Ratios of trimmed sums and order statistics
type: Item
instance of: scholarly article
title: Ratios of trimmed sums and order statistics
zbMATH Open document ID: 0764.60034
author: Harry Kesten
author: Ross A. Maller
published in: The Annals of Probability
publication date: 1993-02-22T00:00:00Z
review text: For i.i.d. \(X_ i\) let \(M_ n^{(n)}\leq\dots\leq M_ n^{(1)}\) be the ordered sample and \(X_ n^{(n)},\dots,X_ n^{(1)}\) be the sample arranged in the increasing order of modulus. Put \(S_ n=X_ 1+\dots+X_ n\), \(^{(r)}S_ n=S_ n-M_ n^{(1)}-\dots-M_ n^{(r)}\). \(^{(r)}\widetilde S_ n\) is defined by substituting \(M_ n^{(i)}\) by \(X_ n^{(i)}\) in \(^{(r)}S_ n\). \textit{R. A. Maller} and \textit{S. I. Resnick} [Proc. Lond. Math. Soc., III. Ser. 49, 385-422 (1984; Zbl 0525.60036)] and \textit{W. E. Pruitt} [Ann. Probab. 15, 885-896 (1987; Zbl 0625.60031)] studied i.i.d. \(X_ i\) for which \(^{(r)}S_ n\) and \(M_ n^{(r)}\) or \(^{(r)}\widetilde S_ n\) and \(X_ n^{(r)}\) are comparable in the sense that the corresponding ratios are bounded away from zero and infinity for large \(n\). In contrast to this the authors are interested in situations where large values of \(X\) are negligible compared with the sum. Roughly speaking this situation holds if \(X_ i\) has light tails whereas the situation studied by Maller and Resnick (loc. cit.) appears for heavy tails. Theorem 2.1 of the paper says that \(^{(r)}S_ n/M_ n^{(r)}\to\infty\) as \(n\to\infty\) holds iff the truncated mean \(\nu(x)=EXI(| X|<x)\) dominates \(xP(| X|>x)\), the truncated moment \(V(x)=EX^ 2I(| X|<x)\) and \((P(| X|\geq x)V(x))^{1/2}\) in the sense that the corresponding ratios tend to infinity. Surprisingly these conditions are independent of \(r\) so that the above convergence holds iff it is true for \(r=1\). The next two theorems concern the a.s. convergence of \(^{(r)}S_ n/X_ n^{(r)}\to\infty\) and \(^{(r)}S_ n/M_ n^{(r)}\to\infty\) for \(r=1,2,\dots\) and \(r=2,3,\dots\), respectively. Now necessary and sufficient conditions to the tails depend on \(r\) and it is shown by an example that the above mentioned equivalence is not true for a.s. convergence.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 120264
zbMATH Keywords: trimmed sums
zbMATH Keywords: order statistics
zbMATH Keywords: relative stability
zbMATH Keywords: strong law
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989530
OpenAlex ID: W1994496254
DOI: 10.1214/AOP/1176989530
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description: scientific article; zbMATH DE number 120264