Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality

Wlodzimierz Bryc; Raouf Fakhfakh; Wojciech Mlotkowski

doi:10.19195/0208-4147.39.2.1

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arXiv · 1708.05343

Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality

Wlodzimierz Bryc·Raouf Fakhfakh·Wojciech Mlotkowski

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Abstract

This paper studies variance functions of Cauchy-Stieltjes Kernel families generated by compactly supported centered probability measures. We describe several operations that allow us to construct additional variance functions from known ones. We construct a class of examples which exhausts all cubic variance functions, and provide examples of polynomial variance functions of arbitrary degree. We also relate Cauchy-Stieltjes Kernel families with polynomial variance functions to generalized orthogonality. Our main results are stated solely in terms of classical probability; some proofs rely on analytic machinery of free probability.

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BibTeX RIS

Wlodzimierz Bryc, Raouf Fakhfakh, Wojciech Mlotkowski. 2019-10-12. Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality. https://doi.org/10.19195/0208-4147.39.2.1

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