Closure properties of the second-order regular variation under convolutions

Second-order regular variation (2RV) is a refinement of the concept of RV which appears in a natural way in applied probability, statistics, risk management, telecommunication networks, and other fields. Let X1, …, Xn be independent and non negative random variables with respective survival functions F‾1,...,F‾n$ {\overline{F}}_1, \ldots , {\overline{F}}_n$, and assume that F‾i$ {\overline{F}}_i$ is of 2RV with the first-order parameter − α and the second-order parameter ρi for each i and that all the F‾i$ {\overline{F}}_i$ are tail-equivalent. It is shown, in this paper, that the survival function of the sum ∑ni = 1Xi is also of 2RV. The main result is applied to establish the 2RV closure property for the randomly weighted sum ∑ni = 1ΘiXi, where the weights Θ1, …, Θn are independent and non negative random variables, independent of X1, …, Xn, and satisfying certain moment conditions.