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

AbstractLet αn={αn(t); t∈(0, 1)} and βn={βn(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 Lφ(dh) induced by these processes. A description of the function h for the central limit theorem in Lφ(dh) for the empirical process αn to hold is given using the probability theory on Banach spaces. To obtain the analogous result for the quantile process βn, it is shown that the Bahadur-Kiefer process αn−βn is negligible in probability in the space Lφ(dh). Similar results for the tail empirical as well as for the tail quantile processes, are given too