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Statistical inference of subcritical strongly stationary Galton–Watson processes with regularly varying immigration

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  • Barczy, Mátyás
  • Basrak, Bojan
  • Kevei, Péter
  • Pap, Gyula
  • Planinić, Hrvoje

Abstract

We describe the asymptotic behavior of the conditional least squares estimator of the offspring mean for subcritical strongly stationary Galton–Watson processes with regularly varying immigration with tail index α∈(1,2). The limit law is the ratio of two dependent stable random variables with indices α∕2 and 2α∕3, respectively, and it has a continuously differentiable density function. We use point process technique in the proofs.

Suggested Citation

  • Barczy, Mátyás & Basrak, Bojan & Kevei, Péter & Pap, Gyula & Planinić, Hrvoje, 2021. "Statistical inference of subcritical strongly stationary Galton–Watson processes with regularly varying immigration," Stochastic Processes and their Applications, Elsevier, vol. 132(C), pages 33-75.
    • Handle: RePEc:eee:spapps:v:132:y:2021:i:c:p:33-75
    DOI: 10.1016/j.spa.2020.10.004
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    References listed on IDEAS

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    1. Basrak, Bojan & Planinić, Hrvoje, 2019. "A note on vague convergence of measures," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 180-186.
    2. Li, Zenghu & Ma, Chunhua, 2015. "Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3196-3233.
    3. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
    4. Davis, Richard & Resnick, Sidney, 1985. "More limit theory for the sample correlation function of moving averages," Stochastic Processes and their Applications, Elsevier, vol. 20(2), pages 257-279, September.
    5. Hu, Yaozhong & Long, Hongwei, 2009. "Least squares estimator for Ornstein-Uhlenbeck processes driven by [alpha]-stable motions," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2465-2480, August.
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