Uniform convergence of sums of order statistics to stable laws (Q1093242)

Let \(X_ 1,X_ 2,..\). 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 \(0<\alpha <1\). Under a von Mises condition we show that the sum of order statistics \[ a_ n^{-1}(\sum ^{k(n)}_{i=1}X_{i:n}+\sum ^{n}_{i=n+1-r(n)}X_{i:n}) \] converges to Q with respect to the norm of total variation if for instance min(k(n),r(n))\(\to \infty\).