Semigroups of Distributions with Linear Jacobi Parameters Academic Article

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.

authors

• Anshelevich, Michael

published proceedings

• JOURNAL OF THEORETICAL PROBABILITY

author list (cited authors)

• Anshelevich, M., & Mlotkowski, W.

citation count

• 6

complete list of authors

• Anshelevich, Michael||Mlotkowski, Wojciech

publication date

• December 2012

publisher

• Springer Nature  Publisher

published in

• Journal of Theoretical Probability  Journal

keywords

• Convolution Semigroups
• Free Convolution
• Jacobi Parameters
• Meixner Class

Digital Object Identifier (DOI)

• 10.1007/s10959-012-0403-x

start page

• 1173

end page

• 1206

volume

• 25

issue

• 4

URL

• http://dx.doi.org/10.1007/s10959-012-0403-x