The Annals of Probability

Ratios of Trimmed Sums and Order Statistics

Harry Kesten and R. A. Maller

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Trimmed sums order statistics relative stability strong law


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.

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