Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas

Abstract: Do there exist circular and spherical copulas in ℝ^d? That is, do there exist circularly symmetric distributions on the unit disk in ℝ² and spherically symmetric distributions on the unit ball in ℝ^d, d ≥ 3, whose one-dimensional marginal distributions are uniform? The answer is yes for d = 2 and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for d ≥ 4. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in ℝ² by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in ℝ^d are also described, and determined explicitly for d = 2.