Moment and geometric probability inequalities arising from arrangement increasing functions
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.
scientific article; zbMATH DE number 4040922