A property of the free Gaussian distribution

Abstract

Let 𝒦 + ⁢ ( σ ) = { ℙ ( ϑ , σ ) ⁢ ( d ⁢ ζ ) : ϑ ∈ ( 0 , ϑ + ⁢ ( σ ) ) } {{\mathcal{K}_{+}}(\sigma)=\{\mathbb{P}_{(\vartheta,\sigma)}(d\zeta):\vartheta% \in(0,\vartheta_{+}(\sigma))\}} be the Cauchy–Stieltjes Kernel (CSK) family generated by a probability measure σ which is non degenerate and has support bounded from above. Consider the concept of V a {V_{a}} -transformation of measures introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545] for a ∈ ℝ {a\in\mathbb{R}} . We prove that V a ⁢ ( ℙ ( ϑ , σ ) ) ∈ 𝒦 + ⁢ ( σ ) {V_{a}(\mathbb{P}_{(\vartheta,\sigma)})\in{\mathcal{K}_{+}}(\sigma)} for all a ∈ ℝ ∖ { 0 } {a\in\mathbb{R}\setminus\{0\}} if and only if the measure σ is of the free Gaussian (semicircle) type law up to affinity.