Intermediate- and extreme-sum processes

Abstract

Let X1,n[les]...[les]Xn,n be the order statistics of n independent random variables with a common distribution function F and let kn be positive numbers such that kn --> [infinity] and kn/n --> 0 as n-->[infinity]. With suitable centering and norming, we investigate the weak convergence of the intermediate-sum process [summation operator]i=[left ceiling]akn[right ceiling]+1[left ceiling]tkn[right ceiling]Xn+1-i,n, a [les] t [les] b, where 0 a b i=1[left ceiling]tkn[right ceiling]Xn+1-i,n, 0 [les] t [les] b. Convergence is with respect to the supremum norm and can take place along a subsequence of the positive integers {n}.