Revisiting Gruss's inequality: covariance bounds,QDE but not QD copulas, and central moments

TL;DR

This paper revisits Gruss's inequality, demonstrating that covariance bounds can be improved under the less restrictive dependence structure of quadrant dependence in expectation (QDE), and introduces new bounds for central moments.

Contribution

It establishes new Gruss-type bounds based on QDE, explores copulas beyond QD, and derives optimal bounds for central moments.

Findings

QDE is a suitable dependence structure for covariance bounds.

New bounds for central moments are derived and shown to be optimal.

Examples with specially devised copulas illustrate the theoretical results.

Abstract

Since the pioneering work of Gerhard Gruss dating back to 1935, Gruss's inequality and, more generally, Gruss-type bounds for covariances have fascinated researchers and found numerous applications in areas such as economics, insurance, reliability, and, more generally, decision making under uncertainly. Gruss-type bounds for covariances have been established mainly under most general dependence structures, meaning no restrictions on the dependence structure between the two underlying random variables. Recent work in the area has revealed a potential for improving Gruss-type bounds, including the original Gruss's bound, assuming dependence structures such as quadrant dependence (QD). In this paper we demonstrate that the relatively little explored notion of `quadrant dependence in expectation' (QDE) is ideally suited in the context of bounding covariances, especially those that appear in the aforementioned areas of application. We explore this research avenue in detail, establish general Gruss-type bounds, and illustrate them with newly constructed examples of bivariate distributions, which are not QD but, nevertheless, are QDE. The examples rely on specially devised copulas. We supplement the examples with results concerning general copulas and their convex combinations. In the process of deriving Gruss-type bounds, we also establish new bounds for central moments, whose optimality is demonstrated.