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 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.