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 naturalway in applied probability, statistics, risk-management, telecommunication networks, and other fields. Let X-1,...,X-n be independent and non negative random variables with respective survival functions (F) over bar (1),..., (F) over bar (n), and assume that (F) over bar (i) is of 2RV with the first-order parameter -alpha and the second-order parameter rho(i) for each i and that all the (F) over bar (i) are tail-equivalent. It is shown, in this paper, that the survival function of the sum Sigma(n)(i=1) X-i is also of 2RV. The main result is applied to establish the 2RV closure property for the randomly weighted sum Sigma(n)(i=1) Theta X-i(i), where the weights Theta(1),..., Theta(n) are independent and non negative random variables, independent of X-1,..., X-n, and satisfying certain moment conditions.