Stable laws and domains of attraction in free probability theory. (English) Zbl 0945.46046

The study of sums of independent random variables in classical versus free probability differs, formally, by employment of the classical convolution \(*\) on the set \(\mathcal M\) of Borel probability measures over \(\mathbb R\) and, respectively, the free (additive) convolution \(\boxplus\) [see, e.g., the authors, Math. Res. Lett. 2, No. 6, 791-795 (1995; Zbl 0846.60023), J. Funct. Anal. 140, No. 2, 359-380 (1996; Zbl 0855.60023), Ann. Probab. 24, No. 1, 453-465 (1996; Zbl 0862.46036]. Having developed an asymptotic formula for Voiculescu’s analogue \(\varphi_{\mu}\) of the Fourier transform of \(\mu\in{\mathcal M}\), the authors translate weak convergence in \(\mathcal M\) into convergence properties of the corresponding \(\varphi_{\mu}\). The Lévy-Khinchin formula and its free counterpart characterize infinitely divisible distributions. A bijection \(\nu\leftrightarrow\nu'\) is established between \(*\)-infinitely divisible \(\nu\in {\mathcal M}\) and \(\boxplus\)-infinitely divisible \(\nu'\in{\mathcal M}\) such that their partial domains of attraction coincide. The same bijection between the \(*\)-stable law \(\nu\) and \(\boxplus\)-stable \(\nu'\) identifies their domains of attraction, \(D_{*}(\nu)=D_{\boxplus}(\nu')\). These domains \(C_{\alpha,\theta}\) in both theories are classified by a pair \((\alpha,\theta)\), the stability index \(\alpha\in(0,2)\) and \(\theta\in[-1,1]\) (except the singleton case \(\alpha=2\) of normal \(\nu\) versus semicircular law \(\nu'\)): there is a unique equivalence class of \(*\)-stable laws \([\nu]\) (respectively, \(\boxplus\)-stable \([\nu']\)) such that \(\nu\in C_{\alpha,\theta}\) (respectively, \(\nu'\in C_{\alpha,\theta}\)) and \(D_{*}(\nu)=D_{\boxplus}(\nu')=C_{\alpha,\theta}\). The measures \(\mu\in C_{\alpha,\theta}\) are characterized in terms of the behaviour of their Cauchy transforms on the imaginary axis. For the other, Boolean, convolution on \(\mathcal M\) introduced by R. Speicher and R. Woroudi [Fields Inst. Commun. 12, 267-279 (1997; 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.


46L54 Free probability and free operator algebras
60E10 Characteristic functions; other transforms
46L53 Noncommutative probability and statistics
60E07 Infinitely divisible distributions; stable distributions
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