  We consider a new approach in the definition of two-dimensional heavy-tailed
distributions. Namely, we introduce the classes of two-dimensional long-tailed,
of twodimensional dominatedly varying and of two-dimensional consistently
varying distributions. Next, we define the closure property with respect to
two-dimensional convolution and to joint max-sum equivalence in order to study
if they are satisfied by these classes. Further we examine the joint behavior
of two random sums, under generalized tail asymptotic independence. Afterward
we study the closure property under scalar product and two dimensional product
convolution and by these results we extended our main result in the case of
jointly randomly weighted sums. Our results contained some applications where
we establish the asymptotic expression of the ruin probability in a
two-dimensional discrete-time risk model.
