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New examples of convolutions and non-commutative central limit theorems. (English) Zbl 0936.46050
Alicki, Robert (ed.) et al., Quantum probability. Workshop, Gdańsk, Poland, July 1-6, 1997. Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 43, 95-103 (1998).
A family of transformations on the set of all probability measures on the real line is introduced so as to define new examples of convolutions. Moreover, the associated noncommutative central limit theorems are studied. More precisely, for a given probability measure $$\mu$$ with compact support on $$\mathbb{R}$$, $G_\mu(z)= \int^{+\infty}_{-\infty} {d\mu(x)\over z- x},\quad z\in\mathbb{C}^+$ gives the definition of Cauchy transform $$G_\mu$$. A direct consequence of the Nevanlinna theorem [cf. e.g. H. Maassen, J. Funct. Anal. 106, 409-438 (1992; Zbl 0784.46047)] allows that the function $$G_{\mu_t}(z)$$, $$(z\in\mathbb{C}^+)$$ defined by the formula: $$1/G_{\mu_t}(z)= t/G_\mu(z)+ (1-t)z$$ turns out to be the Cauchy transform of a probability measure denoted by $$\mu_t$$. The measure $$\mu_t$$ is called the $$t$$-transform of a measure $$\mu$$ and the transformation $${\mathcal U}_t:\mu\mapsto \mu_t$$ is called the $$t$$-transformation. Note that the mapping $${\mathcal U}_t$$ is a multiplicative $$*$$-weakly continuous transformation, which commutes with dilations of measures. In what follows, $$\oplus$$ means a given convolution (for instance, any of the classical convolution, free Voiculescu convolution, Boolean free convolution, and other convolutions may be a possible choice). For two given probability measures $$\mu$$, $$\nu$$ on $$\mathbb{R}$$, a nonnegative number $$t$$, the $$t$$-convoluton $$\oplus_t$$ is defined by $\mu\oplus_t\nu= (\mu_t\oplus \nu_t)_{1/t}={\mathcal U}_{1/t}({\mathcal U}_t(\mu)\oplus{\mathcal U}_t(\nu)).$ The $$t$$-convolution provides a large new class of convolutions. Let $$D_\lambda\mu$$ be the dilation of a measure $$\mu$$ by a number $$\lambda$$, defined as $$D_\lambda\mu(A)= \mu(\lambda^{-1}A)$$ for an arbitrary measurable set $$A$$.
The main theorem asserts: “The sequence of measures $$D_{1/\sqrt n}\mu\oplus_t\cdots \oplus_t D_{1/\sqrt n}\mu$$ tends in the $$*$$-weak topology to a measure $$\nu^{(t)}$$, which is a transformation of the central limit measure $$\nu$$ for the given convolution $$\oplus$$.” Examples of the limit measure, related to the classical, free and Boolean convolutions, are given. For other related works, see e.g. M. Bożejko, M. Leinert and R. Speicher [Pac. J. Math. 175, No. 2, 357-388 (1996; Zbl 0874.60010)].
For the entire collection see [Zbl 0903.00097].

##### MSC:
 46L53 Noncommutative probability and statistics 60F05 Central limit and other weak theorems