Stable laws and domains of attraction in free probability theory

Hari Bercovici and Vittorino Pata

Review from Zentralblatt MATH:

The study of sums of independent random variables in classical versus free probability differs, formally, by employment of the classical convolution $*$ on the set $\cal M$ of Borel probability measures over $\Bbb R$ and, respectively, the free (additive) convolution $\boxplus$ [see, e.g., the authors, Math. Res. Lett. 2, No. 6, 791-795 (1995; Zbl 0872.46033)] it is shown that the limit law can be defined by the classical one, just as in the case of free convolution. In the appendix by P. Biane, the unimodality of free stable distributions and Zolotarev's duality for them are obtained upon a detailed analysis of their densities.

Reviewed by Andrej Bulinski

Keywords: free convolution of probability measures; infinitely divisible law; stable law; (partial) domain of attraction; Voiculescu transformation; convolution; Fourier transform; Lévy-Khinchin formula; *-stable laws; Cauchy transform; Zolotarev's duality; unimodality of free stable distributions

Classification (MSC2000): 46L54 60E10 46L53 60E07

Full text of the article:

• PDF file (267 kilobytes)