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Rate of convergence in the law of large numbers for supercritical general multi-type branching processes

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  • Iksanov, Alexander
  • Meiners, Matthias

Abstract

We provide sufficient conditions for polynomial rate of convergence in the weak law of large numbers for supercritical general indecomposable multi-type branching processes. The main result is derived by investigating the embedded single-type process composed of all individuals having the same type as the ancestor. As an important intermediate step, we determine the (exact) polynomial rate of convergence of Nerman’s martingale in continuous time to its limit. The techniques used also allow us to give streamlined proofs of the weak and strong laws of large numbers and ratio convergence for the processes in focus.

Suggested Citation

  • Iksanov, Alexander & Meiners, Matthias, 2015. "Rate of convergence in the law of large numbers for supercritical general multi-type branching processes," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 708-738.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:2:p:708-738
    DOI: 10.1016/j.spa.2014.10.004
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    References listed on IDEAS

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    1. Nummelin, Esa & Tuominen, Pekka, 1983. "The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 15(3), pages 295-311, August.
    2. Jagers, Peter, 1989. "General branching processes as Markov fields," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 183-212, August.
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    Cited by:

    1. Liu, Rongli & Ren, Yan-Xia & Song, Renming, 2022. "Convergence rate for a class of supercritical superprocesses," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 286-327.
    2. István Fazekas & Attila Barta, 2021. "A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions," Mathematics, MDPI, vol. 9(23), pages 1-26, December.
    3. Kolesko, Konrad & Sava-Huss, Ecaterina, 2023. "Limit theorems for discrete multitype branching processes counted with a characteristic," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 49-75.

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