Abstract
In this paper, we deal with the notion of
Funding statement: This work was funded by the Deanship of Graduate Studies and Scientific Research at Jouf University under grant No. DGSSR-2023-02-02016.
References
[1] 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. 10.2307/121080Search in Google Scholar
[2] H. Bercovici and D. Voiculescu, Lévy–Hinčin type theorems for multiplicative and additive free convolution, Pacific J. Math. 153 (1992), no. 2, 217–248. 10.2140/pjm.1992.153.217Search in Google Scholar
[3] H. Bercovici and D. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), no. 3, 733–773. 10.1512/iumj.1993.42.42033Search in Google Scholar
[4] W. Bryc, Free exponential families as kernel families, Demonstr. Math. 42 (2009), no. 3, 657–672. 10.1515/dema-2009-0320Search in Google Scholar
[5] W. Bryc, R. Fakhfakh and A. Hassairi, On Cauchy–Stieltjes kernel families, J. Multivariate Anal. 124 (2014), 295–312. 10.1016/j.jmva.2013.10.021Search in Google Scholar
[6] W. Bryc, R. Fakhfakh and W. Młotkowski, Cauchy–Stieltjes families with polynomial variance functions and generalized orthogonality, Probab. Math. Statist. 39 (2019), no. 2, 237–258. 10.19195/0208-4147.39.2.1Search in Google Scholar
[7] W. Bryc and A. Hassairi, One-sided Cauchy–Stieltjes kernel families, J. Theoret. Probab. 24 (2011), no. 2, 577–594. 10.1007/s10959-010-0303-xSearch in Google Scholar
[8] R. Fakhfakh, The mean of the reciprocal in a Cauchy–Stieltjes family, Statist. Probab. Lett. 129 (2017), 1–11. 10.1016/j.spl.2017.04.021Search in Google Scholar
[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. 10.1016/j.jmaa.2017.10.003Search in Google Scholar
[10] R. Fakhfakh, Variance function of boolean additive convolution, Statist. Probab. Lett. 163 (2020), Article ID 108777. 10.1016/j.spl.2020.108777Search in Google Scholar
[11] R. Fakhfakh, Boolean multiplicative convolution and Cauchy–Stieltjes kernel families, Bull. Korean Math. Soc. 58 (2021), no. 2, 515–526. Search in Google Scholar
[12] R. Fakhfakh, On some properties of Cauchy–Stieltjes kernel families, Indian J. Pure Appl. Math. 52 (2021), no. 4, 1186–1200. 10.1007/s13226-021-00020-zSearch in Google Scholar
[13] R. Fakhfakh, Explicit free multiplicative law of large numbers, Comm. Statist. Theory Methods 52 (2023), no. 7, 2031–2042. 10.1080/03610926.2021.1944212Search in Google Scholar
[14] R. Fakhfakh, On polynomials associated with Cauchy–Stieltjes kernel families, Comm. Statist. Theory Methods 52 (2023), no. 19, 7009–7021. 10.1080/03610926.2022.2037647Search in Google Scholar
[15] R. Fakhfakh and A. Hassairi, Cauchy–Stieltjes kernel families and multiplicative free convolutions, Convolution. Commun. Math. Stat. (2023), 10.1007/s40304-022-00311-9. 10.1007/s40304-022-00311-9Search in Google Scholar
[16] A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (2005), no. 3, 515–545. 10.1142/S0219025705002104Search in Google Scholar
[17] A. Nica and R. Speicher, On the multiplication of free N-tuples of noncommutative random variables, Amer. J. Math. 118 (1996), no. 4, 799–837. 10.1353/ajm.1996.0034Search in Google Scholar
[18] J.-C. Wang, Limit laws for Boolean convolutions, Pacific J. Math. 237 (2008), no. 2, 349–371. 10.2140/pjm.2008.237.349Search in Google Scholar
[19] L. J. Wojakowski, The Lévy–Khintchine formula and Nica–Speicher property for deformations of the free convolution, Noncommutative Harmonic Analysis with Applications to Probability, Banach Center Publ. 78, Polish Academy of Sciences, Warsaw (2007), 309–314. 10.4064/bc78-0-23Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Numerical approaches for solution of hyperbolic difference equations on circle
- A note on b-generalized (α,β)-derivations in prime rings
- Analytic solution to functional differential equations via Bell’s polynomials
- Floquet theory and stability for a class of first order differential equations with delays
- Representations of a number in an arbitrary base with unbounded digits
- V_a -deformed free convolution and variance function
- Generalized essential spectra involving the class of g-g-Riesz operators
- On perturbation of continuous frames in Hilbert C *-modules
- Almost measurable functions on probability spaces
- On φ-u-S-flat modules and nonnil-u-S-injective modules
- Busemann--Petty-type problem for μ-intersection bodies
- On the representation of solution for the perturbed quasi-linear controlled neutral functional-differential equation with the discontinuous initial condition
- A generalization of Hardy’s inequality to infinite tensors
- A note on maximal estimate for an oscillatory operator
- Some summation theorems and transformations for hypergeometric functions of Kampé de Fériet and Srivastava
- On the correspondence between periodic solutions of differential and dynamic equations on periodic time scales