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April, 1988 The Asymptotic Distribution of Trimmed Sums
Sandor Csorgo, Erich Haeusler, David M. Mason
Ann. Probab. 16(2): 672-699 (April, 1988). DOI: 10.1214/aop/1176991780


Let $X_{1,n} \leq \cdots \leq X_{n,n}$ be the order statistics of $n$ independent and identically distributed random variables and let $m_n$ and $k_n$ be positive integers such that $m_n \rightarrow \infty, k_n \rightarrow \infty, m_n/n \rightarrow 0$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$. We find a necessary and sufficient condition for the existence of normalizing and centering constants $A_n > 0$ and $B_n$ such that the sequence $T_n = A^{-1}_n \big\{\sum_{i=m_n+1}^{n-k_n} X_{i,n} - B_n\big\}$ is stochastically compact and completely describe the possible subsequential limiting distributions. We also give a necessary and sufficient condition for the existence of $A_n$ and $B_n$ such that $T_n$ is asymptotically normal. A variant of Stigler's theorem when $m_n/n \rightarrow \alpha$ and $k_n/n \rightarrow 1 - \beta$, where $0 < \alpha < \beta < 1$, is also obtained as a by-product.


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Sandor Csorgo. Erich Haeusler. David M. Mason. "The Asymptotic Distribution of Trimmed Sums." Ann. Probab. 16 (2) 672 - 699, April, 1988.


Published: April, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0647.62030
MathSciNet: MR929070
Digital Object Identifier: 10.1214/aop/1176991780

Primary: 60F05

Keywords: asymptotic normality , stochastic compactness , trimmed sums

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 2 • April, 1988
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