{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T17:07:37Z","timestamp":1769620057534,"version":"3.49.0"},"reference-count":11,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,1,26]],"date-time":"2026-01-26T00:00:00Z","timestamp":1769385600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Outstanding Youth Innovation Team Program of Shandong Province","award":["2024KJG010"],"award-info":[{"award-number":["2024KJG010"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12371148"],"award-info":[{"award-number":["12371148"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["42007141"],"award-info":[{"award-number":["42007141"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"crossref","award":["ZR2019BA038"],"award-info":[{"award-number":["ZR2019BA038"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this study, we propose a delayed sums method to investigate the convergence rates of partial sums. This approach enables general and systematic treatment of the convergence behavior of partial sums, encompassing and extending classical results such as the law of large numbers, the law of logarithm, and the law of the iterated logarithm, as well as convergence with respect to the general norming factors. By establishing almost sure convergence of appropriately defined delayed sums, the proposed method yields explicit convergence rates across a wide range of probabilistic settings. As a result, many convergence problems that were previously treated in isolation can be analyzed within a single coherent theoretical structure.<\/jats:p>","DOI":"10.3390\/axioms15020092","type":"journal-article","created":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T09:21:40Z","timestamp":1769505700000},"page":"92","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["General Convergence Rates by the Delayed Sums Method"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7991-1469","authenticated-orcid":false,"given":"Cheng","family":"Hu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shangshang","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-1962-4954","authenticated-orcid":false,"given":"Tonghui","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1090\/S0002-9947-1965-0198524-1","article-title":"Convergence rates in the law of large numbers","volume":"120","author":"Baum","year":"1965","journal-title":"Trans. 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Regular Variation, Cambridge University Press."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1007\/PL00008782","article-title":"A non-uniform Berry\u2013Esseen bound via Stein\u2019s method","volume":"120","author":"Chen","year":"2001","journal-title":"Probab. Theory Relat. Fields"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/2\/92\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T05:10:45Z","timestamp":1769577045000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/2\/92"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,26]]},"references-count":11,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["axioms15020092"],"URL":"https:\/\/doi.org\/10.3390\/axioms15020092","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,26]]}}}