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The arithmetic of distributions in free probability theory. (English) Zbl 1239.46046

Summary: We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup \(\mathbb{M}\) equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of \(\mathbb{M}\) contains either indecomposable (“prime”) factors or it belongs to a class, say \(I _{0}\), of distributions without indecomposable factors. In contrast to the classical convolution semigroup, in the free additive and multiplicative convolution semigroups the class \(I _{0}\) consists of units (i.e., Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in \(\mathbb{M}\).

MSC:

46L54 Free probability and free operator algebras
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[1] Akhiezer N.I., The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, 1965; · Zbl 0135.33803
[2] Akhiezer N.I., Glazman I.M., Theory of Linear Operators in Hilbert Space. II, Frederick Ungar, New York, 1963; · Zbl 0098.30702
[3] Belinschi S.T., The atoms of the free multiplicative convolution of two probability distributions, Integral Equations Operator Theory, 2003, 46(4), 377-386 http://dx.doi.org/10.1007/s00020-002-1145-4; · Zbl 1028.46095
[4] Belinschi S.T., Complex Analysis Methods in Noncommutative Probability, Ph.D. thesis, Indiana University, 2005, available at http://arxiv.org/abs/math/0602343v1;
[5] Belinschi S.T., The Lebesgue decomposition of the free additive convolution of two probability distributions, Probab. Theory Related Fields, 2008, 142(1-2), 125-150 http://dx.doi.org/10.1007/s00440-007-0100-3; · Zbl 1390.46059
[6] Belinschi S.T., Bercovici H., Atoms and regularity for measures in a partially defined free convolution semigroup, Math. Z., 2004, 248(4), 665-674 http://dx.doi.org/10.1007/s00209-004-0671-y; · Zbl 1065.46045
[7] Belinschi S.T., Bercovici H., Partially defined semigroups relative to multiplicative free convolution, Int. Math. Res. Not., 2005, 2, 65-101 http://dx.doi.org/10.1155/IMRN.2005.65; · Zbl 1092.46046
[8] Belinschi S.T., Bercovici H., A new approach to subordination results in free probability, J. Anal. Math., 2007, 101, 357-365 http://dx.doi.org/10.1007/s11854-007-0013-1; · Zbl 1142.46030
[9] Belinschi S.T., Bercovici H., Hinčin’s theorem for multiplicative free convolutions, Canad. Math. Bull., 2008, 51(1), 26-31 http://dx.doi.org/10.4153/CMB-2008-004-3; · Zbl 1144.46056
[10] Benaych-Georges F., Failure of the Raikov theorem for free random variables, In: Séminaire de Probabilités XXXVIII, Lecture Notes in Math., 1857, Springer, Berlin, 2005, 313-319; · Zbl 1076.46050
[11] Bercovici H., Pata V., Stable laws and domains of attraction in free probability theory, Ann. of Math., 1999, 149(3), 1023-1060 http://dx.doi.org/10.2307/121080; · Zbl 0945.46046
[12] Bercovici H., Pata V., A free analogue of Hinčins characterization of infinite divisibility, Proc. Amer. Math. Soc, 2000, 128(4), 1011-1015 http://dx.doi.org/10.1090/S0002-9939-99-05087-X; · Zbl 0968.46054
[13] Bercovici H., Voiculescu D., Lévy-Hinčin type theorems for multiplicative and additive free convolution, Pacific J. Math., 1992, 153(2), 217-248; · Zbl 0769.60013
[14] Bercovici H., Voiculescu D., Free convolution of measures with unbounded support, Indiana Univ. Math. J., 1993, 42(3), 733-773 http://dx.doi.org/10.1512/iumj.1993.42.42033; · Zbl 0806.46070
