Limits theorems for tail processes with applications tointermediate quantile estimation

A description of the weak and strong limiting behaviour of weighted uniform tail empirical and tail quantile processes is given. The results for the tail quantile process are applied to obtain weak and strong functional limit theorems for a weighted non-uniform tail-quantile-type process based on a random sample from a distribution that satisfies the so called von Mises sufficient condition for being in the domain of max-attraction of a Fréchet distribution. The functional central limit theorem thus obtained yields asymptotic confidence bands for intermediate quantiles.