Uniform convergence of sums of order statistics to stable laws

Abstract

Let X1, X2,... denote an i.i.d. sequence of real valued random variables which ly in the domain of attraction of a stable law Q with index 0<α<1. Under a von Mises condition we show that the sum of order statistics $$a_n^{ - 1} \left( {\sum\limits_{i = 1}^{k(n)} {X_{i:n} + \sum\limits_{i = n + 1 - r(n)}^n {X_{i:n} } } } \right)$$ converges to Q with respect to the norm of total variation if for instance min(k(n), r(n))→∞.