{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:46:48Z","timestamp":1760244408560,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,23]],"date-time":"2022-12-23T00:00:00Z","timestamp":1671753600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Research Funds for the Central Universities","award":["2232021D-30","cstc2020jcyj-msxmX0762","2104\/09926601"],"award-info":[{"award-number":["2232021D-30","cstc2020jcyj-msxmX0762","2104\/09926601"]}]},{"name":"Natural Science Foundation of Chongqing of China","award":["2232021D-30","cstc2020jcyj-msxmX0762","2104\/09926601"],"award-info":[{"award-number":["2232021D-30","cstc2020jcyj-msxmX0762","2104\/09926601"]}]},{"name":"Initial Funding of Scientific Research for High-level Talents of Chongqing Three Gorges University of China","award":["2232021D-30","cstc2020jcyj-msxmX0762","2104\/09926601"],"award-info":[{"award-number":["2232021D-30","cstc2020jcyj-msxmX0762","2104\/09926601"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>As the classic branching process, the Galton-Watson process has obtained intensive attentions in the past decades. However, this model has two idealized assumptions\u2013discrete states and time-homogeneity. In the present paper, we consider a branching process with continuous states, and for any given n\u2208N, the branching law of every particle in generation n is determined by the population size of generation n. We consider the case that the process is extinct with Probability 1 since in this case the process will be substantially different from the size-dependent branching process with discrete states. We give the extinction rate in the sense of L2 and almost surely by the form of harmonic moments, that is to say, we show how fast {Zn\u22121} grows under a group of sufficient conditions. From the result of the present paper, we observe that the extinction rate will be determined by an asymptotic behavior of the mean of the branching law. The results obtained in this paper have the more superiority than the counterpart from the existing literature.<\/jats:p>","DOI":"10.3390\/axioms12010013","type":"journal-article","created":{"date-parts":[[2022,12,23]],"date-time":"2022-12-23T03:55:21Z","timestamp":1671767721000},"page":"13","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Asymptotic Behavior for Generalized Ji\u0159ina Process"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3313-848X","authenticated-orcid":false,"given":"You","family":"Lv","sequence":"first","affiliation":[{"name":"College of Science, Donghua University, Shanghai 201620, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1836-2873","authenticated-orcid":false,"given":"Huaping","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404020, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Athreya, K.B., and Ney, P. 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J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF02018373","article-title":"Some limit theorems for Ji\u0159ina processes","volume":"10","author":"Pakes","year":"1979","journal-title":"Period. Math. Hungar."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"40","DOI":"10.2307\/3213662","article-title":"Geometric rate of growth in population-size-dependent branching processes","volume":"21","author":"Klebaner","year":"1984","journal-title":"J. Appl. Probab."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"30","DOI":"10.2307\/1427223","article-title":"On population-size-dependent branching processes","volume":"22","author":"Klebaner","year":"1984","journal-title":"Adv. Appl. Probab."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"48","DOI":"10.2307\/3213747","article-title":"A limit theorem for population-size-dependent branching processes","volume":"22","author":"Klebaner","year":"1985","journal-title":"J. Appl. Probab."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1239\/jap\/1143936253","article-title":"On a continuous-state population-size-dependent branching processes and its extinction","volume":"43","author":"Li","year":"2006","journal-title":"J. Appl. Probab."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"576","DOI":"10.1239\/aap\/1246886624","article-title":"The speed of extinction for some generalized Ji\u0159ina processes","volume":"41","author":"Li","year":"2009","journal-title":"Adv. Appl. Probab."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.spl.2008.07.040","article-title":"Limit theorems for generalized Ji\u0159ina process","volume":"79","author":"Li","year":"2009","journal-title":"Statat. Probab. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1016\/j.spl.2017.06.012","article-title":"L2 limits of generalized Ji\u0159ina process","volume":"129","author":"Lv","year":"2017","journal-title":"Statat. Probab. Lett."},{"key":"ref_13","unstructured":"Bass, R.F. (2013). Real Analysis for Graduate Students, CreateSpace Independent Publishing Platform. [2nd ed.]."},{"key":"ref_14","unstructured":"Loeve, M. (1978). Probability Theory, Springer. [2nd ed.]."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/13\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:49:00Z","timestamp":1760147340000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/13"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,23]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["axioms12010013"],"URL":"https:\/\/doi.org\/10.3390\/axioms12010013","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,12,23]]}}}