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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 2314-4785 |