A characteristic property of the Semicircle law

Let ${\mathcal{F}_{+}}(\sigma)=\{\textbf{Q}_m^{\sigma}(dy); \ \ m\in (m_1^{\sigma},m_+^{\sigma})\}$ be the \CSK family induced by a (non-degenerate ) probability measure $\sigma$ possessing a one sided support boundary from above. For $\tau\geq0$, denote by $\widetilde{U}^{\tau}$ the $\tau$-deformation of measures introduced in \cite[Section 5]{Krystek-Yoshida-04}. In this article, a property is presented for the \CSK families based on the stability under $\tau$-deformation of measures. A \CSK family satisfying such stability property is nothing but the \CSK family generated by the Semicircle type law, up to affinity.