<i>V<sub>a</sub></i>-Deformed Free Convolution

<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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Bibliographic Details
Main Authors:	Fahad Alsharari, Raouf Fakhfakh
Format:	Article
Language:	English
Published:	MDPI AG 2025-02-01
Series:	Mathematics
Subjects:	<i>V<sub>a</sub></i>-transformation Marchenko–Pastur law variance function Cauchy–Stieltjes transform
Online Access:	https://www.mdpi.com/2227-7390/13/4/572
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Summary:	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.
ISSN:	2227-7390