A strengthened version of the Glivenko-Cantelli theorem for the uniform empirical distribution function is proved. The strengthened Glivenko-Cantelli theorem is used to establish strong laws of large numbers for linear functions of order statistics.
"A Glivenko-Cantelli Theorem and Strong Laws of Large Numbers for Functions of Order Statistics." Ann. Statist. 5 (3) 473 - 480, May, 1977. https://doi.org/10.1214/aos/1176343844