Bootstrapping the sample quantile based on weakly dependent observations

In this work, we investigate consistency properties of normal approximation and block bootstrap approximations for sample quantiles of weakly dependent data. Under mild weak dependence conditions and mild smoothness conditions on the one-dimensional marginal distribution function, we show that the moving block bootstrap (MBB) method provides a valid approximation to the distribution of normalized sample quantile and the corresponding MBB estimator of the asymptotic variance is also strongly consistent. Along the line, we also examine the rate of convergence of the MBB approximation to the distribution of the sample quantile, and prove a Berry-Esseen Theorem, which indicates that the normal approximation to the distribution of the sample quantile under weak dependence is of order O(n-1/2).