Convex Order Bounds for Sums of Dependent Log-Elliptical Random Variables

In this paper, we construct upper and lower convex order bounds for the distribution of a sum of non-independent log-elliptical random variables. These bounds are applications of the ideas developed in Kaas, Dhaene & Goovaerts (2000). The class of multivariate log-elliptical random variables is an extension of the class of multivariate log-normal random variables. Hence, the results presented here are natural extensions of the results presented in Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002a, 2002b), where bounds for sums of log-normal random variables have been derived. The upper bound is based on the idea of replacing the sum of log-elliptical random variables by a sum of random variables with the same marginals, but with a dependency structure described by the comonotonic copula. Lower bounds and improved upper bounds are constructed by including additional information about the dependency structure by introducing a conditioning random variable, similar to that developed in Vandu¤el, Hoedemakers & Dhaene (2004).