Va-Deformed Free Convolution - Fahad Alsharari; Raouf Fakhfakh (Mathematics)
In this article, we study <semantics>Va</semantics>-transformation of a measure and of a convolution (denoted by <semantics><menclose notation="box">a</menclose></semantics>) defined for <semantics>a∈R</semantics>. We provide significant insights into the stability of the free Meixner family of probability measures under <semantics>Va</semantics>-transformation. We show that the <semantics>Va</semantics>-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 <semantics><menclose notation="box">a</menclose></semantics>-convolution. In addition, some new limit theorems are proved for the <semantics><menclose notation="box">a</menclose></semantics>-convolution incorporating both free and Boolean additive convolutions. Furthermore, some properties related to <semantics>Va</semantics>-deformed free cumulants are presented.