Lower deviation probabilities for supercritical Markov branching processes with immigration

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 $...

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Published inAIMS mathematics Vol. 10; no. 5; pp. 10324 - 10339
Main Authors Wang, Juan, Peng, Chao
Format Journal Article
LanguageEnglish
Published AIMS Press 01.05.2025
Subjects
branching process
cramér method
immigration
lower deviation
supercritical
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ISSN2473-6988
2473-6988
DOI10.3934/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.
AbstractList 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.
Author Wang, Juan
Peng, Chao
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10.1016/0025-5564(70)90038-6
10.1016/j.anihpb.2006.03.001
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Snippet 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...
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SubjectTerms branching process
cramér method
immigration
lower deviation
supercritical
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Title Lower deviation probabilities for supercritical Markov branching processes with immigration
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