The asymptotic behaviour of the tail expectation ?E(Snξ)α?{Snξ>x} is investigated, where exponent α is a nonnegative real number and Snξ=ξ1+…+ξn 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 ?Eξiα?{ξi>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.

In this paper we consider the product $\Pi _{n}=\prod_{k=1}^{n}\xi _{k}$ Π n = ∏ k = 1 n ξ k of n independent normally distributed zero mean random variables $\xi _{1},\dots ,\xi _{n}$ ξ 1 , … , ξ n . We derive an asymptotic formula for the survival probability ${\mathbb{P}}(\Pi _{n}>x)$ P ( Π n > x ) , as $x\to \infty $ x → ∞ , with the first remaining term.