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

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Do there exist circular and spherical copulas in R d ? That is, do there exist circularly symmetric distributions on the unit disk in R 2 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 R 2 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.

KEY WORDS AND PHRASES: Bivariate distribution, multivariate distribution, unit disk, unit ball, circular symmetry, spherical symmetry, circular copula, spherical copula, elliptical copula.


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