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arXiv · 1001.1540

Semigroups of distributions with linear Jacobi parameters

Michael Anshelevich·Wojciech Młotkowski
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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.

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Michael Anshelevich, Wojciech Młotkowski. 2011-07-23. Semigroups of distributions with linear Jacobi parameters. https://doi.org/10.1007/s10959-012-0403-x

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