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Markov branching processes regulated by emigration and large immigration

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  • Chen, Anyue
  • Renshaw, Eric

Abstract

Although simple branching processes play an important role in classical applied probability theory, practical application remains essentially weak since all positive states are transient. A realistic modification which avoids this undesirable feature is to introduce immigration. In this paper we consider a new structure which admits large immigration, i.e. the sum of immigration rates is infinite; excessively high population levels are avoided by allowing the carrying capacity of the system to be controlled by mass emigration. We provide an existence criterion for such models that is easy to check, prove that the corresponding honest process is unique and positive recurrent, and derive the limiting distribution of population size. These results are then illustrated through two interesting examples.

Suggested Citation

  • Chen, Anyue & Renshaw, Eric, 1995. "Markov branching processes regulated by emigration and large immigration," Stochastic Processes and their Applications, Elsevier, vol. 57(2), pages 339-359, June.
    • Handle: RePEc:eee:spapps:v:57:y:1995:i:2:p:339-359
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    References listed on IDEAS

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    1. Chen, Anyue & Renshaw, Eric, 1993. "Recurrence of Markov branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 45(2), pages 231-242, April.
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    Cited by:

    1. Renshaw, Eric, 2004. "Metropolis-Hastings from a stochastic population dynamics perspective," Computational Statistics & Data Analysis, Elsevier, vol. 45(4), pages 765-786, May.
    2. Anyue Chen, 2020. "Resolvent Decomposition Theorems and Their Application in Denumerable Markov Processes with Instantaneous States," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2089-2118, December.
    3. Chen, Anyue & Renshaw, Eric, 2000. "Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration," Stochastic Processes and their Applications, Elsevier, vol. 88(2), pages 177-193, August.
    4. Khrystyna Prysyazhnyk & Iryna Bazylevych & Ludmila Mitkova & Iryna Ivanochko, 2021. "Period-Life of a Branching Process with Migration and Continuous Time," Mathematics, MDPI, vol. 9(8), pages 1-10, April.

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    1. Anyue Chen, 2020. "Resolvent Decomposition Theorems and Their Application in Denumerable Markov Processes with Instantaneous States," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2089-2118, December.
    2. Chen, Anyue & Renshaw, Eric, 2000. "Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration," Stochastic Processes and their Applications, Elsevier, vol. 88(2), pages 177-193, August.

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