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Hinčin's Theorem for Multiplicative Free Convolution

Published online by Cambridge University Press:  20 November 2018

S. T. Belinschi  and
H. Bercovici
S. T. Belinschi
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1 e-mail: sbelinsc@math.uwaterloo.ca
H. Bercovici
Affiliation:
Department of Mathematics, University of Indiana, Bloomington, IN 47405-7000, U.S.A. e-mail: bercovic@indiana.edu
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Abstract

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Hinčin proved that any limit law, associated with a triangular array of infinitesimal random variables, is infinitely divisible. The analogous result for additive free convolution was proved earlier by Bercovici and Pata. In this paper we will prove corresponding results for the multiplicative free convolution of measures defined on the unit circle and on the positive half-line.

Keywords

46L5360E0760E10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 1 , 01 March 2008 , pp. 26 - 31
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Belinschi, S. T. and Bercovici, H., Partially defined semigroups relative to multiplicative free convolution. Int. Math. Res. Not. 2005, no. 2, 54–101.Google Scholar
[2] Bercovici, H. and Pata, V., A free analogue of Hincin's characterization of infinite divisibility. Proc. Amer. Math. Soc. 128(2000), no. 4, 1011–1015.Google Scholar
[3] Bercovici, H. and Pata, V., Limit laws for products of free and independent random variables. Studia Math. 141(2000), no. 1, 43–52.Google Scholar
[4] Bercovici, H. and Voiculescu, D., Lévy-Hinčin type theorems for multiplicative and additive free convolution. Pacific J. Math. 153(1992), no. 2, 217–248.Google Scholar
[5] Bercovici, H. and Voiculescu, D., Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42(1993), no. 3, 733–773.Google Scholar
[6] Gnedenko, B. V. and Kolmogorov, A. N., Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA, 1954.Google Scholar
[7] Voiculescu, D., Multiplication of certain noncommuting random variables. J. Operator Theory 18(1987), no. 2, 223–235.Google Scholar