  This paper is organized in three parts closely related to closure properties
of heavy-tailed distributions and heavy-tailed random vectors. In the first
part we consider two random variables X and Y with distributions F and G
respectively. We assume that these random variables satisfy one type of a weak
dependence structure. Under some mild conditions, we examine whether their
product convolution distribution H belongs in the same distribution class of
the distribution F. Namely we establish the closure property with respect to
the product convolution, under this specific weak dependence structure, in the
classes ERV, C, D, M, OS, OL, PD and K. Further in the second part we introduce
a new distribution class, which satisfies some closure properties such as
convolution, product convolution and mixture. Although the multivariate regular
variation is well-established distribution, it does not happen in other heavy
tailed random vectors. Therefore in the third part we introduced the class of
dominatedly varying vectors and positively decreasing random vectors and we
study the closure property of the scalar product under independent and
dependent cases. Furthermore we study the closure property of the first class
under convolution and mixture, and we study the distribution of stopped sums
where the summands are random vectors which belongs to this class. Some of
these results holds and for positively decreasing random vectors.
