Limit theorems for free multiplicative convolutions
Authors:
Hari Bercovici and JiunChau Wang
Journal:
Trans. Amer. Math. Soc. 360 (2008), 60896102
MSC (2000):
Primary 46L54; Secondary 60F05
Published electronically:
April 25, 2008
MathSciNet review:
2425704
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. The main theorems in this paper extend a result for measures supported on the positive halfline, and provide a new limit theorem for measures on the unit circle with nonzero first moment.
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 [1]
 S. T. Belinschi and H. Bercovici, Atoms and regularity for measures in a partially defined free convolution semigroup, Math. Z. 248 (2004), no. 4, 665674. MR 2103535 (2006i:46095)
 [2]
 , Hinčin's theorem for multiplicative free convolution, Canadian Math. Bulletin, 51 (2008), no. 1, 2631.
 [3]
 H. Bercovici and V. Pata, Stable laws and domain of attraction in free probability, with an appendix by Ph. Biane, Ann. Math. 149 (1999), 10231060. MR 1709310 (2000i:46061)
 [4]
 , Limit laws for products of free and independent random variables, Studia Math. 141 (2000), no. 1, 4352. MR 1782911 (2001i:46105)
 [5]
 H. Bercovici and D. V. Voiculescu, LévyHinčin type theorems for multiplicative and additive free convolution, Pacific J. Math. 153 (1992), no. 2, 217248. MR 1151559 (93k:46052)
 [6]
 , Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), no. 3, 733773. MR 1254116 (95c:46109)
 [7]
 Ph. Biane, Processes with free increments, Math. Z. 227 (1995), no. 1, 143174. MR 1605393 (99e:46085)
 [8]
 G. P. Chistyakov and F. Götze, The arithmetic of distributions in free probability, Arxiv: math. PS/0508245.
 [9]
 , Limit theorems in free probability. I, Ann. Probab. 36 (2008), no. 1, 5490. MR 2370598
 [10]
 D. V. Voiculescu, Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), 223235. MR 915507 (89b:46076)
 [11]
 , The analogues of entropy and of Fisher's information measure in free probability. I, Comm. Math. Phys. 155 (1993), no. 1, 7192. MR 1228526 (94k:46137)
 [12]
 , The coalgebra of the free difference quotient and free probability, Int. Math. Res. Not. 2000 (2000), no. 2, 79106. MR 1744647 (2001d:46096)
 [13]
 , Analytic subordination consequences of free Markovianity, Indiana Univ. Math. J. 51 (2002), no. 5, 11611166. MR 1947871 (2003k:46100)
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Additional Information
Hari Bercovici
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 474054301
Email:
bercovic@indiana.edu
JiunChau Wang
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 474054301
Email:
jiuwang@indiana.edu
DOI:
http://dx.doi.org/10.1090/S0002994708045078
PII:
S 00029947(08)045078
Received by editor(s):
December 20, 2006
Published electronically:
April 25, 2008
Additional Notes:
The first author was supported in part by a grant from the National Science Foundation.
Article copyright:
© Copyright 2008
American Mathematical Society
