Uniform strong consistency of sample quantiles

It is well known that if xq(F) is the unique qth quantile of a distribution function F, then Xk(n):n with k(n)/n --> q is a strongly consistent estimator of xq(F). However, for every [var epsilon] >0 and for every, even very large n, supF[set membership, variant]F,PF{Xk(n):n--Xq(F)>[var epsilon]}=1. This is a consequence of the fact that in the family of all distribution functions with uniquely defined qth quantile the almost sure convergence Xk(n):n --> xq(F) is not uniform. A simple necessary and sufficient condition for the uniform strong consistency of Xk(n):n is given.