V a -Deformed Free Convolution

Fahad Alsharari and Raouf Fakhfakh ()
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Fahad Alsharari: Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi Arabia
Raouf Fakhfakh: Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi Arabia

Mathematics, 2025, vol. 13, issue 4, 1-13

Abstract: In this article, we study V a -transformation of a measure and of a convolution (denoted by a ) defined for a ∈ R . We provide significant insights into the stability of the free Meixner family of probability measures under V a -transformation. We show that the V a -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 a -convolution. In addition, some new limit theorems are proved for the a -convolution incorporating both free and Boolean additive convolutions. Furthermore, some properties related to V a -deformed free cumulants are presented.

Keywords: V a -transformation; Marchenko–Pastur law; variance function; Cauchy–Stieltjes transform (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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