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

Rimas Norvaisa

Stochastic Processes and their Applications, 1993, vol. 46, issue 1, 1-27

Abstract: 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.

Keywords: empirical; processes; Bahadur-Kiefer; processes; central; limit; theorem; Banach; function; space (search for similar items in EconPapers)
Date: 1993
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