Asymptotic independence of distributions of normalized order statistics of the underlying probability measure

Let n denote the sample size, and let ri [set membership, variant] {1,...,n} fulfill the conditions ri - ri-1 >= 5 for i = 1,...,k. It is proved that the joint normalized distribution of the order statistics Zri:n, i = 1,...,k, is independent of the underlying probability measure up to a remainder term of order O((k/n)1/2). A counterexample shows that, as far as central order statistics are concerned, this remainder term is not of the order O((k/n)1/2) if ri - ri-1 = 1 for i = 2,...,k.