Limit theorems in free probability theory II

Gennadii Chistyakov ; Friedrich Götze

Open Mathematics, Tome 6 (2008), p. 87-117 / Harvested from The Polish Digital Mathematics Library

Résumé

Based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line ℝ+ and on the unit circle 𝕋 we prove analogs of limit theorems for nonidentically distributed random variables in classical Probability Theory.

Détail

Publié le : 2008-01-01

Zbl 1148.46035

EUDML-ID : urn:eudml:doc:269629

Bibliographie

[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, Ungar, New York, 1963

[3] Barndorff-Nielsen O.E., Thorbjørnsen S., Selfdecomposability and Levy processes in free probability, Bernoulli, 2002, 8, 323–366 | Zbl 1024.60022

[4] Belinschi S.T., Complex analysis methods on noncommutative probability, preprint available at http://arxlv.org/abs/math/0603104v1

[5] Belinschi S.T., Bercovici H., Hincin’s theorem for multiplicative free convolutions, Canad. Math. Bull., to appear | Zbl 1144.46056

[6] Bercovici H., Voiculescu D., Lévy-Hinčin type theorems for multiplicative and additive free convolution, Pacific J. Math., 1992, 153, 217–248 | Zbl 0769.60013

[7] Bercovici H., Voiculescu D., Free convolution of measures with unbounded support, Indiana Univ. Math. J., 1993, 42, 733–773 http://dx.doi.org/10.1512/iumj.1993.42.42033 | Zbl 0806.46070

[8] Bercovici H., Pata V, Stable laws and domains of attraction in free probability theory, Ann. of Math., 1999, 149, 1023–1060 http://dx.doi.org/10.2307/121080 | Zbl 0945.46046

[9] Bercovici H., Pata V, Limit laws for products of free and independent random variables, Studia Math., 2000, 141, 43–52 | Zbl 0968.46051

[10] Bercovici H., Wang J.C., Limit theorems for free multiplicative convolutions, preprint available at http://arxlv.org/abs/math/0612278v1 | Zbl 1166.46039

[11] Biane Ph., Processes with free increments, Math. Z., 1998, 227, 143–174 http://dx.doi.org/10.1007/PL00004363 | Zbl 0902.60060

[12] Chistyakov C.P., Cötze F., The arithmetic of distributions in free probability theory, preprint available at http://arxlv.org/abs/math/0508245v1

[13] Chistyakov G.P., Götze F., Limit theorems in free probability theory I, Ann. Probab., 2008, 36, 54–90 http://dx.doi.org/10.1214/009117907000000051 | Zbl 1157.46037

[14] Gnedenko B.V., Kolmogorov A.N., Limit distributions for sums of independent random variables, Addison-Wesley Publishing Company, 1968

[15] Hiai F., Petz D., The Semicircle Law, Free Random Variables and Entropy, A.M.S., Providence, Rl, 2000 | Zbl 0955.46037

[16] Loéve M., Probability theory, VNR, New York, 1963

[17] Parthasarathy K.R., Probability measures on metric spaces, Academic Press, New York and London, 1967 | Zbl 0153.19101

[18] Sazonov V.V., Tutubalin V.N., Probability distributions on topological groups, Theory Probab. Appl., 1966, 11, 1–45 http://dx.doi.org/10.1137/1111001 | Zbl 0171.38701

[19] Voiculescu D.V, Multiplication of certain noncommutlng random variavles, J. Operator Theory, 1987, 18, 223–235

[20] Voiculesku D., Dykema K., Nica A., Free random variables, CRM Monograph Series, No 1, A.M.S., Providence, Rl, 1992

[21] Voiculescu D.V., The analogues of entropy and Fisher’s information mesure in free probability theory I, Comm. Math. Phys., 1993, 155, 71–92 http://dx.doi.org/10.1007/BF02100050 | Zbl 0781.60006

[22] Voiculescu D.V., 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

[23] Voiculescu D.V., Analytic subordination consequences of free Markovianity, Indiana Univ. Math. J., 2002, 51, 1161–1166 http://dx.doi.org/10.1512/iumj.2002.51.2252 | Zbl 1040.46044