On the rate of convergence of sums of extremes to a stable law (Q1824272)

Let \(X_ 1,X_ 2,...\), be a sequence of independent and identically distributed random variables in the domain of normal attraction of a non- normal stable law. It is known that only the sum of the \(k_ n\) largest and \(k_ n\) smallest extreme values in the n th partial sum with \(k_ n\to \infty\) and \(k_ n/n\to 0\) are responsible for the asymptotic stable distribution of the whole sum. We investigate the rate at which such sums of extreme values converge to a stable law in conjunction with the rate at which the sums of the middle terms become asymptotically negligible. In terms of rates of convergence our results provide in many cases a quantitative measure of exactly what portion of the sample is asymptotically stable.