Weak Dependence Notions and Their Mutual Relationships

New weak notions of positive dependence between the components X and Y of a random pair (X,Y) have been considered in recent papers that deal with the effects of dependence on conditional residual lifetimes and conditional inactivity times. The purpose of this paper is to provide a structured framework for the definition and description of these notions, and other new ones, and to describe their mutual relationships. An exhaustive review of some well-know notions of dependence, with a complete description of the equivalent definitions and reciprocal relationships, some of them expressed in terms of the properties of the copula or survival copula of (X,Y), is also provided.