Gelfand pairs and zonal polynomials in Multivariate Statistical Analysis

This article shows how Harmonic Analysis on matrix groups can solve specifc problems of Multivariate Statistical Analysis. In particular are studied the properties of zonal polynomials, seen as spherical functions of a classical Gelfand pair. A duality formula allows to write the zonal polynomials as spherical functions of a finite Gelfand pair. Finally a combinatorial technique for the calculation of the zonal polynomials shows that these are the analogous, in the non-central Wishart distribution, of power in central Wishart distribution.