A strong limit theorem for the oscillation modulus of the uniform empirical quantile process

Abstract

Stute (1982) and Mason, Shorack and Wellner (1983) have recently completed a thorough study of the limiting behavior of the oscillation of the uniform empirical process. In this paper, the corresponding oscillation behavior of the uniform empirical quantile process is investigated. It is shown to be closely related to the limiting behavior of the maximum k-spacing of n independent Uniform (0, 1) random variables, where k can possibly be a function of n. Results of this type are directly applicable to the study of the strong consistency properties of various types of density estimators.