Multivariate Bertino copulas

This paper provides a partial answer to an open problem recently posed by R. Mesiar and J. Kalicka 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 D-n 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 D-n+1 is strictly included in D-n, for all n >= 2, and that D-n 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 D-n. (C) 2015 Elsevier Inc. All rights reserved.