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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:
  Akhiezer, N. I.; Glazman, I. M.: Theory of linear operators in Hilbert space. (1963) · Zbl 0098.30702  Arnold, L.: On the asymptotic distribution of the eigenvalues of random matrices. J. math. Anal. appl. 20, 262-268 (1967) · Zbl 0246.60029  Arnold, L.: On Wigner’s semicircle law for the eigenvalues of random matrices. Z. wahrsch. Verw. gebiete 19, 191-198 (1971) · Zbl 0212.51006  Billingsley, P.: Convergence of probability measures. (1968) · Zbl 0172.21201  Ching, W. -M: Free products of von Neumann algebras. Trans. amer. Math. soc. 178, 147-163 (1973) · Zbl 0264.46066  Grenander, U.; Silverstein, J.: Spectral analysis of networks with random topologies. SIAM J. Appl. math. 32, No. No. 2, 449-519 (1977) · Zbl 0355.94043  Glockner, P.; Schürmann, M.; Speicher, R.: Realization of free white noises. SFB-preprint 564 (1990) · Zbl 0724.60104  Jonsson, D.: Some limit theorems for the eigenvalues of a sample covariance matrix. J. multivariate anal. 12, 1-38 (1982) · Zbl 0491.62021  Kümmerer, B.; Speicher, R.: Stochastic integration on the Cuntz algebra o\infty. (1989) · Zbl 0787.46052  Marčenko, V. A.; Pastur, L. A.: Distributions of eigenvalues of some sets of random matrices. Math. USSR-sb. 1, 507-536 (1967)  Speicher, R.: A new example of ”independence” and ”white noise”. Probab. theory related fields 84, 141-159 (1990) · Zbl 0671.60109  Speicher, R.: Stochastic integration on the full Fock space with the help of a kernel calculus. SFB-preprint 556 (1990) · Zbl 0728.60060  Voiculescu, D.: Symmetrices of some reduced free product c\ast-algebras. Lecture notes in mathematics 1132 (1985)  Voiculescu, D.: Addition of certain non-commuting random variables. J. funct. Anal. 66, 323-346 (1986) · Zbl 0651.46063  Voiculescu, D.: Multiplication of certain non-commuting random variables. J. operator theory 18, 223-235 (1987) · Zbl 0662.46069  Voiculescu, D.: Limit laws for random matrices and free products. Invent. math. 104, 201-220 (1991) · Zbl 0736.60007  Voiculescu, D.: Free noncommutative random variables, random matrices and the II1 factors of free groups. (1990)  Wachter, K. W.: The strong limits of random matrix spectra for sample matrices of independent elements. Ann. probab. 6, 1-18 (1978) · Zbl 0374.60039  Wigner, E. P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. of math. 62, 548-564 (1955) · Zbl 0067.08403  Wigner, E. P.: On the distributions of the roots of certain symmetric matrices. Ann. of math. 67, 325-327 (1958) · Zbl 0085.13203
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