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Occupation times of subcritical branching immigration systems with Markov motions

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  • Milos, Piotr

Abstract

We consider a branching system consisting of particles moving according to a Markov family in and undergoing subcritical branching with a constant rate V>0. New particles immigrate to the system according to a homogeneous space-time Poisson random field. The process of the fluctuations of the rescaled occupation time is studied with very mild assumptions on the Markov family. In this general setting a functional central limit theorem is proved. The subcriticality of the branching law is crucial for the limit behaviour and in a sense overwhelms the properties of the particles' motion. It is for this reason that the limit is the same for all dimensions and can be obtained for a wide class of Markov processes. Another consequence is the form of the limit --an -valued Wiener process with a simple temporal structure and a complicated spatial one. This behaviour contrasts sharply with the case of critical branching systems.

Suggested Citation

  • Milos, Piotr, 2009. "Occupation times of subcritical branching immigration systems with Markov motions," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3211-3237, October.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:10:p:3211-3237
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    References listed on IDEAS

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    1. Bojdecki, T. & Gorostiza, L.G. & Talarczyk, A., 2006. "Limit theorems for occupation time fluctuations of branching systems I: Long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 1-18, January.
    2. Bojdecki, T. & Gorostiza, L.G. & Talarczyk, A., 2008. "Occupation time limits of inhomogeneous Poisson systems of independent particles," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 28-52, January.
    3. Bojdecki, T. & Gorostiza, L.G. & Talarczyk, A., 2006. "Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 19-35, January.
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    Cited by:

    1. Sun, Hongyan, 2013. "A large deviation theorem for a branching Brownian motion with random immigration," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1559-1566.

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