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January, 1992 Asymptotic Behavior of Self-Normalized Trimmed Sums: Nonnormal Limits
Marjorie G. Hahn, Daniel C. Weiner
Ann. Probab. 20(1): 455-482 (January, 1992). DOI: 10.1214/aop/1176989937


Let $\{X_j\}$ be independent, identically distributed random variables with continuous nondegenerate distribution $F$ which is symmetric about the origin. Let $\{X_n(1), X_n(2),\ldots, X_n(n)\}$ denote the arrangement of $\{X_1,\ldots, X_n\}$ in decreasing order of magnitude, so that with probability 1, $|X_n(1)| > |X_n(2)| > \cdots > |X_n(n)|$. For integers $r_n \rightarrow \infty$ such that $r_n/n \rightarrow 0$, define the self-normalized trimmed sum $T_n = \sum^n_{i=r_n}X_n(i)/\{\sum^n_{i=r_n}X^2_n(i)\}^{1/2}$. The asymptotic behavior of $T_n$ is studied. Under a probabilistically meaningful analytic condition generalizing the asymptotic normality criterion for $T_n$, various interesting nonnormal limit laws for $T_n$ are obtained and represented by means of infinite random series. In general, moreover, criteria for degenerate limits and stochastic compactness for $\{T_n\}$ are also obtained. Finally, more general results and technical difficulties are discussed.


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Marjorie G. Hahn. Daniel C. Weiner. "Asymptotic Behavior of Self-Normalized Trimmed Sums: Nonnormal Limits." Ann. Probab. 20 (1) 455 - 482, January, 1992.


Published: January, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0743.60024
MathSciNet: MR1143431
Digital Object Identifier: 10.1214/aop/1176989937

Primary: 60F05
Secondary: 62G05 , 62G30

Keywords: Infinitely divisible laws , magnitude order statistics , nonnormal limits , self-normalization and studentization , series representations , stochastic compactness , symmetry , trimmed sums , weak convergence

Rights: Copyright © 1992 Institute of Mathematical Statistics


Vol.20 • No. 1 • January, 1992
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