Optimal Rate of Convergence for Empirical Quantiles and Distribution Functions for Time Series

Abstract

Given a stationary sequence ¹X k º k2Z , we are interested in the rate of convergence in the central limit theorem of the empirical quantiles and the empirical distribution function. Under a general notion of weak dependence, we show a Berry-Esseen result with optimal rate n 1=2 . The setup includes many prominent time series models, such as functions of ARMA or (augmented) GARCH processes. In this context, optimal Berry-Esseen rates for empirical quantiles appear to be novel.