On the rate of convergence of sums of extremes to a stable law

Abstract

LetX1,X2, ..., be a sequence of independent and identically distributed random variables in the domain of normal attraction of a nonnormal stabler law. It is known that only the sum of thek n largest andk n smallest extreme values in thenth partial sum withk n →∞ andk n /n→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.