On the invariant properties of notions of positive dependence and copulas under increasing transformations

► We provide two counterexamples for Theorem 3.10.19 of Müller and Stoyan (2002). ► We prove the invariant properties of CIS, CI, PDS, and PDUO. ► A continuous random vector is PDS (PDUO) if and only of its copula is PDS (PDUO). ► A rigorous proof for the invariant of copulas under non-decreasing transformations. ► We give a characterization of a continuous PDUO random vector by its survival copula.