摘要
Let {X<sub>n</sub>, n≥1} be a sequence of iidrvs with a common df F and for every n, let X<sub>n,1</sub>≤…≤X<sub>n,n</sub> denote the order statistics of X<sub>1</sub>,…, X<sub>n</sub>. Consider the sums S<sub>n</sub><sup>*</sup> =sum from k=k<sub>n</sub>+1 to l<sub>n</sub>（X<sub>n</sub>,k）, n≥1, where k<sub>n</sub> and l<sub>n</sub> satisfy k<sub>n</sub>/（n+1）→a and l<sub>n</sub>/（n+1）→b for some 0【a【b【1.This paper gives necessary and sufficient conditions for (S<sub>n</sub><sup>*</sup>- （n+1） M（k<sub>n</sub>/（n+1）, l<sub>n</sub>/（n+1））/（n+1）<sup>1/2</sup>to converge weakly to a df G, whereM（s, t） = integral from n=s to t（F<sup>-</sup>（w）dw） forr 0【s【t【1;F<sup>-</sup>（t） = inf{x:F（x）≥t}.
Let {X_n, n≥1} be a sequence of iidrvs with a common df F and for every n, let X_(n,1)≤…≤X_(n,n) denote the order statistics of X_1,…, X_n. Consider the sums S_n~* =sum from k=k_n+1 to l_n(X_n,k), n≥1, where k_n and l_n satisfy k_n/(n+1)→a and l_n/(n+1)→b for some 0<a<b<1.This paper gives necessary and sufficient conditions for (S_n~*- (n+1) M(k_n/(n+1), l_n/(n+1))/(n+1)^(1/2)to converge weakly to a df G, whereM(s, t) = integral from n=s to t(F^-(w)dw) forr 0<s<t<1;F^-(t) = inf{x:F(x)≥t}.
