Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

BOOLEAN MULTIPLICATIVE CONVOLUTION AND CAUCHY-STIELTJES KERNEL FAMILIES

  • Fakhfakh, Raouf (Mathematics Department College of Science and Arts in Gurayat Jouf University)
  • Received : 2020.04.24
  • Accepted : 2020.11.09
  • Published : 2021.03.31
Download PDF
⟨ Previous Next ⟩

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

References

  1. 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
  2. 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
  3. 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
  4. H. Bercovici, On Boolean convolutions, in Operator theory 20, 7-13, Theta Ser. Adv. Math., 6, Theta, Bucharest, 2006.
  5. 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
  6. 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
  7. W. Bryc, Free exponential families as kernel families, Demonstratio Math. 42 (2009), no. 3, 657-672. https://doi.org/10.1515/dema-2009-0320
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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.
  15. A. Hassairi and R. Fakhfakh, Cauchy-Stieltjes kernel families and multiplicative free convolutions, arXiv:2004.07191 [math.PR], 2020.
  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
  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. J. Wesolowski, Kernels families, Unpublished manuscript, 1999.