{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:32:41Z","timestamp":1760059961104,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T00:00:00Z","timestamp":1753228800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Hebei Province","award":["A2025501005"],"award-info":[{"award-number":["A2025501005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let (Zn)n\u22650 be a supercritical branching process with offspring distribution pr=P(Z1=r),r\u22650, such that p0=p1=\u2026=pk0\u22121=0 and pk0&gt;0, where k0\u22652. The Lotka\u2013Nagaev estimator Zn+1\/Zn is an important estimator for the offspring mean m=\u2211i=k0\u221eipi. In this paper, we establish a self-normalized large deviation result and self-normalized Cram\u00e9r type moderate deviations for the Lotka\u2013Nagaev estimator. An application to constructing confidence intervals for m is also discussed.<\/jats:p>","DOI":"10.3390\/axioms14080556","type":"journal-article","created":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T14:22:44Z","timestamp":1753280564000},"page":"556","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Self-Normalized Large Deviation Principle and Cram\u00e9r Type Moderate Deviations for a Supercritical Branching Process"],"prefix":"10.3390","volume":"14","author":[{"given":"Peishuang","family":"Duan","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China"}]},{"given":"Haijuan","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China"}]},{"given":"Tingyue","family":"Mei","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China"}]},{"given":"Chongxian","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,23]]},"reference":[{"key":"ref_1","first-page":"123","article-title":"Theorie analytique des assiciation biologiques","volume":"780","author":"Lotka","year":"1939","journal-title":"Actualit\u00e9s Sci. Ind."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1137\/1112037","article-title":"On estimating the expected number of direct descendants of a particle in a branching process","volume":"12","author":"Nagaev","year":"1967","journal-title":"Theory Probab. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1214\/aoap\/1177004971","article-title":"Large deviation rates for branching processes. I. Single type case","volume":"4","author":"Athreya","year":"1994","journal-title":"Ann. Appl. Probab."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1214\/aoap\/1050689589","article-title":"Harmonic moments and large deviation rates for supercritical branching processes","volume":"13","author":"Ney","year":"2003","journal-title":"Ann. Appl. Probab."},{"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-Watson processes","volume":"48","author":"He","year":"2016","journal-title":"Adv. Appl. Probab."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1848","DOI":"10.1214\/07-AAP506","article-title":"Exponential inequalities for self-normalized martingales with applications","volume":"18","author":"Bercu","year":"2008","journal-title":"Ann. Appl. Probab."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1017\/jpr.2018.29","article-title":"Self-normalized large deviation for supercritical branching processes","volume":"55","author":"Chu","year":"2018","journal-title":"J. Appl. Probab."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.1017\/jpr.2022.134","article-title":"Self-normalized Cram\u00e9r moderate deviations for a supercritical Galton-Watson process","volume":"60","author":"Fan","year":"2023","journal-title":"J. Appl. Probab."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"130","DOI":"10.3150\/24-BEJ1722","article-title":"Self-normalized Cram\u00e9r type moderate deviations for martingales and applications","volume":"31","author":"Fan","year":"2025","journal-title":"Bernoulli"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"109711","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":"Statist. Probab. Letters"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Petrov, V.V. (1975). Sums of Independent Random Variables, Springer.","DOI":"10.1515\/9783112573006"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1017\/jpr.2019.15","article-title":"Cram\u00e9r type moderate deviations for random fields","volume":"56","author":"Beknazaryan","year":"2019","journal-title":"J. Appl. Probab."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2046","DOI":"10.1214\/23-AIHP1372","article-title":"Cram\u00e9r\u2019s moderate deviations for martingales with applications","volume":"60","author":"Fan","year":"2024","journal-title":"Ann. Inst. H. Poincar\u00e9 Probab. Statist."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"1865","DOI":"10.1007\/s11425-022-2161-4","article-title":"Normalized and self-normalized Cram\u00e9r-type moderate deviations for Euler-Maruyama scheme for SDE","volume":"67","author":"Fan","year":"2024","journal-title":"Sci. China Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1214\/aop\/1024404289","article-title":"Self-normalized large deviations","volume":"25","author":"Shao","year":"1997","journal-title":"Ann. Probab."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/556\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:14:50Z","timestamp":1760033690000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/556"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,23]]},"references-count":17,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080556"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080556","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,23]]}}}