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arXiv · 1805.01133

Some properties of the free stable distributions

Takahiro Hasebe·Thomas Simon·Min Wang
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Abstract

We investigate certain analytical properties of the free $\alpha-$stable densities on the line. We prove that they are all classically infinitely divisible when $\alpha\le 1$, and that they belong to the extended Thorin class when $\alpha \leq 3/4.$ The L\'evy measure is explicitly computed for $\alpha =1,$ showing that the free 1-stable random variables are not Thorin except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when $\alpha > 1.$ In the one-sided case we prove, refining unimodality, that the densities are whale-shaped that is their successive derivatives vanish exactly once. Finally, we derive a collection of results connected to the fine structure of the one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, a new identity for the Beta-Gamma algebra, and several intrinsic properties of whale-shaped densities.

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Takahiro Hasebe, Thomas Simon, Min Wang. 2018-05-03. Some properties of the free stable distributions. https://arxiv.org/abs/1805.01133

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