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Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration

Author

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  • Juan Wang
  • Xiaojuan Wang
  • G. Muhiuddin

Abstract

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.

Suggested Citation

  • Juan Wang & Xiaojuan Wang & G. Muhiuddin, 2022. "Large Deviation Rates for the Continuous-Time Supercritical Branching Processes with Immigration," Journal of Mathematics, Hindawi, vol. 2022, pages 1-10, June.
    • Handle: RePEc:hin:jjmath:8314977
    DOI: 10.1155/2022/8314977
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