Asymptotics of set - indexed conditional empiricalprocesses based on dependent data

We study the asymptotic properties of the conditional empirical process based on b F n (Cjx) = n X t=1 ! n (X t ? x)I fYt2Cg indexed by C 2 C, where f(X t ; Y t); t = 1; : : :; ng are observations from a strong mixing stochas-tic process, f! n (X t ? x)g denote some kernel weights, and C is a class of sets. Under the assumption on the richness of the index class C in terms of metric entropy with bracketing, we have established uniform convergence, and asymptotic normality for b F n (jx). The key technical result gives rates of convergences for the sup-norm of the conditional empirical process over a sequence of classes C with decreasing maximum L 1-norm. The results are then applied to derive Bahadur-Kiefer type approximations for a generalized conditional quantile process which is closely related to the minimum volume sets. The potential applications in the areas of estimation of level sets and testing for unimodality of conditional distributions are discussed.