Lower deviation probabilities for supercritical Markov branching processes with immigration

Juan Wang; Chao Peng; Juan Wang; Chao Peng

doi:10.3934/math.2025470

AIMS Mathematics

2025, Volume 10, Issue 5: 10324-10339. doi: 10.3934/math.2025470

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Research article

Lower deviation probabilities for supercritical Markov branching processes with immigration

Juan Wang ^,,
Chao Peng

College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

Received: 15 November 2024 Revised: 01 April 2025 Accepted: 07 April 2025 Published: 06 May 2025
MSC : 60J27, 60J35

Let $ \{Z(t); t\geq 0\} $ be a continuous-time supercritical branching process with immigration (MBPI) with the offspring mean $ m(t) $. In this paper, we mainly research the lower deviation probabilities $ {\rm P}(Z(t) = k_t) $ and $ {\rm P}(0\leq Z(t)\leq k_t) $ with $ k_t/e^{m(t)}\rightarrow 0 $ as $ t\rightarrow \infty $. Moreover, we present the local limit theorem and some related estimates of the MBPIs. For our proofs, we use the well-known Cramér method to prove the large deviation of the sum of independent variables to satisfy our needs.
- lower deviation,
- supercritical,
- branching process,
- immigration,
- Cramér method
Citation: Juan Wang, Chao Peng. Lower deviation probabilities for supercritical Markov branching processes with immigration[J]. AIMS Mathematics, 2025, 10(5): 10324-10339. doi: 10.3934/math.2025470

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Abstract

Let $ \{Z(t); t\geq 0\} $ be a continuous-time supercritical branching process with immigration (MBPI) with the offspring mean $ m(t) $. In this paper, we mainly research the lower deviation probabilities $ {\rm P}(Z(t) = k_t) $ and $ {\rm P}(0\leq Z(t)\leq k_t) $ with $ k_t/e^{m(t)}\rightarrow 0 $ as $ t\rightarrow \infty $. Moreover, we present the local limit theorem and some related estimates of the MBPIs. For our proofs, we use the well-known Cramér method to prove the large deviation of the sum of independent variables to satisfy our needs.

References

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