Multidimensional probability inequalities via spherical symmetry

TL;DR

This paper introduces a spherical symmetry-based method to extend probability inequalities from real-valued variables to Hilbert space-valued vectors, providing new formulas and optimal constants for moments and inequalities.

Contribution

It develops a general framework using spherical symmetry to translate inequalities for real variables into those for Hilbert space vectors, including moment expressions and optimal constants.

Findings

Derived a formula for the $p$th moment of the norm of Hilbert space-valued random vectors.

Extended von Bahr--Esseen-type inequalities with optimal constants to Hilbert space vectors.

Generalized inequalities relating population contrast and spread to Hilbert space-valued random vectors.

Abstract

Spherical symmetry arguments are used to produce a general device to convert identities and inequalities for the $p$th absolute moments of real-valued random variables into the corresponding identities and inequalities for the $p$th moments of the norms of random vectors in Hilbert spaces. Particular results include the following: (i) an expression of the $p$th moment of the norm of such a random vector $X$ in terms of the characteristic functional of $X$; (ii) an extension of a previously obtained von~Bahr--Esseen-type inequality for real-valued random variables with the best possible constant factor to random vectors in Hilbert spaces, still with the best possible constant factor; (iii) an extension of a previously obtained inequality between measures of "contrast between populations" and "spread within populations" to random vectors in Hilbert spaces.