Characterization of weak convergence for smoothed empirical and quantile processes under phi-mixing

Let {X(n), n greater than or equal to 1} be a sequence of phi-mixing random variables having a smooth common distribution function F. The smoothed empirical distribution function is obtained by integrating a kernel type density estimator. In this paper we provide necessary and sufficient conditions for the central limit theorem to hold for smoothed empirical distribution functions and smoothed sample quantiles. Also, necessary and sufficient conditions are given for weak convergence of the smoothed empirical process and the smoothed uniform quantile process.