erdös, P. and Rényi, A. (1970) On a new law of large numbers. J. Analyse Math. 23, 103–111.Google Scholar