Laws of Large Numbers for Sums of Extreme Values

Let X₁, X₂, , be a sequence of nonnegative i. i. d. random variables with common distribution F, and for each n 1 let X₁₍ X₍₍ denote the order statistics based on X₁, , Xₙ. Necessary and sufficient conditions are obtained for averages of the extreme values X₍+₁-₈, ₍i = 1, , kₙ + 1 of the form: k^-1ₙ ^kₙ₈ = ₁ (X₍+₁-₈, ₍ - X₍-₊䂸, ₍), 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.