Gerard Letac (Toulouse)
Jacobi polynomials and joint distributions in Rn with Beta margins and prescribed correlation
matrices.
If one calls copula in Rn any joint distribution of X in Rn such that the margins of X are uniform on (0,1) it
is a challenge to prove that for any correlation matrix R of order n then there exists a copula with this
correlation matrix R. We shall prove that this conjecture is correct for n<6 even while replacing the
uniform distribution by a beta distribution. We use for this Jacobi polynomials, a famous result due to
Gasper (1971) and the work of Angelo Koudou on Lancaster probabilities. The difficulty lies in the lack of
an easy characterization of the extreme points of the convex set of correlation matrices of order n.