The Annals of Probability

Ratios of Trimmed Sums and Order Statistics

Harry Kesten and R. A. Maller

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Abstract

Let $X_i$ be independent and identically distributed random variables with distribution $F$. Let $M^{(n)}_n \leq \cdots \leq M^{(1)}_n$ be the sample $X_1, X_2,\ldots, X_n$ arranged in increasing order, with a convention for the breaking of ties, and let $X^{(n)}_n,\ldots, X^{(1)}_n$ be the sample arranged in increasing order of modulus, again with a convention to break ties. Let $S_n = X_1 + \cdots + X_n$ be the sample sum. We consider sums trimmed by large values, $^{(r)}S_n = S_n - M^{(1)}_n - \cdots - M^{(r)}_n, r = 1,2,\ldots, n, ^{(0)}S_n = S_n,$ and sums trimmed by values large in modulus, $^{(r)}\tilde{S}_n = S_n - X^{(1)}_n - \cdots - X^{(r)}_n, r = 1,2,\ldots, n, ^{(0)}\tilde{S}_n = S_n.$ In this paper we give necessary and sufficient conditions for $^{(r)}\tilde{S}_n/|X^{(r)}_n| \rightarrow \infty$ and $^{(r)}S_n/M^{(r)}_n \rightarrow \infty$ to hold almost surely or in probability, when $r = 1,2,\ldots$. These express the dominance of the sum over the large values in the sample in various ways and are of interest in relation to the law of large numbers and to central limit behavior. Our conditions are related to the relative stability almost surely or in probability of the trimmed sum and, hence, to analytic conditions on the tail of the distribution of $X_i$ which give relative stability.

Article information

Source
Ann. Probab., Volume 20, Number 4 (1992), 1805-1842.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989530

Digital Object Identifier
doi:10.1214/aop/1176989530

Mathematical Reviews number (MathSciNet)
MR1188043

Zentralblatt MATH identifier
0764.60034

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60J15 60F05: Central limit and other weak theorems 62G30: Order statistics; empirical distribution functions

Keywords
Trimmed sums order statistics relative stability strong law

Citation

Kesten, Harry; Maller, R. A. Ratios of Trimmed Sums and Order Statistics. Ann. Probab. 20 (1992), no. 4, 1805--1842. doi:10.1214/aop/1176989530. https://projecteuclid.org/euclid.aop/1176989530


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