Semigroups of Distributions with Linear Jacobi Parameters

Michael Anshelevich () and Wojciech Młotkowski ()
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Michael Anshelevich: Texas A&M University
Wojciech Młotkowski: University of Wrocław

Journal of Theoretical Probability, 2012, vol. 25, issue 4, 1173-1206

Abstract: Abstract We show that a convolution semigroup {μ 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.

Keywords: Convolution semigroups; Jacobi parameters; Free convolution; Meixner class; 46L54; 33C45; 46L53; 05A18; 60J25 (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-012-0403-x

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