On a construction of multivariate distributions given some multidimensional marginals

This paper was written during an invited stay of the authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors warmly thank the VIASM institute for his support. The authors are very grateful to the anonymous reviewers, the associate editor andthe editor, for their careful reading and their many insightful comments and suggestions.International audienceIn this paper, we investigate the link between the joint law of a d-dimensional random vector and the law of some of its multivariate marginals. We introduce and focus on a class of distributions, that we call projective, for which we give detailed properties. This allows us to obtain necessary conditions for a given construction to be projective. We illustrate our results by proposing some theoretical projective distributions, as elliptical distributions or a new class of distribution having given bivariate margins. In the case where the data do not necessarily correspond to a projective distribution, we also explain how to build proper distributions while checking that the distance to the prescribed projections is small enough