Skip to main content
Article
One-Sided Refinements of the Strong Law of Large Numbers and the Glivenko-Cantelli Theorem
The Annals of Probability
  • David Gilat, Tel Aviv University
  • Theodore P. Hill, Georgia Institute of Technology - Main Campus
Publication Date
1-1-1992
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.

Disciplines
Citation Information
David Gilat and Theodore P. Hill. "One-Sided Refinements of the Strong Law of Large Numbers and the Glivenko-Cantelli Theorem" The Annals of Probability Vol. 20 Iss. 3 (1992) p. 1213 - 1221
Available at: http://works.bepress.com/tphill/38/