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

Abstract

Let n denote the sample size, and let r i ∈ {1,…, n} fulfill the conditions r i − r i−1 ≥ 5 for i = 1,…, k. It is proved that the joint normalized distribution of the order statistics Z r i : 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 r i − r i−1 = 1 for i = 2,…, k.