Cauchy-Stieltjes Kernel families: Some properties

Abstract

In the setting of the theory of Cauchy-Stieltjes Kernel (CSK) families, we present in this article some properties related to the Marchenko–Pastur and Semicircle laws. Let K+(μ)={Qmμ(dx);m∈(m1μ,m+μ)} be the CSK family generated by a non degenerate probability measure μ with support bounded from above. Firstly, for 0<t≠1 such that Qmμ⊞t is well defined, we prove that if Qmμ⊞t∈K+(μ) then the measure μ is a scale transformation of the Marchenko–Pastur law. Secondly, consider Tβ:x⟼x+β, for β∈R∖{0}. We prove that if Tβ(Qmμ)∈K+(μ), then the measure μ is an affine transformation of the Semicircle law.