{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:00:15Z","timestamp":1760144415923,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,19]],"date-time":"2024-04-19T00:00:00Z","timestamp":1713484800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Sciences Foundation of China","doi-asserted-by":"publisher","award":["11901392"],"award-info":[{"award-number":["11901392"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Consider a supercritical Galton\u2013Watson process with immigration (Xn;n\u22650). The Lotka\u2013Nagaev estimator Xn+1Xn is a common estimator for the offspring mean. In this work, we used the Martingale method to establish several types of Cram\u00e9r moderate deviation results for the Lotka\u2013Nagaev estimator. To satisfy our needs, we employed the well-known Cram\u00e9r approach for our proofs, which establishes the moderate deviation of the sum of the independent variables. Simultaneously, we provided a concrete example of its applicability in constructing confidence intervals.<\/jats:p>","DOI":"10.3390\/axioms13040272","type":"journal-article","created":{"date-parts":[[2024,4,19]],"date-time":"2024-04-19T08:44:31Z","timestamp":1713516271000},"page":"272","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Cram\u00e9r Moderate Deviations for a Supercritical Galton\u2013Watson Process with Immigration"],"prefix":"10.3390","volume":"13","author":[{"given":"Juan","family":"Wang","sequence":"first","affiliation":[{"name":"College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chao","family":"Peng","sequence":"additional","affiliation":[{"name":"College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,19]]},"reference":[{"key":"ref_1","first-page":"5","article-title":"Sur un nouveau th\u00e9or\u00e8me-limite de la th\u00e9orie des probabilit\u00e9s","volume":"736","year":"1938","journal-title":"Actualite\u2019s Sci. Indust."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1214\/aoap\/1177004971","article-title":"Large Deviation Rates for Branching Processes\u2013I. Single Type Case","volume":"4","author":"Athreya","year":"1994","journal-title":"Ann. Appl. Probab."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Athreya, K.B., and Vidyashankar, A.N. (1997). Classical and Modern Branching Processes, Springer.","DOI":"10.1007\/978-1-4612-1862-3"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1007\/s00440-007-0090-1","article-title":"Large deviations for sums indexed by the generations of a Galton-Watson process","volume":"141","author":"Fleischmann","year":"2008","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1214\/105051604000000242","article-title":"Local limit theory and large deviations for supercritical branching processes","volume":"14","author":"Ney","year":"2004","journal-title":"Ann. Appl. Probab."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"672","DOI":"10.1017\/apr.2016.22","article-title":"On large deviation rates for sums associated with Galton\u2013Watson processes","volume":"48","author":"He","year":"2016","journal-title":"Adv. Appl. Probab."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1007\/s10114-016-5437-z","article-title":"Large deviation for supercritical branching processes with immigration","volume":"32","author":"Liu","year":"2016","journal-title":"Acta Math. Sin."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2465","DOI":"10.1007\/s11425-017-9144-9","article-title":"Sharp large deviation results for sums of bounded from above random variables","volume":"60","author":"Fan","year":"2017","journal-title":"Sci. China Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1201","DOI":"10.1007\/s11464-017-0642-3","article-title":"Harmonic moments and large deviations for supercritical branching processes with immigration","volume":"12","author":"Sun","year":"2017","journal-title":"Front. Math. China"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1007\/s10959-019-00968-z","article-title":"Large deviation rates for supercritical branching processes with immigration","volume":"34","author":"Li","year":"2021","journal-title":"J. Theoret. Probab."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2167","DOI":"10.1214\/aop\/1068646382","article-title":"Self-normalized Cram\u00e9r-type large deviations for independent random variables","volume":"31","author":"Jing","year":"2003","journal-title":"Ann. Probab."},{"key":"ref_12","first-page":"314","article-title":"Harmonic moments and large deviation rates for supercritical branching processes","volume":"12","author":"Ney","year":"2003","journal-title":"Ann. Appl. Probob."},{"key":"ref_13","first-page":"1848","article-title":"Exponential inequalities for self-normalized martingales with applications","volume":"18","author":"Bercu","year":"2008","journal-title":"Ann. Appl. Probob."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1080\/03610926.2019.1635703","article-title":"Moderate deviations for the total population arising from a nearly unstable sub-critical Galton-Watson process with immigration","volume":"50","author":"Chen","year":"2021","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.spl.2022.109711","article-title":"Cram\u00e9r moderate deviations for a supercritical Galton-Watson process","volume":"192","author":"Doukhan","year":"2023","journal-title":"Stat. Probabil. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2793","DOI":"10.3150\/18-BEJ1071","article-title":"Self-normalized Cram\u00e9r type moderate deviations for martingales","volume":"25","author":"Fan","year":"2019","journal-title":"Bernoulli"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s40304-017-0122-9","article-title":"Berry-Essseen bounds for self-normalized martingales","volume":"6","author":"Fan","year":"2018","journal-title":"Comm. Math.-Statist."},{"key":"ref_18","first-page":"1281","article-title":"Self-normalized Cram\u00e9r moderate deviation for a supercritical Galton-Watson process","volume":"60","author":"Fan","year":"2023","journal-title":"Math. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1214\/22-AAP1840","article-title":"A Kesten\u2013Stigum Type Theorem for A Supercritical Multitype Branching Process in A Random Environment","volume":"33","author":"Grama","year":"2023","journal-title":"Ann. Appl. Probab."},{"key":"ref_20","first-page":"233","article-title":"Lower deviation probabilities for supercritical Galton-Watson processes","volume":"43","author":"Fleischmann","year":"2007","journal-title":"Ann. l\u2019IHP Probab. Stat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"100603","DOI":"10.1103\/PhysRevLett.110.100603","article-title":"Ergodicity Breaking in Geometric Brownian Motion","volume":"110","author":"Peters","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2481","DOI":"10.1007\/s11425-016-9017-7","article-title":"Limit theorems for a supercritical branching process with immigration in a random environment","volume":"60","author":"Wang","year":"2017","journal-title":"Sci. China Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"969","DOI":"10.4310\/AJM.2019.v23.n6.a5","article-title":"Quenched weighted moments of a supercritical branching process in a random environment","volume":"23","author":"Wang","year":"2019","journal-title":"Asian J. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"L012106","DOI":"10.1103\/PhysRevE.105.L012106","article-title":"Nonergodicity of reset geometric Brownian motion","volume":"105","author":"Vinod","year":"2022","journal-title":"Phys. Rev. E."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/272\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:31:01Z","timestamp":1760106661000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/272"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,19]]},"references-count":24,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["axioms13040272"],"URL":"https:\/\/doi.org\/10.3390\/axioms13040272","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,4,19]]}}}