Work

Notes on Cauchy–Stieltjes Kernel Families

Year: 2025

Type: article

Abstract: The free Meixner family (FMF) is the family of measures that produces quadratic Cauchy–Stieltjes Kernel (CSK) families (i.e., meaning that the associated variance function (VF) is a polynomial with degree ≤2 in the mean). Furthermore, a cubic class is introduced in the context of CSK families and is connected to the quadratic class via a reciprocity relation. The associated probability measures are the so-called free analog of the Letac–Mora class (with VF of degree 3). In free probability theory, these two classes of probabilities are crucial. However, a novel transformation of measures is introduced in the setting of free probability, known as the Ta-transformation of probability measures. Denote by P the set of (non-degenerate) real probabilities. For ν∈P and a∈R, consider the transformation of measure ν, denoted Ta(ν), defined by FTa(ν)(w)=Fν(w−a)+a, where Fν(·) is the inverse of the Cauchy–Stieltjes transformation of ν. In this study, we provide important insights into the notion of the Ta-transformation of probabilities. We demonstrate that the FMF (respectively, the free counterpart of the Letac–Mora class of measures) is invariant under the Ta-transformation. Furthermore, we develop additional characteristics of the Ta-transformation, which yield intriguing findings for significant free probability distributions such as the free Poisson and free Gamma distributions.

Source: Axioms

DOI: 10.3390/axioms14030189

Authors: Shokrya S. Alshqaq, Raouf Fakhfakh, Fatimah Alshahrani

Institutions: Jazan University, Jouf University, Princess Nourah bint Abdulrahman University

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,896

Funder: Princess Nourah Bint Abdulrahman University