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July, 1992 One-Sided Refinements of the Strong Law of Large Numbers and the Glivenko-Cantelli Theorem
David Gilat, T. P. Hill
Ann. Probab. 20(3): 1213-1221 (July, 1992). DOI: 10.1214/aop/1176989688


A one-sided refinement of the strong law of large numbers is found for which the partial weighted sums not only converge almost surely to the expected value, but also the convergence is such that eventually the partial sums all exceed the expected value. The new weights are distribution-free, depending only on the relative ranks of the observations. A similar refinement of the Glivenko-Cantelli theorem is obtained, in which a new empirical distribution function not only has the usual uniformly almost-sure convergence property of the classical empirical distribution function, but also has the property that all its quantiles converge almost surely. A tool in the proofs is a strong law of large numbers for order statistics.


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David Gilat. T. P. Hill. "One-Sided Refinements of the Strong Law of Large Numbers and the Glivenko-Cantelli Theorem." Ann. Probab. 20 (3) 1213 - 1221, July, 1992.


Published: July, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0762.60025
MathSciNet: MR1175259
Digital Object Identifier: 10.1214/aop/1176989688

Primary: 60F15
Secondary: 62G30

Keywords: convergence of medians , Glivenko-Cantelli theorem , one-sided strong laws , order statistics , Strong law of large numbers

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 3 • July, 1992
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