On aggregation sets and lower-convex sets

It has been a challenge to characterize the set of all possible sums of random variables with given marginal distributions, referred to as an aggregation set in this paper. We study the aggregation set via its connection to the corresponding lower-convex set, which is the set of all sums of random variables that are smaller than the respective marginal distributions in convex order. Theoretical properties of the two sets are discussed, assuming that all marginal distributions have finite mean. In particular, an aggregation set is always a subset of its corresponding lower-convex set, and the two sets are identical in the asymptotic sense after scaling. We also show that a lower-convex set is identical to the set of comonotonic sums with the same marginal constraint. The main theoretical results contribute to the field of multivariate distributions with fixed margins. Key-words: aggregation set; convex order; comonotonicity; dependence uncertainty; Fréchet classes.