Positively Decreasing and Related Distributions under Dependence

We consider closure properties in the class of positively decreasing distributions. Our results stem from different types of dependence, but each type belongs in the family of asymptotically independent dependence structures. Namely, we examine the closure property with respect to minimum, maximum, convolution, convolution roots, and convolution product. Furthermore, we take into account some closure properties of the class of generalized subexponential positively decreasing distributions, as also we introduce and study the class of the generalized long-tailed positively decreasing distributions. Further, we consider the convolution closure problem of subexponentiality in the case of a subexponential positively decreasing class. In some classes we discuss the closure property of randomly stopped sums. Finally, we revisit some problems of infinity divisibility distributions in a subexponential positively decreasing class of distributions, and we study the asymptotic relation between jump measure and Lévy measure of superpositions of the Ornstein--Uhlenbeck process in the case where jump measure has positive and finite Matuszewska indexes.