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Inhomogeneous Markov branching processes: Supercritical case

Author

Listed:
  • Cohn, H.
  • Hering, H.

Abstract

It is our aim to improve the limit theory of supercritical Bienaymé--Galton--Watson processes with varying environment (cf. [3], [4]), and to simultaneously construct its continuous-time analogue. We give a necessary and sufficient condition for the existence of a non-degenerate limit with expectation norming and show that if the limit is non-degenerate, its expectation must be equal to 1. We continue with a result on the expectation of the limit in case of general norming and conditions for the continuity and strict monotonicity of the limiting distribution function. Finally, we reformulate the non-degeneracy condition.

Suggested Citation

  • Cohn, H. & Hering, H., 1983. "Inhomogeneous Markov branching processes: Supercritical case," Stochastic Processes and their Applications, Elsevier, vol. 14(1), pages 79-91, January.
    • Handle: RePEc:eee:spapps:v:14:y:1983:i:1:p:79-91
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    Citations

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    Cited by:

    1. Zhang, Hanjun & Mo, Yongxiang, 2023. "Domain of attraction of quasi-stationary distribution for absorbing Markov processes," Statistics & Probability Letters, Elsevier, vol. 192(C).
    2. Mailler, Cécile & Mörters, Peter & Senkevich, Anna, 2021. "Competing growth processes with random growth rates and random birth times," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 183-226.
    3. 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.
    4. Li, Junping & Meng, Weiwei, 2017. "Regularity criterion for 2-type Markov branching processes with immigration," Statistics & Probability Letters, Elsevier, vol. 121(C), pages 109-118.
    5. Liu, Rongli & Ren, Yan-Xia & Song, Renming & Sun, Zhenyao, 2021. "Quasi-stationary distributions for subcritical superprocesses," Stochastic Processes and their Applications, Elsevier, vol. 132(C), pages 108-134.
    6. Ren, Yan-Xia & Song, Renming & Sun, Zhenyao, 2020. "Limit theorems for a class of critical superprocesses with stable branching," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4358-4391.
    7. Ren, Yan-Xia & Song, Renming & Zhang, Rui, 2015. "Central limit theorems for supercritical superprocesses," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 428-457.
    8. Wang, Juan & Wang, Xueke & Li, Junping, 2023. "Asymptotic behavior for supercritical branching processes," Statistics & Probability Letters, Elsevier, vol. 195(C).
    9. Sagitov, Serik, 2017. "Tail generating functions for extendable branching processes," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1649-1675.

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