{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T04:09:29Z","timestamp":1745467769069,"version":"3.40.4"},"reference-count":33,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T00:00:00Z","timestamp":1728259200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000755_inline1.png\"\/><jats:tex-math>\n$(0,+\\infty)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for such CB processes. We give a representation of the stationary measure in terms of scale functions of related L\u00e9vy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom distribution in the subcritical case. Finally we explore some further properties of the limit distributions.<\/jats:p>","DOI":"10.1017\/jpr.2024.75","type":"journal-article","created":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T04:45:37Z","timestamp":1728276337000},"page":"576-602","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Stationary measures and the continuous-state branching process conditioned on extinction"],"prefix":"10.1017","volume":"62","author":[{"given":"Rongli","family":"Liu","sequence":"first","affiliation":[]},{"given":"Yan-Xia","family":"Ren","sequence":"additional","affiliation":[]},{"given":"Ting","family":"Yang","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2024,10,7]]},"reference":[{"key":"S0021900224000755_ref23","first-page":"297","article-title":"The","volume":"24","author":"Maillard","year":"2018","journal-title":"Bernoulli"},{"doi-asserted-by":"publisher","key":"S0021900224000755_ref17","DOI":"10.1016\/j.spa.2012.03.012"},{"doi-asserted-by":"publisher","key":"S0021900224000755_ref12","DOI":"10.2307\/3212550"},{"doi-asserted-by":"publisher","key":"S0021900224000755_ref16","DOI":"10.1007\/978-3-319-41598-7"},{"doi-asserted-by":"publisher","key":"S0021900224000755_ref2","DOI":"10.1016\/j.anihpb.2005.05.003"},{"doi-asserted-by":"publisher","key":"S0021900224000755_ref27","DOI":"10.2977\/prims\/1195192172"},{"doi-asserted-by":"publisher","key":"S0021900224000755_ref21","DOI":"10.1017\/S1446788700001580"},{"key":"S0021900224000755_ref28","first-page":"83","article-title":"On invariant measures of critical multitype Galton\u2013Watson processes","volume":"13","author":"Ogura","year":"1976","journal-title":"Osaka J. 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