On the construction of radially symmetric copulas in higher dimensions

We prove a representation theorem for copulas that are (simultaneously) symmetric and radially symmetric. We use this representation theorem to propose a method to construct an n-ary symmetric function that is radially symmetric, starting from an (n−1)-dimensional copula and an n-ary auxiliary function. We study the necessary and sufficient conditions on this auxiliary function that guarantee our construction method to result in a symmetric and radially symmetric n-dimensional copula. We examine several options for defining the auxiliary function in the trivariate case, inspired by the nesting of copulas, the lifting of copulas and product-like extensions. For each choice of auxiliary function, we provide several examples for different families of copulas.