A note on quantile estimation for long-range dependent stochastic processes

This note investigates the consistency properties of the kernel-type estimator of a quantile, in the setting of a long memory stationary stochastic process. Under a general long-range dependence situation (without any restriction of gaussian type) we give consistency results, and rates of convergence. An interesting by-product of this paper is a new consistency result for kernel-type estimator of a smooth distribution function (with rates) over the whole real line.