On a class of free Lévy laws related to a regression problem |
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Authors: | Marek Bo?ejko W?odzimierz Bryc |
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Affiliation: | a Instytut Matematyczny, Uniwersytet Wroc?awski, Pl. Grunwaldzki 2/4, 50-384 Wroc?aw, Poland b Department of Mathematics, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA |
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Abstract: | The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum. We apply this result to describe free Lévy processes with quadratic conditional variances, and to prove a converse implication related to asymptotic freeness of random Wishart matrices. |
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Keywords: | Free Meixner law Free cumulants |
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