Abstract: Let {X_n, n≥1}be a sequence of i.i.d.random variables with acommon nondegenerate d.f., for each n≥1, denote X_{n, 1}≤...≤X_{n, n} as the order statistics of X₁，…，X_n and for integers 1≤l_n≤r_n≤n and nonnegative real numbers p_n and q_n, define S_n(l_n, r_n)=∑^r_n-1_i=l _n+1X_{n, i} +p_nX_{n, ln}+q_nX_{n, rn}. Assume that { l_n, n≥1} satisfies either l_n=l for all n≥1 (l is a fixed positive integer), or l_n→∞ and l_n/(n+1)→0 and that {r_n, n≥1} satisfies n-r_n+1→∞　and r_n/(n+1>→λ∈(0, 1]. We will discuss asymptotic distributions of normalized sums {(S_n(l_n, r_n)-β _n)/α _n, n≥1}. Results on trimmed sums and winsorized sums will be obtained as special cases of the above sums. Especially, we will improve a Griffin's result on asymptotic normality of winsorized sums and give a positive reply for one of his conjectures.

Key words: Trimmed sums, winsorized sums, stochastic compactness, asymptotic normality

摘要： 设{X_n, n≥1}是独立同分布随机变量列，X_{n, 1}≤...≤X_{n, n}是X₁，…，X_n的次序统计量。对非负实数ｐ_ｎ，ｑ_ｎ和满足1≤l_n≤r_n≤n的整数l_n, r_n，令S_n(l_n, r_n)=∑^r_n-1_i=l _n+1 X_{n, i} +p_nX_{n, ln}+q_nX_{n, rn}。当{l_n), n≥1}满足ｌ_ｎ≡ｌ（ｌ是一给定的正整数）或ｌ_ｎ→∞但ｌ_ｎ/(n+1)→0，同时{r_n), n≥1}，满足n-r_n+1→∞但r_n/(n+1)→λ∈(0, 1]时，我们讨论了标准化后之和{(S_n(l_n, r_n)-β_n)/α_n, n≥1}的渐近分布问题。关于截断和及修正截断和的结果将作为特例给出。特别地，我们改进了格里芬关于修正截断和渐近正态性的结论；对于他的一个猜测也作出了正面的回答。

关键词: 截断和, 修正截断和, 随机紧性, 渐近正态性