Upper and lower bounds for sums of random variables.

In this contribution, the upper bounds for sums of dependent random variables X-1 + X-2 + ... + X-n derived by using comonotonicity are sharpened for the case when there exists a random variable Z such that the distribution functions of the X-i, 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. (C) 2000 Elsevier Science B.V. All rights reserved.dependent risks; comonotonicity; convex order; cash-flows; present values; stochastic annuities; risks;