Characterizations of Multivariate Implicit Dependence Copulas

The copula C of continuously distributed random variables X-1,..., X-d is said to be an implicit dependence copula if there are Borel functions alpha(1),..., alpha(d) such that alpha(1)(X-1),..., alpha(d)(X-d) are equal almost surely and continuously distributed, that is their common distribution function is continuous. Bivariate implicit dependence copulas have recently been characterized in terms of a generalized Markov product. In this manuscript, the characterizations are extended to the multivariate case in terms of a product of d copulas, called A -product where A is a class of copulas At, t is an element of [0, 1]. The class of implicit dependence d-copulas are characterized as A -products of d complete dependence copulas. Explicit forms of the joining copulas A(t) are obtained when the functions ai are countably piecewise monotonic surjections.