PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2011 | 9 | 5 | 997-1050
Tytuł artykułu

The arithmetic of distributions in free probability theory

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 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 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 M.
Wydawca
Czasopismo
Rocznik
Tom
9
Numer
5
Strony
997-1050
Opis fizyczny
Daty
wydano
2011-10-01
online
2011-07-26
Bibliografia
  • [1] Akhiezer N.I., The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, 1965
  • [2] Akhiezer N.I., Glazman I.M., Theory of Linear Operators in Hilbert Space. II, Frederick Ungar, New York, 1963
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [13] Bercovici H., Voiculescu D., Lévy-Hinčin type theorems for multiplicative and additive free convolution, Pacific J. Math., 1992, 153(2), 217–248
  • [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
  • [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
  • [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
  • [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
  • [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
  • [19] Biane Ph., Processes with free increments, Math. Z., 1998, 227(1), 143–174 http://dx.doi.org/10.1007/PL00004363
  • [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)
  • [22] Chistyakov G.P., Götze F., Limit theorems in free probability theory. I, 2006, preprint available at http://arxiv.org/abs/math/0602219
  • [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
  • [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
  • [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
  • [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
  • [27] Goluzin G.M., Geometric Theory of Functions of a Complex Variable, Transl. Math. Monogr., 26, American Mathe-matical Society, Providence, 1969
  • [28] Hiai F., Petz D., The Semicircle Law, Free Random Variables and Entropy, Math. Surveys Monogr., 77, American Mathematical Society, Providence, 2000
  • [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
  • [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
  • [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
  • [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
  • [37] Markushevich A.I., Theory of Functions of a Complex Variable. II&III, Prentice-Hall, Englewood Cliffs, 1965&1967
  • [38] Nevanlinna R., Paatero V, Introduction to Complex Analysis, Addison-Wesley, Reading-London-Don Mills, 1969
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [51] Voiculescu D.V., Dykema K.J., Nica A., Free Random Variables, CRM Monogr. Ser., 1, American Mathematical Society, Providence, 1992
  • [52] Williams J.D., A Khintchine decomposition in free probability, preprint available at http://arxiv.org/abs/1009.4955
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0049-4
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.