Threshold copulas and positive dependence

Starting with a notion of positive dependence PP and with the family of the lower threshold copulas CtCt associated with a bivariate distribution having copula CC, we define different notions of positive dependence for CC, reflecting the dependence properties of the copulas CtCt for some tt.Then, we analyze some structural aspects of lower threshold copulas and of the given definitions. Furthermore we consider several specific cases arising from relevant special choices of PP (e.g., PQD, LTD, TP22 and PLR). Our analysis, in particular, allows us to present a number of relevant examples and counter-examples, which can be useful in the study of the tail dependence for a bivariate distribution.