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Measure-valued branching processes with immigration

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  • Li, Zeng-Hu

Abstract

Starting from the cumulant semigroup of a measure-valued branching process, we construct the transition probabilities of some Markov process Y([beta])=(Y([beta])t, t [epsilon] , which we call a measure-valued branching process with discrete immigration of unit[beta]. The immigration of Y([beta]) is governed by a Poisson random measure [rho] on the time-distribution space and a probability generating function h, both depending on [beta]. It is shown that, under suitable hypotheses, Y([beta]) approximates to a Markov process Y=(Yt, t [epsilon] as [beta]-->0+. The latter is the one we call a measure-valued branching process with immigration. The convergence of branching particle systems with immigration is also studied.

Suggested Citation

  • Li, Zeng-Hu, 1992. "Measure-valued branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 43(2), pages 249-264, December.
    • Handle: RePEc:eee:spapps:v:43:y:1992:i:2:p:249-264
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    Citations

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    Cited by:

    1. Li, Zeng-Hu, 1996. "Immigration structures associated with Dawson-Watanabe superprocesses," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 73-86, March.
    2. Li Wang, 2018. "Central Limit Theorems for Supercritical Superprocesses with Immigration," Journal of Theoretical Probability, Springer, vol. 31(2), pages 984-1012, June.
    3. Zenghu Li & Chunhua Ma, 2008. "Catalytic Discrete State Branching Models and Related Limit Theorems," Journal of Theoretical Probability, Springer, vol. 21(4), pages 936-965, December.
    4. Hong, Wenming, 2002. "Longtime behavior for the occupation time process of a super-Brownian motion with random immigration," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 43-62, November.
    5. Wenming Hong, 2003. "Large Deviations for the Super-Brownian Motion with Super-Brownian Immigration," Journal of Theoretical Probability, Springer, vol. 16(4), pages 899-922, October.
    6. Xiong, Jie & Yang, Xu, 2016. "Superprocesses with interaction and immigration," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3377-3401.
    7. Hong, Wenming & Li, Zenghu, 2001. "Fluctuations of a super-Brownian motion with randomly controlled immigration," Statistics & Probability Letters, Elsevier, vol. 51(3), pages 285-291, February.

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