{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:26:53Z","timestamp":1760059613137,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,25]],"date-time":"2025-06-25T00:00:00Z","timestamp":1750809600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271029","11901570","KM202411417001"],"award-info":[{"award-number":["12271029","11901570","KM202411417001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"R&amp;D Program of Beijing Municipal Education Commission","award":["12271029","11901570","KM202411417001"],"award-info":[{"award-number":["12271029","11901570","KM202411417001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We establish a Perron\u2013Frobenius-type theorem for the subdominant eigenvalue of M\u00f6bius monotone transition matrices defined on partially ordered state spaces. This result extends the classical work of Keilson and Kester, where they considered stochastically monotone transition matrices in a totally ordered setting. Furthermore, we show that this subdominant eigenvalue is the geometric ergodicity rate.<\/jats:p>","DOI":"10.3390\/axioms14070493","type":"journal-article","created":{"date-parts":[[2025,6,25]],"date-time":"2025-06-25T03:53:02Z","timestamp":1750823582000},"page":"493","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Subdominant Eigenvalue of M\u00f6bius Monotone Transition Probability Matrix"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3949-9529","authenticated-orcid":false,"given":"Pei-Sen","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100872, China"}]},{"given":"Pan","family":"Zhao","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Physics, Beijing Union University, Beijing 100101, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,25]]},"reference":[{"key":"ref_1","unstructured":"Chen, M.F. (2005). Eigenvalues, Inequalities, and Ergodic Theory, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Levin, D.A., and Peres, Y. (2017). Markov Chains and Mixing Times, American Mathematical Society. [2nd ed.].","DOI":"10.1090\/mbk\/107"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Seneta, E. (1981). Non-negative Matrices and Markov Chains, Springer.","DOI":"10.1007\/0-387-32792-4"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/0304-4149(77)90033-3","article-title":"Monotone matrices and monotone Markov processes","volume":"5","author":"Keilson","year":"1977","journal-title":"Stoch. Process. Appl."},{"key":"ref_5","unstructured":"Kester, A. (1977). Preservation of Cone Characterizing Properties in Markov Chains. [Ph.D. Thesis, University of Rochester]."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1287\/moor.12.2.350","article-title":"Stochastic ordering for Markov processes on partially ordered spaces","volume":"12","author":"Massey","year":"1987","journal-title":"Math. Oper. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s11134-012-9284-z","article-title":"Strong stationary duality for M\u00f6bius monotone Markov chains","volume":"71","author":"Lorek","year":"2012","journal-title":"Queueing Syst."},{"key":"ref_8","first-page":"75","article-title":"Strong stationary duality for M\u00f6bius monotone Markov chains: Examples","volume":"36","author":"Lorek","year":"2016","journal-title":"Probab. Math. Statist."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1007\/s11009-016-9507-6","article-title":"Generalized Gambler\u2019s ruin problem: Explicit formulas via Siegmund duality","volume":"19","author":"Lorek","year":"2017","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1017\/S0269964817000341","article-title":"Siegmund duality for Markov chains on partially ordered state spaces","volume":"32","author":"Lorek","year":"2018","journal-title":"Probab. Eng. Inform. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1007\/BF00531932","article-title":"On the foundations of combinatorial theory I. theory of M\u00f6bius functions","volume":"2","author":"Rota","year":"1964","journal-title":"Z. Wahrsch. Verw. Geb."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1214\/aop\/1176995936","article-title":"The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes","volume":"4","author":"Siegmund","year":"1976","journal-title":"Ann. Probab."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1483","DOI":"10.1214\/aop\/1176990628","article-title":"Strong stationary times via a new form of duality","volume":"18","author":"Diaconis","year":"1990","journal-title":"Ann. Probab."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1239\/jap\/1208358960","article-title":"The limit behavior of dual Markov branching processes","volume":"45","author":"Li","year":"2008","journal-title":"J. Appl. Prob."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.R. (2012). Matrix Analysis, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9781139020411"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/493\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:58:04Z","timestamp":1760032684000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/493"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,25]]},"references-count":15,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070493"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070493","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,6,25]]}}}