  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 structure.
Namely we examine the closure property with respect to minimum, maximum,
convolution product and convolution. 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. Also we revisited
the (independent) convolution closure problem of subexponentiality, in the case
of subexponential positively decreasing class. In some classes we discuss the
closedness of randomly stopped sums. In the last section we study the closure
property with respect to minimum for two classes of random vectors.
