{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:17:13Z","timestamp":1760145433044,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,7,24]],"date-time":"2024-07-24T00:00:00Z","timestamp":1721779200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","award":["PNURSP2024R734"],"award-info":[{"award-number":["PNURSP2024R734"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let K+(\u03bci)={Qsi\u03bci,si\u2208(m0\u03bci,m+\u03bci)}, i=1,2, be two CSK families generated by the nondegenerate probability measures \u03bc1 and \u03bc2 with support bounded from above. Define the set of measures L=K+(\u03bc1)\u2022K+(\u03bc2)={Qs1\u03bc1\u2022Qs2\u03bc2,s1\u2208(m0\u03bc1,m+\u03bc1)ands2\u2208(m0\u03bc2,m+\u03bc2)}, where Qs1\u03bc1\u2022Qs2\u03bc2 denotes the Fermi convolution of Qs1\u03bc1 and Qs2\u03bc2. We prove that if L is still a CSK family (that is, L=K+(\u03c3) for some nondegenerate probability measure ()\u03c3), then the probability measures \u03c3, \u03bc1 and \u03bc2 are of the free Poisson type and follow the free Poisson law up to affinity. The same result, regarding the free Poisson measure, is obtained if we consider the t-deformed free convolution t replacing the Fermi convolution \u2022 in the family of measures L.<\/jats:p>","DOI":"10.3390\/axioms13080496","type":"journal-article","created":{"date-parts":[[2024,7,24]],"date-time":"2024-07-24T14:55:47Z","timestamp":1721832947000},"page":"496","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Some Results on the Free Poisson Distribution"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5337-2276","authenticated-orcid":false,"given":"Ayed. R. A.","family":"Alanzi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia"},{"name":"Department of Mathematics, College of Science and Arts in Gurayat, Jouf University, Gurayat 77454, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-4020-6647","authenticated-orcid":false,"given":"Ohud A.","family":"Alqasem","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"given":"Maysaa Elmahi Abd","family":"Elwahab","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2233-4899","authenticated-orcid":false,"given":"Raouf","family":"Fakhfakh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Arts in Gurayat, Jouf University, Gurayat 77454, Saudi Arabia"},{"name":"Laboratory of Probability and Statistics, Faculty of Sciences of Sfax, University of Sfax, Sfax 3000, Tunisia"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,24]]},"reference":[{"key":"ref_1","first-page":"137","article-title":"A constant regression characterization of the Marchenko-Pastur law","volume":"36","author":"Szpojankowski","year":"2016","journal-title":"Probab. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"109660","DOI":"10.1016\/j.spl.2022.109660","article-title":"A characterization of the Marchenko-Pastur probability measure","volume":"19","author":"Fakhfakh","year":"2022","journal-title":"Stat. Probab. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.crma.2015.12.008","article-title":"A short proof of the Marchenko-Pastur theorem","volume":"354","author":"Yascov","year":"2016","journal-title":"Comptes Rendus Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1850020","DOI":"10.1142\/S0219025718500200","article-title":"Approximation of a free Poisson process by systems of freely independent particles","volume":"21","author":"Kuna","year":"2018","journal-title":"Infin. Dimens. Anal. Quantum Probab. Relat. Top."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"285","DOI":"10.4064\/cm123-2-11","article-title":"Free powers of the free Poisson measure","volume":"123","author":"Hinz","year":"2011","journal-title":"Colloq. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2139","DOI":"10.3150\/14-BEJ638","article-title":"Poisson convergence on the free Poisson algebra","volume":"21","author":"Bourguin","year":"2015","journal-title":"Bernoulli"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1016\/S0246-0203(01)01084-6","article-title":"Remarks on t-transformations of measures and convolutions","volume":"37","year":"2001","journal-title":"Ann. Inst. Poincar Probab. Statist."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"95","DOI":"10.4064\/-43-1-95-103","article-title":"New Examples of Onvolutions and Non-Commutative Central Limit Theorems","volume":"43","year":"1998","journal-title":"Banach Cent. Publ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1142\/S0219025702000791","article-title":"Fermi convolution","volume":"2","author":"Oravecz","year":"2002","journal-title":"Infin. Dimens. Anal. Quantum Probab. Relat. Top."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1515\/dema-2009-0320","article-title":"Free exponential families as kernel families","volume":"42","author":"Bryc","year":"2009","journal-title":"Demonstr. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1007\/s10959-010-0303-x","article-title":"One-sided Cauchy-Stieltjes kernel families","volume":"24","author":"Bryc","year":"2011","journal-title":"J. Theor. Probab."},{"key":"ref_12","first-page":"97","article-title":"Generalized t-transformations of probability measures and deformed convolution","volume":"24","author":"Krystek","year":"2004","journal-title":"Probab. Math. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/j.jmaa.2017.10.003","article-title":"Characterization of quadratic Cauchy-Stieltjes Kernel families based on the orthogonality of polynomials","volume":"459","author":"Fakhfakh","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_14","first-page":"3","article-title":"Fermi convolution and variance function","volume":"24","author":"Fakhfakh","year":"2023","journal-title":"Proc. Rom. Acad. Ser. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"769","DOI":"10.2307\/2310775","article-title":"The functional equation f (x + y) = g(x) + h(y)","volume":"69","author":"Friedman","year":"1962","journal-title":"Am. Math. Mon."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/496\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:22:35Z","timestamp":1760109755000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/496"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,24]]},"references-count":15,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080496"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080496","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,7,24]]}}}