Laws of Large Numbers for Sums of Extreme Values

Abstract

Let X1,X2,⋯, be a sequence of nonnegative i.i.d. random variables with common distribution F, and for each n≥1 let X1n≤⋯≤Xnn denote the order statistics based on X1,⋯,Xn. Necessary and sufficient conditions are obtained for averages of the extreme values Xn+1−i,ni=1,⋯,kn+1 of the form: k−1n∑kni=1(Xn+1−i,n−Xn−kn,n), where kn→∞ and n−1kn→0, to converge in probability or almost surely to a finite positive constant. In the process, characterizations are given of the classes of distributions with regularly varying upper tails and of distributions with "exponential-like" upper tails.