Published in The Annals of Probability, Volume 20, Issue 3, January 1, 1992, pages 1213-1221.
Copyright © 1992 Institute of Mathematical Statistics.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
The definitive version is available at https://doi.org/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.