Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product
Abstract
:1. Introduction
- (ii)
- Let be the image of μ by where and . Then, ∀m close to ,
- (iii)
2. Main Results
2.1. Stability of CSK Families by Free Additive Convolution Product
2.2. Stability of CSK Families by Boolean Additive Convolution Product
2.3. Stability of CSK Families by Multiplicative Free Convolution Product
- (ii)
- For , we have , and . Thus, the parameters and must strictly be positive. In that case, the measures , and σ are of free Gamma type law up to affinity. For more details about free Gamma CSK family, see ([4], [Theorem 3.2]).
- (iii)
2.4. Stability of CSK Families by Boolean Multiplicative Convolution Product
- (i)
- (ii)
- at least one of the first moments of one of the probabilities or is finite, the measure is well defined.
- (ii)
- Note that for , we have , and . Thus, the parameters and must be positive strictly. In that case the measures , and σ are of Marchenko–Pastur type law up to affinity.
- (iii)
3. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Koudou, A.E.; Pommeret, D. A Characterization of Poisson-Gaussian Families by Convolution-Stability. J. Multivar. Anal. 2002, 81, 120–127. [Google Scholar] [CrossRef]
- Letac, G. Lectures on Natural Exponential Families and Their Variance Functions; Instituto de Matemtica Pura e Aplicada: Monografias de Matemtica, Rio de Janeiro, Brazil, 1992; Volume 50. [Google Scholar]
- Pommeret, D. Stabilité des familles exponentielles naturelles par convolution. C. R. Acad. Sci. Sr. I 1999, 328, 929–933. [Google Scholar]
- Bryc, W. Free exponential families as kernel families. Demonstr. Math. 2009, 42, 657–672. [Google Scholar] [CrossRef]
- Bryc, W.; Hassairi, A. One-sided Cauchy-Stieltjes kernel families. J. Theoret. Probab. 2011, 24, 577–594. [Google Scholar] [CrossRef]
- Bercovici, H.; Voiculescu, D. Free convolution of measures with unbounded support. Indiana Univ. Math. J. 1993, 42, 733–773. [Google Scholar] [CrossRef]
- Alanzi, A.R.A.; Alqasem, O.A.; Elwahab, M.E.A.; Fakhfakh, R. Some Results on the Free Poisson Distribution. Axioms 2024, 13, 496. [Google Scholar] [CrossRef]
- Speicher, R.; Woroudi, R. Boolean convolution. Fields Inst. Commun. 1997, 12, 26727. [Google Scholar] [CrossRef]
- Fakhfakh, R. Variance function of boolean additive convolution. Stat. Probab. Lett. 2020, 163, 108777. [Google Scholar] [CrossRef]
- Friedman, D. The functional equation f(x+y)=g(x)+h(y). Am. Math. Mon. 1962, 69, 769–772. [Google Scholar] [CrossRef]
- Fakhfakh, R. A characterization of the Marchenko-Pastur probability measure. Stat. Probab. Lett. 2022, 19, 109660. [Google Scholar] [CrossRef]
- Belinschi, S.T. Complex Analysis Methods in Noncommutative Probability; ProQuest LLC.: Ann Arbor, MI, USA, 2005. [Google Scholar]
- Noriyoshi, S.; Hiroaki, Y. New limit theorems related to free multiplicative convolution. Stud. Math. 2013, 214, 251–264. [Google Scholar]
- Bercovici, H. On Boolean convolutions. In Operator Theory 20; MR2276927l; Theta: Bucharest, Romania, 2006; Volume 6, pp. 7–13. [Google Scholar]
- Anshelevich, M.; Wang, J.C.; Zhong, P. Local limit theorems for multiplicative free convolutions. J. Funct. Anal. 2014, 267, 3469–3499. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Alanzi, A.R.A.; Alshqaq, S.S.; Fakhfakh, R. Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product. Mathematics 2024, 12, 3465. https://doi.org/10.3390/math12223465
Alanzi ARA, Alshqaq SS, Fakhfakh R. Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product. Mathematics. 2024; 12(22):3465. https://doi.org/10.3390/math12223465
Chicago/Turabian StyleAlanzi, Ayed. R. A., Shokrya S. Alshqaq, and Raouf Fakhfakh. 2024. "Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product" Mathematics 12, no. 22: 3465. https://doi.org/10.3390/math12223465
APA StyleAlanzi, A. R. A., Alshqaq, S. S., & Fakhfakh, R. (2024). Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product. Mathematics, 12(22), 3465. https://doi.org/10.3390/math12223465