Bootstrapping the sample quantile of a weakly dependent sequence

In this paper, we investigate consistency properties of 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 method provides a valid approximation to the distribution of normalized sample quantile in the almost sure sense. Strong consistency of the block bootstrap estimator of the asymptotic variance of the sample quantile is also established under similar conditions. For the proof, we develop some exponential inequalities for block bootstrap moments and also develop some almost sure bounds on the oscillations of the empirical distribution function of strongly mixing random variables, which may be of some independent interest.