Long-time behaviour of supercritical finite circular mechanism branching processes

Long-time behaviour of supercritical finite circular mechanism branching processes | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Long-time behaviour of supercritical finite circular mechanism branching processes Junping Li, Mixuan Hou This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4423930/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Let {Zna*b:n≥ 0} be a discrete-time branching process with circular mechanism a*b. For mechanism a, the offspring distribution is {aj:j≥ 0}. For mechanism b, the offspring distribution is {bj:j ≥0}. Let ma=∑j≥0 jaj and mb=∑j≥0 jbj . The extinction property of such branching processes is first studied. It is proved that Wn=Zna*b/γn ( γn=(mamb)k for n=2k and γn=(mamb)kma for n=2k+1) is an integrable martingale and hence converges to some random variable W. Then, under assumption that a0=b0=0, a1,b1>0 and aj,bj≠1 for any j≠ 1, we study the rates of convergence to zero as k→∞ of P(|Z2k+1a*b/Z2ka*b-ma |>ε), P(|Z2ka*b/Z2k-1a*b-mb |>ε), P(|Wk-W|>ε) P(|Z2k+1a*b/Z2ka*b-ma |>ε|W>δ), P(|(Z2ka*b/Z2k-1a*b-mb |>ε|W>δ) for ε>0 and δ>0 under various moment conditions on {aj} and {bj}. It is shown that the rates for the first two are geometric while the last three rates are always supergeometric under a finite moment generating function hypothesis. Branching processes with circular mechanism Extinction Large deviation Geometric Supergeometric. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4423930","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":305026846,"identity":"d504acb4-599c-4508-9a87-e33a2aeaabee","order_by":0,"name":"Junping Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1UlEQVRIiWNgGAWjYBACxgbmhgMfGBh4+OAChLUwNhycAdTCRrQWsD08QIp4LcyzGxsP2+bYybAxMD/+zMNgI7vhAPOzB3jtmHOw4XDutmSgw9jMpHkY0ow3HGAzN8CrZUYiSAszUAsPG9CFhxM3HOBhkyCoxXJbPUgLM9Bh/4nUwrjtMEgLA9BhB4jQAvTLwd5tx4GuYjOTnGOQbDzzMJsZXi2Gs5sPf/i5rdqen7358Yc3FXayfcebn+HXMgPGYgYRBjAGHiCP18BRMApGwSgYBSAAAAg1QeGW06t/AAAAAElFTkSuQmCC","orcid":"","institution":"Guangdong University of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Junping","middleName":"","lastName":"Li","suffix":""},{"id":305026847,"identity":"a5918f40-82d6-4bfb-9701-99fffa1078cc","order_by":1,"name":"Mixuan Hou","email":"","orcid":"","institution":"Central South University","correspondingAuthor":false,"prefix":"","firstName":"Mixuan","middleName":"","lastName":"Hou","suffix":""}],"badges":[],"createdAt":"2024-05-15 08:59:10","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4423930/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4423930/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":62849641,"identity":"d1ec6ef8-df37-4339-9244-906b7cff7cd7","added_by":"auto","created_at":"2024-08-20 08:17:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":231170,"visible":true,"origin":"","legend":"","description":"","filename":"BIGW20230408.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4423930/v1_covered_6de6018e-b16d-452e-a09f-482d015e5969.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Long-time behaviour of supercritical finite circular mechanism branching processes","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Branching processes with circular mechanism, Extinction, Large deviation, Geometric, Supergeometric.","lastPublishedDoi":"10.21203/rs.3.rs-4423930/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4423930/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eLet {Zna*b:n≥\u0026nbsp;0} be a discrete-time branching process with circular mechanism a*b. For mechanism a, the offspring distribution is {aj:j≥ 0}. For mechanism b, the offspring distribution is {bj:j ≥0}. Let ma=∑j≥0 jaj and mb=∑j≥0 jbj . The extinction property of such branching processes is first studied. It is proved that Wn=Zna*b/γn \u0026nbsp;( γn=(mamb)k for n=2k and γn=(mamb)kma for n=2k+1) is an integrable martingale and hence converges to some random variable W. Then, under assumption that a0=b0=0, a1,b1\u0026gt;0 and aj,bj≠1 for any j≠ 1, we study the rates of convergence to zero as k→∞ of \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;P(|Z2k+1a*b/Z2ka*b-ma |\u0026gt;ε),\u0026nbsp; \u0026nbsp; \u0026nbsp; P(|Z2ka*b/Z2k-1a*b-mb |\u0026gt;ε),\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp;P(|Wk-W|\u0026gt;ε)\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;P(|Z2k+1a*b/Z2ka*b-ma |\u0026gt;ε|W\u0026gt;δ),\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; P(|(Z2ka*b/Z2k-1a*b-mb |\u0026gt;ε|W\u0026gt;δ)\u0026nbsp;\u003c/p\u003e\n\u003cp\u003efor ε\u0026gt;0 and δ\u0026gt;0 under various moment conditions on {aj} and {bj}. It is shown that the rates for the first two are geometric while the last three rates are always supergeometric under a finite moment generating function hypothesis.\u003c/p\u003e","manuscriptTitle":"Long-time behaviour of supercritical finite circular mechanism branching processes","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-06-03 03:06:50","doi":"10.21203/rs.3.rs-4423930/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"66dd42bc-7607-4113-bcd7-78005466c2b2","owner":[],"postedDate":"June 3rd, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-08-20T08:08:56+00:00","versionOfRecord":[],"versionCreatedAt":"2024-06-03 03:06:50","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4423930","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4423930","identity":"rs-4423930","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}