{"title":"A property of the free Gaussian distribution","authors":"Raouf Fakhfakh, Fatimah Alshahrani","doi":"10.1515/gmj-2024-2037","DOIUrl":null,"url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant=\"script\">𝒦</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>σ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mrow> <m:msub> <m:mi>ℙ</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ϑ</m:mi> <m:mo>,</m:mo> <m:mi>σ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>d</m:mi> <m:mo></m:mo> <m:mi>ζ</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:mi>ϑ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mrow> <m:msub> <m:mi>ϑ</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>σ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2037_eq_0162.png\"/> <jats:tex-math>{{\\mathcal{K}_{+}}(\\sigma)=\\{\\mathbb{P}_{(\\vartheta,\\sigma)}(d\\zeta):\\vartheta% \\in(0,\\vartheta_{+}(\\sigma))\\}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>V</m:mi> <m:mi>a</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2037_eq_0072.png\"/> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>a</m:mi> <m:mo>∈</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2037_eq_0131.png\"/> <jats:tex-math>{a\\in\\mathbb{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mi>a</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>ℙ</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ϑ</m:mi> <m:mo>,</m:mo> <m:mi>σ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:msub> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi mathvariant=\"script\">𝒦</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>σ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2037_eq_0068.png\"/> <jats:tex-math>{V_{a}(\\mathbb{P}_{(\\vartheta,\\sigma)})\\in{\\mathcal{K}_{+}}(\\sigma)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>a</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>ℝ</m:mi> <m:mo>∖</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mn>0</m:mn> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2037_eq_0130.png\"/> <jats:tex-math>{a\\in\\mathbb{R}\\setminus\\{0\\}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> if and only if the measure σ is of the free Gaussian (semicircle) type law up to affinity.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Georgian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2024-2037","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let 𝒦+(σ)={ℙ(ϑ,σ)(dζ):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 Va{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 Va(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.
期刊介绍:
The Georgian Mathematical Journal was founded by the Georgian National Academy of Sciences and A. Razmadze Mathematical Institute, and is jointly produced with De Gruyter. The concern of this international journal is the publication of research articles of best scientific standard in pure and applied mathematics. Special emphasis is put on the presentation of results obtained by Georgian mathematicians.