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Semigroups of Distributions with Linear Jacobi Parameters

Author

Listed:
  • Michael Anshelevich

    (Texas A&M University)

  • Wojciech Młotkowski

    (University of Wrocław)

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.

Suggested Citation

  • Michael Anshelevich & Wojciech Młotkowski, 2012. "Semigroups of Distributions with Linear Jacobi Parameters," Journal of Theoretical Probability, Springer, vol. 25(4), pages 1173-1206, December.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:4:d:10.1007_s10959-012-0403-x
    DOI: 10.1007/s10959-012-0403-x
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    Cited by:

    1. Wiktor Ejsmont, 2014. "Characterizations of Some Free Random Variables by Properties of Conditional Moments of Third Degree Polynomials," Journal of Theoretical Probability, Springer, vol. 27(3), pages 915-931, September.

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