Multivariate Bertino copulas

This paper provides a partial answer to an open problem recently posed by R. Mesiar and J. Kalická regarding the existence of an n-dimensional Bertino copula with a given diagonal section for any n⩾2. It is known that for any 2-diagonal function, there exists a 2-dimensional Bertino copula that has the given 2-diagonal function as diagonal section. In the present paper, we introduce the notion of a regular n-diagonal function and we characterise for any n⩾3 the sets Dn of regular n-diagonal functions for which there exists an n-dimensional Bertino copula whose diagonal section coincides with the given n-diagonal function. We prove that Dn+1 is strictly included in Dn, for all n⩾2, and that Dn is the set of all increasing n/(n−1)-Lipschitz continuous n-diagonal functions. As a by-product, we show that all marginal copulas of an n-dimensional Bertino copula are Bertino copulas themselves. Examples are given to illustrate the construction of an n-dimensional Bertino copula with a given diagonal section and the characterisation of the sets Dn.