The normal distribution is $\boxplus$-infinitely divisible

Details The normal distribution is $\boxplus$-infinitely divisible The normal distribution is $\boxplus$-infinitely divisible

2009-10-22

arxiv.org/abs/0910.4263v2

Mathematics

Operator Algebras

AMS LaTeX, 29 pages, using tikz and 3 eps figures; new proof including infinite divisibility of certain Askey-Wilson-Kerov dis

Scientific paper

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 subfamily 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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