Semigroups of Distributions with Linear Jacobi Parameters Academic Article uri icon

abstract

  • We show that a convolution semigroup { t} of measures has Jacobi parameters polynomial in the convolution parameter t if and only if the measures come from the Meixner class. Moreover, we prove the parallel result, in a more explicit way, for the free convolution and the free Meixner class. We then construct the class of measures satisfying the same property for the two-state free convolution. This class of two-state free convolution semigroups has not been considered explicitly before. We show that it also has Meixner-type properties. Specifically, it contains the analogs of the normal, Poisson, and binomial distributions, has a Laha-Lukacs-type characterization, and is related to the q=0 case of quadratic harnesses. 2012 Springer Science+Business Media, LLC.

published proceedings

  • Journal of Theoretical Probability

author list (cited authors)

  • Anshelevich, M., & Motkowski, W.

citation count

  • 4

complete list of authors

  • Anshelevich, Michael||MÅ‚otkowski, Wojciech

publication date

  • December 2012