One-Sided Refinements of the Strong Law of Large Numbers and the Glivenko-Cantelli Theorem

David Gilat, Tel Aviv University
Theodore P. Hill, Georgia Institute of Technology - Main Campus

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Copyright © 1992 Institute of Mathematical Statistics. The definitive version is available at http://dx.doi.org/10.1214/aop/1176989688.

NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.

Abstract

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.