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

摘要: 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, decision making under uncertainly. been established mainly most general dependence structures, meaning no restrictions on structure between two underlying random variables. Recent area has revealed a potential improving bounds, including original bound, assuming structures quadrant (QD). In this paper we demonstrate that relatively little explored notion `quadrant expectation' (QDE) is ideally suited context bounding covariances, especially those appear aforementioned application. We explore research avenue detail, establish illustrate them with newly constructed examples bivariate distributions, which are not QD but, nevertheless, QDE. The rely specially devised copulas. supplement results concerning copulas their convex combinations. process deriving also new central moments, whose optimality demonstrated.