[15] Bercovici H., Voiculescu D., Superconvergence to the central limit and failure of the Cramér theorem for free random variables, Probab. Theory Related Fields, 1995, 103(2), 215-222 http://dx.doi.org/10.1007/BF01204215; · Zbl 0831.60036
[16] Bercovici H., Voiculescu D., Regularity questions for free convolution, In: Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics, Oper. Theory Adv. Appl., 104, Birkhäuser, Basel, 1998, 37-47; · Zbl 0927.46048
[17] Bercovici H., Wang J.-Ch., On freely indecomposable measures, Indiana Univ. Math. J., 2008, 57(6), 2601-2610 http://dx.doi.org/10.1512/iumj.2008.57.3662; · Zbl 1171.46043
[18] Biane Ph., On the free convolution with a semi-circular distribution, Indiana Univ. Math. J., 1997, 46(3), 705-718 http://dx.doi.org/10.1512/iumj.1997.46.1467; · Zbl 0904.46045
[19] Biane Ph., Processes with free increments, Math. Z., 1998, 227(1), 143-174 http://dx.doi.org/10.1007/PL00004363; · Zbl 0902.60060
[20] Chistyakov G.P., Götze F., The arithmetic of distributions in free probability theory, Bielefed University, 2005, #05-001, preprint available at http://www.mathematik.uni-bielefeld.de/fgweb/Preprints/fg05001.pdf;
[21] Chistyakov G.P., Götze F., The arithmetic of distributions in free probability theory, preprint available at http://arxiv.org/abs/math/0508245v1 (version 2005), http://arxiv.org/abs/math/0508245v2 (version 2010); · Zbl 1239.46046
[22] Chistyakov G.P., Götze F., Limit theorems in free probability theory. I, 2006, preprint available at http://arxiv.org/abs/math/0602219; · Zbl 1157.46037
[23] Chistyakov G.P., Götze F., Limit theorems in free probability theory. I, Ann. Probab., 2008, 36(1), 54-90 http://dx.doi.org/10.1214/009117907000000051; · Zbl 1157.46037
[24] Davidson R., Arithmetic and other properties of certain Delphic semigroups. I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1968, 10(2), 120-145 http://dx.doi.org/10.1007/BF00531845; · Zbl 0164.46502
[25] Davidson R., Arithmetic and other properties of certain Delphic semigroups. II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1968, 10(2), 146-172 http://dx.doi.org/10.1007/BF00531846; · Zbl 0164.46502
[26] Davidson R., More Delphic theory and practice, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1969, 13(3-4), 191-203 http://dx.doi.org/10.1007/BF00539200; · Zbl 0181.20101
[27] Goluzin G.M., Geometric Theory of Functions of a Complex Variable, Transl. Math. Monogr., 26, American Mathe-matical Society, Providence, 1969; · Zbl 0183.07502
[28] Hiai F., Petz D., The Semicircle Law, Free Random Variables and Entropy, Math. Surveys Monogr., 77, American Mathematical Society, Providence, 2000; · Zbl 0955.46037
[29] Kendall DG., Delphic semigroups, Bull. Amer. Math. Soc, 1967, 73(1), 120-121 http://dx.doi.org/10.1090/S0002-9904-1967-11673-2; · Zbl 0178.19804
[30] Kendall D.G., Delphic semi-groups, infinitely divisible regenerative phenomena, and the arithmetic of p-functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1968, 9(3), 163-195 http://dx.doi.org/10.1007/BF00535637; · Zbl 0159.46902
[31] Khintchine A., Contribution à larithmétique des lois de distribution, Bull. Univ. État Moscou, Sér. Int., Sect. A, Math. et Mécan., 1937, 1(1), 6-17;
[32] Kreĭn M.G., Nudelman A.A., The Markov Moment Problem and Extremal Problems, Transl. Math. Monogr., 50, American Mathematical Society, Providence, 1977;
