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

Let [alpha]n={[alpha]n(t); t[set membership, variant](0, 1)} and [beta]n={[beta]n(t); t[set membership, variant](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 [phi], consider probability distributions on the Banach space L[phi](dh) induced by these processes. A description of the function h for the central limit theorem in L[phi](dh) for the empirical process [alpha]n to hold is given using the probability theory on Banach spaces. To obtain the analogous result for the quantile process [beta]n, it is shown that the Bahadur-Kiefer process [alpha]n-[beta]n is negligible in probability in the space L[phi](dh). Similar results for the tail empirical as well as for the tail quantile processes, are given too.