The central limit theorem for empirical and quantile processes in some Banach spaces

Abstract

Let α n ={ α n ( t); t∈(0, 1)} and β n ={ β n ( t); t∈(0, 1)} be the uniform empirical process and the uniform quantile process, respectively. For given increasing continuous function h on (0, 1) and Orlicz function φ, consider probability distributions on the Banach space L φ(d h) induced by these processes. A description of the function h for the central limit theorem in L φ(d h) for the empirical process α n to hold is given using the probability theory on Banach spaces. To obtain the analogous result for the quantile process β n , it is shown that the Bahadur-Kiefer process α n − β n is negligible in probability in the space L φ(d h). Similar results for the tail empirical as well as for the tail quantile processes, are given too.