Va-Deformed Free Convolution

Abstract: In this article, we study Va-transformation of a measure and of a convolution (denoted by a)definedfor a ∈ R. Weprovidesignificant insights into the stability of the free Meixner family of probability measures under Va-transformation. We show that the Va-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 Va-deformed free cumulants are presented.