×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akhiezer, N. I.; Glazman, I. M.: Theory of linear operators in Hilbert space. (1963) · Zbl 0098.30702
[2] Arnold, L.: On the asymptotic distribution of the eigenvalues of random matrices. J. math. Anal. appl. 20, 262-268 (1967) · Zbl 0246.60029
[3] Arnold, L.: On Wigner’s semicircle law for the eigenvalues of random matrices. Z. wahrsch. Verw. gebiete 19, 191-198 (1971) · Zbl 0212.51006
[4] Billingsley, P.: Convergence of probability measures. (1968) · Zbl 0172.21201
[5] Ching, W. -M: Free products of von Neumann algebras. Trans. amer. Math. soc. 178, 147-163 (1973) · Zbl 0264.46066
[6] 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
[7] Glockner, P.; Schürmann, M.; Speicher, R.: Realization of free white noises. SFB-preprint 564 (1990) · Zbl 0724.60104
[8] Jonsson, D.: Some limit theorems for the eigenvalues of a sample covariance matrix. J. multivariate anal. 12, 1-38 (1982) · Zbl 0491.62021
[9] Kümmerer, B.; Speicher, R.: Stochastic integration on the Cuntz algebra o\infty. (1989) · Zbl 0787.46052
[10] Marčenko, V. A.; Pastur, L. A.: Distributions of eigenvalues of some sets of random matrices. Math. USSR-sb. 1, 507-536 (1967)
[11] Speicher, R.: A new example of ”independence” and ”white noise”. Probab. theory related fields 84, 141-159 (1990) · Zbl 0671.60109
[12] Speicher, R.: Stochastic integration on the full Fock space with the help of a kernel calculus. SFB-preprint 556 (1990) · Zbl 0728.60060
[13] Voiculescu, D.: Symmetrices of some reduced free product c\ast-algebras. Lecture notes in mathematics 1132 (1985)
[14] Voiculescu, D.: Addition of certain non-commuting random variables. J. funct. Anal. 66, 323-346 (1986) · Zbl 0651.46063
[15] Voiculescu, D.: Multiplication of certain non-commuting random variables. J. operator theory 18, 223-235 (1987) · Zbl 0662.46069
[16] Voiculescu, D.: Limit laws for random matrices and free products. Invent. math. 104, 201-220 (1991) · Zbl 0736.60007
[17] Voiculescu, D.: Free noncommutative random variables, random matrices and the II1 factors of free groups. (1990)
[18] 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
[19] Wigner, E. P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. of math. 62, 548-564 (1955) · Zbl 0067.08403
[20] Wigner, E. P.: On the distributions of the roots of certain symmetric matrices. Ann. of math. 67, 325-327 (1958) · Zbl 0085.13203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.