On matrix variance inequalities

TL;DR

This paper extends matrix variance inequalities, including Chernoff, Poincare, and Bessel types, to matrices of any size and broad distribution classes, generalizing prior results for Normal and Gamma distributions.

Contribution

It introduces generalized matrix variance inequalities applicable to a wide range of distributions and matrix sizes, expanding the scope of previous univariate results.

Findings

Derived matrix Poincare-type inequalities

Established Bessel-type inequalities for matrices

Generalized Chernoff's inequality for broader distributions

Abstract

Olkin and Shepp (2005, J. Statist. Plann. Inference, vol. 130, pp. 351--358) presented a matrix form of Chernoff's inequality for Normal and Gamma (univariate) distributions. We extend and generalize this result, proving Poincare-type and Bessel-type inequalities, for matrices of arbitrary order and for a large class of distributions.