Characterization of product probability measures on ℝd in terms of their orthogonal polynomials

In paper [1] the d-dimensional analogue of the Jacobi parameters has been individuated in a pair of sequences ((a.n0),(Ω∼n)), where (a.n0) is a sequence of Hermitean matrices and Ω∼n(n ℕ) a positive definite kernel with values in the linear operators on the n-th space of the orthogonal gradation. In this paper we prove that product measures on ℝd are characterized by the property that the (a.n0) are diagonal and the (Ω∼n) quasidiagonal (see Definition 2 below) in the orthogonal polynomial basis.