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Asymptotic behavior for supercritical branching processes

Author

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  • Wang, Juan
  • Wang, Xueke
  • Li, Junping

Abstract

Let {Z(t);t⩾0} be a Markov branching process (MBP). There exists a well-known sequence {C(t);t⩾0} such that W(t)≔Z(t)/C(t) a.s. converges to a non-degenerate random variable W as t→∞. This paper attempts to study the asymptotic behavior of P(Z(t)=kt) and P(0⩽Z(t)⩽kt) with kte−λt→0 as t→∞ for MBPs, which helps to study large deviations of Z(t+s)/Z(t). Moreover, we obtain the local limit theorem of this process as an additional finding. During the argumentation, the Cramér method is applied to analyze the large deviation of the sum of random variables.

Suggested Citation

  • Wang, Juan & Wang, Xueke & Li, Junping, 2023. "Asymptotic behavior for supercritical branching processes," Statistics & Probability Letters, Elsevier, vol. 195(C).
    • Handle: RePEc:eee:stapro:v:195:y:2023:i:c:s0167715223000068
    DOI: 10.1016/j.spl.2023.109782
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    References listed on IDEAS

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    1. Cohn, H. & Hering, H., 1983. "Inhomogeneous Markov branching processes: Supercritical case," Stochastic Processes and their Applications, Elsevier, vol. 14(1), pages 79-91, January.
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