The asymptotic distribution of sums of extreme values from a regularly varying distribution. (English) Zbl 0593.60034

Csörgö, Sándor; Mason, David M.

Ann. Probab. 14, 974-983 (1986).

Summary: Let \(X_{1,n}\leq...\leq X_{n,n}\) be the order statistics of n independent and identically distributed positive random variables with common distribution function F satisfying \(1-F(x)=x^{- 1/\alpha}L^*(x),\) \(x>0\), where \(\alpha\) is any positive number and \(L^*\) is any function slowly varying at infinity. We give a complete description of the asymptotic distribution of the sum of the top \(k_ n\) extreme values \(X_{n+1-k_ n,n},X_{n+2-k_ n,n},...,X_{n,n}\) for any sequence of positive integers \(k_ n\) such that \(k_ n\to \infty\) and \(k_ n/n\to 0\) as \(n\to \infty\).

MSC:

60F05	Central limit and other weak theorems
62G30	Order statistics; empirical distribution functions

Keywords:

regular variation; order statistics; slowly varying; asymptotic distribution