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http://dx.doi.org/10.4134/BKMS.b200380

BOOLEAN MULTIPLICATIVE CONVOLUTION AND CAUCHY-STIELTJES KERNEL FAMILIES  

Fakhfakh, Raouf (Mathematics Department College of Science and Arts in Gurayat Jouf University)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.2, 2021 , pp. 515-526 More about this Journal
Abstract
Denote by ��+ the set of probability measures supported on ℝ+. Suppose V�� is the variance function of the Cauchy-Stieltjes Kernel (CSK) family ��-(��) generated by a non degenerate probability measure �� ∈ ��+. We determine the formula for variance function under boolean multiplicative convolution power. This formula is used to identify the relation between variance functions under the map ${\nu}{\mapsto}{\mathbb{M}}_t({\nu})=({\nu}^{{\boxtimes}(t+1)})^{{\uplus}{\frac{1}{t+1}}}$ from ��+ onto itself.
Keywords
Variance function; Cauchy-Stieltjes kernel; boolean multiplicative convolution;
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1 S. T. Belinschi and A. Nica, On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution, Indiana Univ. Math. J. 57 (2008), no. 4, 1679-1713. https://doi.org/10.1512/iumj.2008.57.3285   DOI
2 H. Bercovici, On Boolean convolutions, in Operator theory 20, 7-13, Theta Ser. Adv. Math., 6, Theta, Bucharest, 2006.
3 H. Bercovici and D. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), no. 3, 733-773. https://doi.org/10.1512/iumj.1993.42.42033   DOI
4 W. Bryc, Free exponential families as kernel families, Demonstratio Math. 42 (2009), no. 3, 657-672.   DOI
5 W. Bryc, R. Fakhfakh, and A. Hassairi, On Cauchy-Stieltjes kernel families, J. Multivariate Anal. 124 (2014), 295-312. https://doi.org/10.1016/j.jmva.2013.10.021   DOI
6 W. Bryc, R. Fakhfakh, and W. M lotkowski, Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality, Probab. Math. Statist. 39 (2019), no. 2, 237-258. https://doi.org/10.19195/0208-4147.39.2.1   DOI
7 W. Bryc and A. Hassairi, One-sided Cauchy-Stieltjes kernel families, J. Theoret. Probab. 24 (2011), no. 2, 577-594. https://doi.org/10.1007/s10959-010-0303-x   DOI
8 R. Fakhfakh, The mean of the reciprocal in a Cauchy-Stieltjes family, Statist. Probab. Lett. 129 (2017), 1-11. https://doi.org/10.1016/j.spl.2017.04.021   DOI
9 R. Fakhfakh, Characterization of quadratic Cauchy-Stieltjes kernel families based on the orthogonality of polynomials, J. Math. Anal. Appl. 459 (2018), no. 1, 577-589. https://doi.org/10.1016/j.jmaa.2017.10.003   DOI
10 H. Bercovici and V. Pata, Stable laws and domains of attraction in free probability theory, Ann. of Math. (2) 149 (1999), no. 3, 1023-1060. https://doi.org/10.2307/121080   DOI
11 R. Fakhfakh, Variance function of boolean additive convolution, Statist. Probab. Lett. 163 (2020), 108777, 9 pp. https://doi.org/10.1016/j.spl.2020.108777   DOI
12 J. Wesolowski, Kernels families, Unpublished manuscript, 1999.
13 U. Franz, Boolean convolution of probability measures on the unit circle, in Analyse et probabilites, 83-94, Semin. Congr., 16, Soc. Math. France, Paris, 2008.
14 M. Anshelevich, J.-C. Wang, and P. Zhong, Local limit theorems for multiplicative free convolutions, J. Funct. Anal. 267 (2014), no. 9, 3469-3499. https://doi.org/10.1016/j.jfa.2014.08.015   DOI
15 O. Arizmendi and T. Hasebe, Semigroups related to additive and multiplicative, free and Boolean convolutions, Studia Math. 215 (2013), no. 2, 157-185. https://doi.org/10.4064/sm215-2-5   DOI
16 N. Sakuma and H. Yoshida, New limit theorems related to free multiplicative convolution, Studia Math. 214 (2013), no. 3, 251-264. https://doi.org/10.4064/sm214-3-4   DOI
17 R. Speicher and R. Woroudi, Boolean convolution, in Free probability theory (Waterloo, ON, 1995), 267-279, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997.
18 A. Hassairi and R. Fakhfakh, Cauchy-Stieltjes kernel families and multiplicative free convolutions, arXiv:2004.07191 [math.PR], 2020.