Tails of the Moments for Sums with Dominatedly Varying Random Summands

The asymptotic behaviour of the tail expectation E((S-n(xi))(alpha) 1({Sn xi>x})) is investigated, where exponent a is a nonnegative real number and S-n(xi) = xi(1) + ... + xi(n) is a sum of dominatedly varying and not necessarily identically distributed random summands, following a specific dependence structure. It turns out that the tail expectation of such a sum can be asymptotically bounded from above and below by the sums of expectations E(xi(alpha)(i) 1({xi i>x})) with correcting constants. The obtained results are extended to the case of randomly weighted sums, where collections of random weights and primary random variables are independent. For illustration of the results obtained, some particular examples are given, where dependence between random variables is modelled in copulas framework.