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October, 1992 Ratios of Trimmed Sums and Order Statistics
Harry Kesten, R. A. Maller
Ann. Probab. 20(4): 1805-1842 (October, 1992). DOI: 10.1214/aop/1176989530

## 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.

## Citation

Harry Kesten. R. A. Maller. "Ratios of Trimmed Sums and Order Statistics." Ann. Probab. 20 (4) 1805 - 1842, October, 1992. https://doi.org/10.1214/aop/1176989530

## Information

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

zbMATH: 0764.60034
MathSciNet: MR1188043
Digital Object Identifier: 10.1214/aop/1176989530

Subjects:
Primary: 60F15
Secondary: 60F05, 60J15, 62G30