The contribution to the sum of the summand of maximum modulus
type: Item
instance of: scholarly article
title: The contribution to the sum of the summand of maximum modulus
zbMATH Open document ID: 0625.60031
author: William E. Pruitt
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: Limiting behaviour of sums and the term of maximal modulus is investigated. Let \(X_ n^{(r)}\) denote the rth largest in modulus of \(\{X_ 1,...,X_ n\}\). Next \(^{(r)}S_ n\) will be the trimmed sum with the largest summands discarded, i.e. \(^{(r)}S_ n=S_ n-X_ n^{(1)}-...-X_ n^{(r)}\), \(S_ n=\sum^{n}_{k=1}X_ k\), where \(X,X_ 1,X_ 2,..\). is a sequence of i.i.d. random variables. For \(\epsilon\in (0,1)\), let \[ u_ k=P[2^ k<| X| \leq 2^{k+1}| \quad | X| >2^ k],\quad v_ k=v_ k(\epsilon)=P[\epsilon^{-k}<| X| \leq \epsilon^{-k-1}| \quad | X| >\epsilon^{-k}\}. \] It is proved, among other things, that the following statements are equivalent: \(\sum^{\infty}_{k=1}(v_ k(\epsilon))^{r+1}<\infty\) for every (for some) \(\epsilon\in (0,1);\) \(\sum^{\infty}_{k=1}u_ k^{r+1}<\infty\); \(^{(r)}S_ n/X_ n^{(1)}\to 0\) a.s., \(n\to \infty\); \(X_ n^{(r+1)}/X_ n^{(1)}\to 0\) a.s., \(n\to \infty;\) \(\limsup_{n\to \infty}(X_ n^{(r+1)}/X_ n^{(1)})<1\) a.s.; \(\liminf_{n\to \infty}(| X_ n^{(1)}| /| S_ n|)\geq 1/r\) a.s.; \(\liminf_{n\to \infty}(| X_ n^{(1)}| /| S_ n|)>1/(r+1)\) a.s. The results of this paper extend or complete the ones by \textit{D. Z. Arov} and \textit{A. A. Bobrow}, Teor. Verojatn. Primen. 5, 415-434 (1960; Zbl 0098.112) and \textit{R. A. Maller} and \textit{S. I. Resnick}, Proc. Lond. Math. Soc., III. Ser. 49, 385-422 (1984; Zbl 0525.60036).
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4015811
zbMATH Keywords: dominance of maximal summand
zbMATH Keywords: slowly varying tails
zbMATH Keywords: maximal modulus
zbMATH Keywords: trimmed sum
reviewed by: Dominik Szynal
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992071
OpenAlex ID: W2063164203
DOI: 10.1214/AOP/1176992071
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description: scientific article; zbMATH DE number 4015811