B t -Transformation and Variance Function

Abdulmajeed Albarrak, Raouf Fakhfakh () and Ghadah Alomani
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Abdulmajeed Albarrak: Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72388, Saudi Arabia
Raouf Fakhfakh: Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72388, Saudi Arabia
Ghadah Alomani: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Mathematics, 2025, vol. 13, issue 21, 1-15

Abstract: This study investigates the B t -transformation of probability measures within the framework of free probability. A primary focus is the invariance under this transformation of two fundamental families: the free Meixner family and the free analog of the Letac–Mora class. In addition, we introduce novel characteristics associated with the B t -transformation, offering refined analytical tools to probe its structural and functional properties. These tools allow us to uncover new and significant properties of several distributions in free probability, including the semicircle, the Marchenko–Pastur, and the free Gamma laws, yielding explicit invariance results and stability conditions. Our findings extend the theoretical understanding of the B t -transformation and provide practical methods for analyzing the dynamics and stability of classical free distributions under this operator.

Keywords: variance function; ? t -transformation of measures; free Meixner family (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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