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Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration

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  • Chen, Anyue
  • Renshaw, Eric

Abstract

Attention has recently focussed on stochastic population processes that can undergo total annihilation followed by immigration into state j at rate [alpha]j. The investigation of such models, called Markov branching processes with instantaneous immigration (MBPII), involves the study of existence and recurrence properties. However, results developed to date are generally opaque, and so the primary motivation of this paper is to construct conditions that are far easier to apply in practice. These turn out to be identical to the conditions for positive recurrence, which are very easy to check. We obtain, as a consequence, the surprising result that any MBPII that exists is ergodic, and so must possess an equilibrium distribution. These results are then extended to more general MBPII, and we show how to construct the associated equilibrium distributions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:88:y:2000:i:2:p:177-193
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Pakes, Anthony G., 1993. "Absorbing Markov and branching processes with instantaneous resurrection," Stochastic Processes and their Applications, Elsevier, vol. 48(1), pages 85-106, October.
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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.

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    5. Renshaw, Eric, 2004. "Metropolis-Hastings from a stochastic population dynamics perspective," Computational Statistics & Data Analysis, Elsevier, vol. 45(4), pages 765-786, May.

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