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Limit Theorems for Additive Conditionally Free Convolution

Published online by Cambridge University Press:  20 November 2018

Jiun-Chau Wang
Jiun-Chau Wang*
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK, S7N 5E6freeprobability@gmail.com
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Abstract

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In this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of $\text{c}$-free convolution.

Keywords

46L5360F05additive c-free convolutionlimit theoremsinfinitesimal arrays
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 1 , 01 February 2011 , pp. 222 - 240
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Achieser, N. I., The classical moment problem and some related questions in analysis. Hafner Publishing Co., New York, 1965.Google Scholar
[2] Belinschi, S. T., C-free convolution for measures with unbounded support. In: Von Neumann algebras in Sibiu, Theta Ser. Adv. Math., 10, Theta, Bucharest, 2008, pp. 1-7.Google Scholar
[3] Belinschi, S. T. and Bercovici, H., A new approach to subordination results in free probability. J. Anal. Math. 101 (2007), 357-365. doi:10.1007/s11854-007-0013-1Google Scholar
[4] Bercovici, H., On Boolean convolutions. In: Operator theory 20, Theta Ser. Adv. Math., 6, Theta, Bucharest, 2006, pp. 7-13.Google Scholar
[5] Bercovici, H. and, Pata, V., Stable laws and domains of attraction in free probability theory. Ann. of Math. 149 (1999), no. 3, 1023-1060. doi:10.2307/121080Google Scholar
[6] Bercovici, H. and, Pata, V., A free analogue of Hinčin's characterization of infinite divisibility. Proc. Amer. Math. Soc. 128 (2000), no. 4, 1011-1015. doi:10.1090/S0002-9939-99-05087-XGoogle Scholar
[7] Bercovici, H. and, Voiculescu, D., Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42 (1993), no. 3, 733-773. doi:10.1512/iumj.1993.42.42033Google Scholar
[8] Bercovici, H. and, J.-C. Wang, The asymptotic behavior of free additive convolution. Oper. Matrices 2 (2008), no. 1, 115-124.Google Scholar
[9] Bożejko, M., Leinert, M., and Speicher, R., Convolution and limit theorems for conditionally free random variables. Pacific J. Math. 175 (1996), no. 2, 357-388.Google Scholar
[10] Bożejko, M. and Bryc, W., A quadratic regression problem for two-state algebras with application to the central limit theorem. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), no. 2, 213-249. doi:10.1142/S0219025709003616Google Scholar
[11] Biane, P., Processes with free increments. Math. Z. 227 (1998), no. 1, 143-174. doi:10.1007/PL00004363Google Scholar
[12] Billingsley, P., Probability and measure. Third ed., Wiley Series in Probability and Mathematical Statistics, John Wiley ' Sons, New York, 1995.Google Scholar
[13] Chistyakov, G. P. and F. Götze, Limit theorems in free probability theory. I. Ann. Probab. 36 (2008), no. 1, 54-90. doi:10.1214/009117907000000051Google Scholar
[14] Gnedenko, B. V. and Kolmogorov, A. N., Limit distributions for sums of independent random variables. Addison-Wesley Publishing Company, Cambridge, MA, 1954.Google Scholar
[15] Khintchine, A., Zur Theorie der unbeschränkt teilbaren Verteilungsgesetze. Rec. Math. [Mat. Sbornik] 44 (1937), no. 1, 79-119.Google Scholar
[16] Krystek, A. D., Infinite divisibility for the conditionally free convolution. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007), no. 4, 499-522. doi:10.1142/S0219025707002919Google Scholar
[17] Pata, V., Lèvy type characterization of stable laws for free random variables. Trans. Amer. Math. Soc. 347 (1995), no. 7, 2457-2472. doi:10.2307/2154831Google Scholar
[18] Popa, M. and, Wang, J. C., Multiplicative c-free convolution. arXiv.0805.0257v2, 2008.Google Scholar
[19] Speicher, R. and, R., Woroudi, Boolean convolution. In: Free probability theory, Fields Inst. Commun., 12, American Mathematical Society, Providence, RI, 1997, pp. 267-279.Google Scholar
[20] Voiculescu, D. V., Addition of certain noncommuting random variables. J. Funct. Anal. 66 (1986), no. 3, 323-346. doi:10.1016/0022-1236(86)90062-5Google Scholar
[21] Voiculescu, D. V., Dykema, K. J., and Nica, A., Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. CRM Monograph Series, 1, American Mathematical Society, Providence, RI, 1992.Google Scholar
[22] Wang, J.-C., Limit laws for Boolean convolutions. Pacific J. Math. 237 (2008), no. 2, 349-371. doi:10.2140/pjm.2008.237.349Google Scholar