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

David Gilat, T. P. Hill
1992 Annals of Probability  
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 refine ment of the Glivenko-Cantelli theorem is obtained, in which a new empiri cal distribution function not only has the usual
more » ... 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. o. Introduction. The classical strong law of large numbers of Kolmogorov (1933) says that if Xl' X 2 , ••. are independent and identically distributed with finite mean IL, then the weighted partial sums (n -IX 1 + ... +n-1 X n ) converge almost surely to IL, and the recurrence of mean-zero
doi:10.1214/aop/1176989688 fatcat:5w76sxhvazcgdmj3wvqi4aw7e4