On a class of free Levy laws related to a regression problem

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 Levy processes with quadratic conditional variances, and to prove a converse implication related to asymptotic freeness of random Wishart matrices.

LaTeX, v2: strengthened main theorem

References in corpus (6)

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• Cumulants in noncommutative probability II. Generalized Gaussian random variables
• Free martingale polynomials
• Conditional moments of q-Meixner processes
• Orthogonal polynomials with a resolvent-type generating function
• Classical and free infinitely divisible distributions and random matrices

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