Bounds for functions of multivariate risks

Li et al. [Distributions with Fixed Marginals and Related Topics, vol. 28, Institute of Mathematics and Statistics, Hayward, CA, 1996, pp. 198–212] provide bounds on the distribution and on the tail for functions of dependent random vectors having fixed multivariate marginals. In this paper, we correct a result stated in the above article and we give improved bounds in the case of the sum of identically distributed random vectors. Moreover, we provide the dependence structures meeting the bounds when the fixed marginals are uniformly distributed on the k-dimensional hypercube. Finally, a definition of a multivariate risk measure is given along with actuarial/financial applications.