{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T03:56:35Z","timestamp":1774583795401,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,7,29]],"date-time":"2024-07-29T00:00:00Z","timestamp":1722211200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["PNURSP2024R358"],"award-info":[{"award-number":["PNURSP2024R358"]}]},{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["RGP1\/163\/45"],"award-info":[{"award-number":["RGP1\/163\/45"]}]},{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","award":["PNURSP2024R358"],"award-info":[{"award-number":["PNURSP2024R358"]}]},{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","award":["RGP1\/163\/45"],"award-info":[{"award-number":["RGP1\/163\/45"]}]},{"name":"Deanship of Research and Graduate Studies at King Khalid University","award":["PNURSP2024R358"],"award-info":[{"award-number":["PNURSP2024R358"]}]},{"name":"Deanship of Research and Graduate Studies at King Khalid University","award":["RGP1\/163\/45"],"award-info":[{"award-number":["RGP1\/163\/45"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper introduces an innovative concentration inequality for Extended Negative Dependence (END) random variables, providing new insights into their almost complete convergence. We apply this inequality to analyze END variable sequences, particularly focusing on the first-order auto-regressive (AR(1)) model. This application highlights the dynamics and convergence properties of END variables, expanding the analytical tools available for their study. Our findings contribute to both the theoretical understanding and practical applications of END variables in fields such as finance and machine learning, where understanding variable dependencies is crucial.<\/jats:p>","DOI":"10.3390\/axioms13080511","type":"journal-article","created":{"date-parts":[[2024,7,29]],"date-time":"2024-07-29T12:27:43Z","timestamp":1722256063000},"page":"511","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Advancing the Analysis of Extended Negative Dependence Random Variables: A New Concentration Inequality and Its Applications for Linear Models"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7621-6711","authenticated-orcid":false,"given":"Zouaoui","family":"Chikr Elmezouar","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1666-1044","authenticated-orcid":false,"given":"Abderrahmane","family":"Belguerna","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sciences Institute, S.A University Center, P.O. Box 66, Naama 45000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hamza","family":"Daoudi","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, College of Technology, Tahri Mohamed University, Al-Qanadisa Road, P.O. Box 417, Bechar 08000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1306-7688","authenticated-orcid":false,"given":"Fatimah","family":"Alshahrani","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zoubeyr","family":"Kaddour","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sciences Institute, S.A University Center, P.O. Box 66, Naama 45000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1290","DOI":"10.1016\/j.spl.2009.02.001","article-title":"Precise large deviations for dependent random variables with heavy tails","volume":"79","author":"Liu","year":"2009","journal-title":"Stat. Probab. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1186\/1029-242X-2011-98","article-title":"Probability inequalities for END sequence and their applications","volume":"2011","author":"Shen","year":"2011","journal-title":"J. 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