{"title":"V_a - 变形自由卷积和方差函数","authors":"Raouf Fakhfakh","doi":"10.1515/gmj-2024-2004","DOIUrl":null,"url":null,"abstract":"In this paper, we deal with the notion 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-2024-2004_eq_0134.png\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free convolution, 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], from a point of view related to the theory of Cauchy–Stieltjes kernel (CSK) families and their corresponding variance functions. We determine the formula for variance function under a power 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-2024-2004_eq_0134.png\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free convolution. Then we provide an approximation of elements of the CSK family generated by <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-2024-2004_eq_0134.png\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free Poisson distribution.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"181 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"V_a -deformed free convolution and variance function\",\"authors\":\"Raouf Fakhfakh\",\"doi\":\"10.1515/gmj-2024-2004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we deal with the notion 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-2024-2004_eq_0134.png\\\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free convolution, 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], from a point of view related to the theory of Cauchy–Stieltjes kernel (CSK) families and their corresponding variance functions. We determine the formula for variance function under a power 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-2024-2004_eq_0134.png\\\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free convolution. Then we provide an approximation of elements of the CSK family generated by <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-2024-2004_eq_0134.png\\\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free Poisson distribution.\",\"PeriodicalId\":55101,\"journal\":{\"name\":\"Georgian Mathematical Journal\",\"volume\":\"181 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-01-31\",\"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-2024-2004\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Georgian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2024-2004","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
摘要
本文将讨论 V a {V_{a}} 变形自由卷积的概念。 -变形自由卷积的概念。D. Krystek 和 L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin.Dimens.Anal.Quantum Probab.Relat.Top.8 2005, 3, 515-545], 从与 Cauchy-Stieltjes 核（CSK）族及其相应方差函数理论相关的角度出发。我们确定了 V a {V_{a}} 的幂下的方差函数公式。 -变形自由卷积下的方差函数公式。然后，我们提供了由 V a {V_{a}} 变形自由泊松分布生成的 CSK 族元素的近似值。 -变形自由泊松分布产生的 CSK 族元素的近似值。
V_a -deformed free convolution and variance function
In this paper, we deal with the notion of Va{V_{a}}-deformed free convolution, 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], from a point of view related to the theory of Cauchy–Stieltjes kernel (CSK) families and their corresponding variance functions. We determine the formula for variance function under a power of Va{V_{a}}-deformed free convolution. Then we provide an approximation of elements of the CSK family generated by Va{V_{a}}-deformed free Poisson distribution.
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.
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