<i>V<sub>a</sub></i>-Deformed Free Convolution
In this article, we study <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation of a...
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| author | Fahad Alsharari Raouf Fakhfakh |
| author_facet | Fahad Alsharari Raouf Fakhfakh |
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| description | In this article, we study <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation of a measure and of a convolution (denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><menclose notation="box"><mi>a</mi></menclose></mrow></semantics></math></inline-formula>) defined for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. We provide significant insights into the stability of the free Meixner family of probability measures under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation. We show that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation of measures (of convolutions) of any member of the free Meixner family remains in the free Meixner family. We also present some properties of the Marchenko–Pastur law in connection with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><menclose notation="box"><mi>a</mi></menclose></mrow></semantics></math></inline-formula>-convolution. In addition, some new limit theorems are proved for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><menclose notation="box"><mi>a</mi></menclose></mrow></semantics></math></inline-formula>-convolution incorporating both free and Boolean additive convolutions. Furthermore, some properties related to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>a</mi></msub></semantics></math></inline-formula>-deformed free cumulants are presented. |
| format | Article |
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| language | English |
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| spelling | doaj-art-50d73009db3d4d2a879b3d165a904b812025-08-20T02:03:25ZengMDPI AGMathematics2227-73902025-02-0113457210.3390/math13040572<i>V<sub>a</sub></i>-Deformed Free ConvolutionFahad Alsharari0Raouf Fakhfakh1Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi ArabiaDepartment of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi ArabiaIn this article, we study <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation of a measure and of a convolution (denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><menclose notation="box"><mi>a</mi></menclose></mrow></semantics></math></inline-formula>) defined for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. We provide significant insights into the stability of the free Meixner family of probability measures under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation. We show that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>a</mi></msub></semantics></math></inline-formula>-transformation of measures (of convolutions) of any member of the free Meixner family remains in the free Meixner family. We also present some properties of the Marchenko–Pastur law in connection with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><menclose notation="box"><mi>a</mi></menclose></mrow></semantics></math></inline-formula>-convolution. In addition, some new limit theorems are proved for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><menclose notation="box"><mi>a</mi></menclose></mrow></semantics></math></inline-formula>-convolution incorporating both free and Boolean additive convolutions. Furthermore, some properties related to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>a</mi></msub></semantics></math></inline-formula>-deformed free cumulants are presented.https://www.mdpi.com/2227-7390/13/4/572<i>V<sub>a</sub></i>-transformationMarchenko–Pastur lawvariance functionCauchy–Stieltjes transform |
| spellingShingle | Fahad Alsharari Raouf Fakhfakh <i>V<sub>a</sub></i>-Deformed Free Convolution Mathematics <i>V<sub>a</sub></i>-transformation Marchenko–Pastur law variance function Cauchy–Stieltjes transform |
| title | <i>V<sub>a</sub></i>-Deformed Free Convolution |
| title_full | <i>V<sub>a</sub></i>-Deformed Free Convolution |
| title_fullStr | <i>V<sub>a</sub></i>-Deformed Free Convolution |
| title_full_unstemmed | <i>V<sub>a</sub></i>-Deformed Free Convolution |
| title_short | <i>V<sub>a</sub></i>-Deformed Free Convolution |
| title_sort | i v sub a sub i deformed free convolution |
| topic | <i>V<sub>a</sub></i>-transformation Marchenko–Pastur law variance function Cauchy–Stieltjes transform |
| url | https://www.mdpi.com/2227-7390/13/4/572 |
| work_keys_str_mv | AT fahadalsharari ivsubasubideformedfreeconvolution AT raouffakhfakh ivsubasubideformedfreeconvolution |