BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

One dimensional stable distributions are back! Fourati, Sonia


It has been known for some time that extremal $\alpha$ stable variables ($S_{\alpha}$) are Generalized Gamma Convolutions. Recently, L. Bondesson has shown that the family of GGC distributions is stable by multiplicative convolution. I will exhibit a first consequence of this result with a short proof of the fact that $\alpha$-stable densities are hyperbolically completely monotone (HCM in short). This result was first obtained by T. Simon and P. Bosch. I will give a general representation of all positive $\alpha$-densities. This result can be used for numerical reasons and gives a new point of view on the famous Zolotarev result on unimodality problem. Also we obtain the optimal $\beta$ such that $S_{\alpha}^{\beta}$ has an HCM density.

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