[33] Linnik Ju.V, Ostrovs’kiĭ Ĭ.V., Decomposition of Random Variables and Vectors, Transl. Math. Monogr., 48, American Mathematical Society, Providence, 1977;
[34] Livshic L.Z., Ostrovskii I.V., Chistyakov G.P., The arithmetic of probability laws, In: Probability Theory, Mathematical Statistic, Theoretical Cybernetics, 12, Akad. Nauk SSSR Vsesojuz. Inst. Nauch. i Tehn. Informacii, Moscow, 1975, 5-42 (in Russian);
[35] Loève M., Probability Theory, 3rd ed., Van Nostrand, Princeton-Toronto-London, 1963; · Zbl 0108.14202
[36] Maassen H., Addition of freely independent random variables, J. Funct. Anal., 1992, 106(2), 409-438 http://dx.doi.org/10.1016/0022-1236(92)90055-N; · Zbl 0784.46047
[37] Markushevich A.I., Theory of Functions of a Complex Variable. II&III, Prentice-Hall, Englewood Cliffs, 1965&1967; · Zbl 0135.12002
[38] Nevanlinna R., Paatero V, Introduction to Complex Analysis, Addison-Wesley, Reading-London-Don Mills, 1969; · Zbl 0169.09001
[39] Nica A., Speicher R., On the multiplication of free N-tuples of noncommutative random variables, Amer. J. Math., 1996, 118(4), 799-837 http://dx.doi.org/10.1353/ajm.1996.0034; · Zbl 0856.46035
[40] Ostrovskiĭ I.V., The arithmetic of probability distributions, J. Multivariate Anal., 1977, 7(4), 475-490 http://dx.doi.org/10.1016/0047-259X(77)90061-6;
[41] Ostrovskiĭ I.V., The arithmetic of probability distributions, Teor. Veroyatn. Primen., 1986, 31(1), 3-30;
[42] Pastur L, Vasilchuk V, On the law of addition of random matrices, Comm. Math. Phys., 2000, 214(2), 249-286 http://dx.doi.org/10.1007/s002200000264; · Zbl 1039.82020
[43] Speicher R., Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Mem. Amer. Math. Soc., 627, American Mathematical Society, Providence, 1998; · Zbl 0935.46056
[44] Speicher R., Woroudi R., Boolean convolution, In: Free Probability Theory, Waterloo, 1995, Fields Inst. Commun., 12, American Mathematical Society, Providence, 1997, 267-279;
[45] Vasilchuk V., On the law of multiplication of random matrices, Math. Phys. Anal. Geom., 2001, 4(1), 1-36 http://dx.doi.org/10.1023/A:1011807424118; · Zbl 0992.15021
[46] Voiculescu D., Addition of certain noncommuting random variables, J. Funct. Anal., 1986, 66(3), 323-346 http://dx.doi.org/10.1016/0022-1236(86)90062-5;
[47] Voiculescu D., Multiplication of certain noncommuting random variables, J. Operator Theory, 1987, 18(2), 223-235; · Zbl 0662.46069
[48] Voiculescu D., The analogues of entropy and of Fisher’s information measure in free probability theory. I, Comm. Math. Phys., 1993, 155(1), 71-92 http://dx.doi.org/10.1007/BF02100050; · Zbl 0781.60006
[49] Voiculescu D., The coalgebra of the free difference quotient and free probability, Int. Math. Res. Not., 2000, 2, 79-106 http://dx.doi.org/10.1155/S1073792800000064; · Zbl 0952.46038
[50] Voiculescu D.V., Analytic subordination consequences of free Markovianity, Indiana Univ. Math. J., 2002, 51(5), 1161-1166 http://dx.doi.org/10.1512/iumj.2002.51.2252; · Zbl 1040.46044
[51] Voiculescu D.V., Dykema K.J., Nica A., Free Random Variables, CRM Monogr. Ser., 1, American Mathematical Society, Providence, 1992; · Zbl 0795.46049
[52] Williams J.D., A Khintchine decomposition in free probability, preprint available at http://arxiv.org/abs/1009.4955; · Zbl 1260.60185
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