Upper and lower bounds for sums of random variables

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.