A characterization of the Wigner’s semicircle law

Ayed. R.A. Alanzi, Shokrya S. Alshqaq and Raouf Fakhfakh

Statistics & Probability Letters, 2025, vol. 222, issue C

Abstract: This paper present a new characterization of the Wigner’s semicircle law. Denote by P (respectively by Pba) the set of non-degenerate real probabilities (respectively, with one sided support boundary from above). 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. On the other hand, let F+(μ)={Qlμ(dx):l∈(m0μ,m+μ)} be the (CSK) family induced by μ∈Pba with finite first moment m0μ. Define a novel family of probabilities Ta(F+(μ))={Ta(Qlμ)(dx):l∈(m0μ,m+μ)}. For a≠0, we prove that Ta(F+(μ))=F+(Ta(μ)), (with Ta(x)=x+a), if and only if μ is of the Wigner’s type measure up to affinity.

Keywords: Variance function; Transformation of measures; Wigner’s semicircle law (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spl.2025.110397

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