Moment and geometric probability inequalities arising from arrangement increasing functions
type: Item
instance of: scholarly article
title: Moment and geometric probability inequalities arising from arrangement increasing functions
zbMATH Open document ID: 0638.60013
author: Philip J. Boland
author: Frank Proschan
author: Y. L. Tong
published in: The Annals of Probability
publication date: 1988-01-01T00:00:00Z
review text: For a given vector \(x=(x_ 1,...,x_ n)\in R^ n\) let \(x\uparrow\) and \(x\downarrow\) denote, respectively, the vector with the components of x arranged in increasing (decreasing) order. For any permutation \(\pi\) of \(\{1,2,...,n\}\) let \(x_{\pi}=(x_{\pi (1)},...,x_{\pi (n)})\). For vectors \(x,y,u,v\in R^ n\) we write \((x,y)=(u,v)\) if there exists a permutation \(\pi\) of \(\{\) 1,...,n\(\}\) such that \(x_{\pi}=u\) and \(y_{\pi}=v\). Further, (x,y)\(\leq^{a}(u,v)\) if there exists a finite number of vectors z 1,...,z k such that \[ (i)\quad (x,y)=^{a}(x\uparrow,z^ 1)\text{ and } (z\uparrow,z^ k)=^{a}(u,v), \] and (ii) \(z^{i-1}\) can be obtained from z i by interchanging two components of \(z^ i\), the first of which is less than second. A function g of two vectors arguments x and y for which \(g(x,y)\leq g(u,v)\) when \((x,y)\leq^{a}(u,v)\) is said to be arrangement increasing (or AI). \textit{M. Hollander}, \textit{F. Proschan} and \textit{I. Sethuraman} [Ann. Stat. 5, 722-733 (1977; Zbl 0356.62043)] have shown, that the class of AI functions is closed under some basic operations. In particular, the convolution of AI functions is AI. In the present paper, the authors prove some corollaries of this result of Hollander, Proschman and Sethuraman. In section 2, the authors show that many geometric probabilities of random vectors X with exchangeable densities f(x) (that is, the distribution of \(x_{\pi}\) does not depend on the permutation \(\pi)\) are AI functions. In section 3 they reduce examples AI functions of geometric type from AI families of densities. Some moment inequalities are also obtained.
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60G09
Mathematics Subject Classification ID: 62H10
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 4040922
zbMATH Keywords: arrangement increasing
zbMATH Keywords: exchangeable densities
zbMATH Keywords: moment inequalities
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991911
OpenAlex ID: W2038862688
DOI: 10.1214/AOP/1176991911
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description: scientific article; zbMATH DE number 4040922