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Addition of freely independent random variables. (English) Zbl 0784.46047
The important problem of Voiculescu’s free probability calculus is to find the probability distribution $$\mu$$ of the sum $$X_ 1+ X_ 2$$ of two freely independent random variables, given the distributions $$\mu_ 1$$ and $$\mu_ 2$$ of the summands. Here, $$X_ 1$$ and $$X_ 2$$ are linear operators on a Hilbert space, and the distributions are taken with respect to a fixed vector state. In the case of bounded random variables, the above problem was solved by D. Voiculescu in J. Funct. Anal. 66, 323-346 (1986; Zbl 0651.46063). The author relaxes the assumptions on $$X_ i$$, considering (self-adjoint) variables of finite variance. The operator $$X_ 1+ X_ 2$$ is then essentially self-adjoint and the distribution $$\mu$$ of its closure is $$\mu_ 1\boxplus \mu_ 2$$, a measure whose reciprocal Cauchy transform is $$F_ 1\boxplus F_ 2$$ – the free convolution product of the reciprocal Cauchy transforms $$F_ 1$$, $$F_ 2$$ of the distributions $$\mu_ 1$$, $$\mu_ 2$$. A beautiful geometric definition of the free convolution product is preceded by a careful examination of the properties of the reciprocal Cauchy transform. A free central limit theorem and the Lévy-Khinchin formula, corresponding to the case considered, are also proved.

MSC:
 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras
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References:
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