A probabilistic counterpart of the Askey scheme for continuous polynomials
Consider the part of the Askey scheme of orthogonal polynomials which belongs to orthogonality measures with Lebesgue densities. For all polynomials there (Wilson, continuous Hahn, continuous dual Hahn, Jacobi, Meixner-Pollaczek, Laguerre, Hermite), the densities of the associated orthogonality measures are known where these measures are normalized as probability measures. The author checks by hand for each limit transition for the polynomials that the densities converge (after suitable rescaling) pointwise and that hence, by Scheffe's theorem, the associated probability measures converge w.r.t. total variation norm.NEWLINENEWLINEFor the more classical limits these results are known for a long time (even with estimates for the rate of convergence). The reviewer expects that at least a part of the limits should be obtained by a general argument from the convergence of the polynomials.
scientific article; zbMATH DE number 6031455