Asymptotic Normality of Linear Combinations of Functions of Order Statistics

Abstract

FN'(t) = inf {x:FN(x) > t}; and we write g o FN-l for the composed function g (FN-1). Note that TN = aO g o FN-1 dVN when the signed measure VN puts mass CNi/N at i/N for i = 1, * * , N and puts 0 mass elsewhere. Let v denote a signed measure on (0, 1). (The signed measures VN will not be used, but v is in some sense their limit.) For technical reasons to be displayed below, we bound ourselves away from 0 and 1 by an amount #N where NiN -3 0 as N - oo at a rate to be specified later. Let (1.2) 1N - fANd (1.2) Y~~~~~~.N = fAN g dv