Squaring the circle and cubing the sphere: Circular and spherical copulas (Q527578)

Summary: Do there exist circular and spherical copulas in \(\mathbb R^d\)? That is, do there exist circularly symmetric distributions on the unit disk in \(\mathbb R^2\) and spherically symmetric distributions on the unit ball in \(\mathbb R^d\), \(d\geq 3\), 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 \(d\geq 4\). A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in \(\mathbb R^2\) by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in \(\mathbb R^dx\) are also described, and determined explicitly for \(d=2\).