Cauchy–Stieltjes families with polynomial variance functions and generalized orthogonality

This paper studies variance functions of Cauchy–Stieltjes Kernel CSK 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 c...

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Bibliographic Details
Published in:Probability and mathematical statistics 2019-01, Vol.39 (2), p.237-258
Main Authors: Bryc, Włodzimierz, Fakhfakh, Raouf, Młotkowski, Wojciech
Format: Article
Language:English
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Summary:This paper studies variance functions of Cauchy–Stieltjes Kernel CSK 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 CSK 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.
ISSN:0208-4147
2300-8113
DOI:10.19195/0208-4147.39.2.1