Semigroups of distributions with linear Jacobi parameters

Abstract

We show that a convolution semigroup 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.

v3: the article is merged back together with arXiv:1003.4025. A significant revision following suggestions by the referee. 2 pdf figures

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Cited by in corpus (2)

• Characterizations of some free random variables by properties of conditional moments of third degree polynomials
• New characterization of two-state normal distribution