Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration
Let Yt;t≥0 be a supercritical continuous-time branching process with immigration; our focus is on the large deviation rates of Yt and thus extending the results of the discrete-time Galton–Watson process to the continuous-time case. Firstly, we prove that Zt=e−mtYt−emt+1−1/em−1ea+m is a submartingal...
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2022/8314977 |
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| _version_ | 1832565497383616512 |
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| author | Juan Wang Xiaojuan Wang |
| author_facet | Juan Wang Xiaojuan Wang |
| author_sort | Juan Wang |
| collection | DOAJ |
| description | Let Yt;t≥0 be a supercritical continuous-time branching process with immigration; our focus is on the large deviation rates of Yt and thus extending the results of the discrete-time Galton–Watson process to the continuous-time case. Firstly, we prove that Zt=e−mtYt−emt+1−1/em−1ea+m is a submartingale and converges to a random variable Z. Then, we study the decay rates of PZt−Z>ε as t⟶∞ and PYt+v/Yt−emv>ε|Z≥α as t⟶∞ for α>0 and ε>0 under various moment conditions on bk;k≥0 and aj;j≥0. We conclude that the rates are supergeometric under the assumption of finite moment generation functions. |
| format | Article |
| id | doaj-art-337f3ce73c174f45bf06c314aa538952 |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-337f3ce73c174f45bf06c314aa5389522025-02-03T01:07:31ZengWileyJournal of Mathematics2314-47852022-01-01202210.1155/2022/8314977Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with ImmigrationJuan Wang0Xiaojuan Wang1School of ScienceSchool of ScienceLet Yt;t≥0 be a supercritical continuous-time branching process with immigration; our focus is on the large deviation rates of Yt and thus extending the results of the discrete-time Galton–Watson process to the continuous-time case. Firstly, we prove that Zt=e−mtYt−emt+1−1/em−1ea+m is a submartingale and converges to a random variable Z. Then, we study the decay rates of PZt−Z>ε as t⟶∞ and PYt+v/Yt−emv>ε|Z≥α as t⟶∞ for α>0 and ε>0 under various moment conditions on bk;k≥0 and aj;j≥0. We conclude that the rates are supergeometric under the assumption of finite moment generation functions.http://dx.doi.org/10.1155/2022/8314977 |
| spellingShingle | Juan Wang Xiaojuan Wang Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration Journal of Mathematics |
| title | Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration |
| title_full | Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration |
| title_fullStr | Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration |
| title_full_unstemmed | Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration |
| title_short | Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration |
| title_sort | large deviation rates for the continuous time supercritical branching processes with immigration |
| url | http://dx.doi.org/10.1155/2022/8314977 |
| work_keys_str_mv | AT juanwang largedeviationratesforthecontinuoustimesupercriticalbranchingprocesseswithimmigration AT xiaojuanwang largedeviationratesforthecontinuoustimesupercriticalbranchingprocesseswithimmigration |