On the asymptotic joint distribution of an unbounded number of sample extremes

Summary Convergence of the sample maximum to a nondegenerate random variable, as the sample sizen, implies the convergence in distribution of thek largest sample extremes to ak-dimensional random vectorM _k , for all fixedk. If we letk=k(n),k/n0, then a question arises in a natural way: how fast cank grow so that asymptotic probability statements are unaffected when sample extremes are replaced byM _k . We answer this question for two classes of events-the class of all Lebesgue sets inR ^k and the class of events of the form .