On Generalized V a -Transformation of Measures

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, 1-14

Abstract: In this study, we introduce a novel transformation of probability measures that unifies two significant transformations in free probability theory: the t -transformation and the V a -transformation. Our unified transformation, denoted U ( a , t ) , is defined analytically via a modified functional equation involving the Cauchy transform, and reduces to the t -transformation when a = 0 , and to the V a -transformation when t = 1 . We investigate some properties of this new transformation from the lens of Cauchy–Stieltjes kernel (CSK) families and the corresponding variance functions (VFs). We derive a general expression for the VF resulting from the U ( a , t ) -transformation. This new expression is applied to prove a central result: the free Meixner family (FMF) of measures is invariant under this transformation. Furthermore, novel limiting theorems involving U ( a , t ) -transformation are proved providing new insights into the relations between some important measures in free probability such as the semicircle, Marchenko–Pastur, and free binomial measures.

Keywords: variance function; limit theorems; semicircle law; Cauchy–Stieltjes transform (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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