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A characterization of the Wigner’s semicircle law

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

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.

Suggested Citation

  • Alanzi, Ayed. R.A. & Alshqaq, Shokrya S. & Fakhfakh, Raouf, 2025. "A characterization of the Wigner’s semicircle law," Statistics & Probability Letters, Elsevier, vol. 222(C).
    • Handle: RePEc:eee:stapro:v:222:y:2025:i:c:s0167715225000422
    DOI: 10.1016/j.spl.2025.110397
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    References listed on IDEAS

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    1. Włodzimierz Bryc & Abdelhamid Hassairi, 2011. "One-Sided Cauchy–Stieltjes Kernel Families," Journal of Theoretical Probability, Springer, vol. 24(2), pages 577-594, June.
    2. Bryc, Włodek & Fakhfakh, Raouf & Hassairi, Abdelhamid, 2014. "On Cauchy–Stieltjes kernel families," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 295-312.
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    1. Abdulmajeed Albarrak & Raouf Fakhfakh & Ghadah Alomani, 2025. "B t -Transformation and Variance Function," Mathematics, MDPI, vol. 13(21), pages 1-15, October.

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