On T -Transformation of Probability Measures

Shokrya S. Alshqaq, Ohud A. Alqasem and Raouf Fakhfakh ()
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Shokrya S. Alshqaq: Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia
Ohud A. Alqasem: Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Raouf Fakhfakh: Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi Arabia

Mathematics, 2025, vol. 13, issue 5, 1-10

Abstract: The concept of the ( T = ( s , t ) ) -transformation of probability measures, introduced for s > 0 and t ∈ R , is examined in this work from the perspective of Cauchy–Stieltjes kernel (CSK) families and their related variance functions (VFs). We calculate the VF formula under the T -transformation of measures. Furthermore, the stability of the free Meixner family ( FMF ) of probability measures under the ( T = ( s , t ) ) -transformation is significantly shown based on this formula. Additionally, the Wigner’s semicircle CSK family is given a novel characterization based on the ( 1 , t ) -transformation of probability measures.

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