The normal distribution is ?-infinitely divisible |
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| Authors: | Serban T Belinschi Marek Bo?ejko Franz Lehner Roland Speicher |
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| Institution: | aDepartment of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK, S7N, Canada;bInstitute of Mathematics of the Romanian Academy, Bucharest, Romania;cInstytut Matematyczny, Uniwersytet Wroc?awski, Pl. Grunwaldzki 2/4, 50-384 Wroc?aw, Poland;dInstitute for Mathematical Structure Theory, Graz Technical University, Steyrergasse 30, 8010 Graz, Austria;eQueen's University, Department of Mathematics and Statistics, Jeffery Hall, Kingston, ON K7L 3N6, Canada;fUniversität des Saarlandes, FR 6.1 – Mathematik, Campus E2 4, 66123 Saarbrücken, Germany |
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| Abstract: | We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a sub-family of Askey–Wimp–Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution. |
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| Keywords: | MSC: primary 46L54 secondary 05C30 |
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