Inequalities for probability contents of convex sets via geometric average
It is shown that: If \((X_ 1,X_ 2)\) is a permutation-invariant central convex unimodal random vector and if A is a symmetric (about 0), permutation-invariant convex set then \(P\{(aX_ 1,X_ 2/a)\in A\}\) is non-decreasing as a varies from \(0+\) to 1, and is nonincreasing as a varies from 1 to \(\infty\) (that is, \(P\{(a_ 1X_ 1,a_ 2X_ 2)\in A\}\) is a Schur-concave function of (log \(a_ 1,\log a_ 2))\). Some extensions of this result for the n-dimensional case are discussed. Applications are given for elliptically contoured distributions and scale parameter families.
scientific article; zbMATH DE number 4044841