Sharp Lower and Upper Bounds for the Covariance of Bounded Random Variables

TL;DR

This paper establishes precise bounds for the covariance of bounded random variables based on known expectations and variances, extending existing inequalities and providing standardized measures relevant in genetics.

Contribution

It introduces sharp bounds for covariance with known moments and offers standardized measures, including a novel one aligning with genetic dependence metrics.

Findings

Derived sharp covariance bounds based on expectations and variances

Extended Bhatia-Davis Inequality for variances

Proposed standardized covariance measures, including a genetic dependence measure

Abstract

In this paper we derive sharp lower and upper bounds for the covariance of two bounded random variables when knowledge about their expected values, variances or both is available. When only the expected values are known, our result can be viewed as an extension of the Bhatia-Davis Inequality for variances. We also provide a number of different ways to standardize covariance. For a binary pair random variables, one of these standardized measures of covariation agrees with a frequently used measure of dependence between genetic variants.