On T-Transformation of Probability Measures

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.