Upper and Lower Bounds for Sums of Random Variables.

Abstract

In this contribution, the upper bounds for sums of dependent random variables X1 + X2 +...+ Xn derived by using comonotonicity are sharpened for the case when there exists a random variable Z such that the distribution functions of the Xi, given Z = z, are known. By a similar technique, lower bounds are derived. A numerical application for the case of lognormal random variables is given.

Keywords: Dependent risks, Comonotonicity, Convex order, Cash-flows, Present values, Stochastic annuities