次序统计量和的渐近分布（Ⅱ）

The Asymptotic Distributions for Sums of Order Statistics(Ⅰ

记｛Ｘ＿ｎ，ｎ≥１｝为独立冈分布的随机变量列。以Ｘ＿（ｎ．１）≤…≤Ｘ＿（ｎ，ｎ）记Ｘ＿１，…，Ｘ＿ｎ的次序统计量。作者将陆续给出３篇文章来讨论和Ｓ＿ｎ（ｌ＿ｎ，ｒ＿ｎ）当ｎ→∞时的渐近分布．这些文章所使用的方法是统一旦初等的。作为上述和的特例，本文将改进文献中关于截断和及修正截断和的某些结果，还将讨论一类新的截断和──边项次序统计量的和。本文先列举了所要讨论的问题和给出一般性的引理，然后讨论当ｌ＿ｎ≡ｌ和ｎ—ｒ＿ｎ＋ｌ≡ｒ时上述和在适当标准化以后的渐近分布。

Let(X_n,n≥1}be i.i.d. random variables and for each n≥1,denote X_(n,1)≤…≤X_(n,n)as the order statistics of X_1,…,X_n.We will devote three papers to discussing asymptotic distributions of sums where l_n and r_n are integers satisfying 1≤l_n≤r_n≤n and both p_n and q_n are nonnegative real numbers,The methods we used here are unified and elemen-tary. As special cases of such sums,we improve some results on trimmed sums and winsorized sums ap-peared in literature and discuss a new type of trimn1ed sums-sums of intermediate order statistics.At first,in present paper,we list the problems we will discuss and give some lemmas for general use. Then the asymptotic distributions of the normalized sums {S_n(l_n,r_n) β_n/α_n,n≥1}are discussed for the cases of l_n≡l and n r_n+1≡r.