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Convergence rate for a class of supercritical superprocesses

Author

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  • Liu, Rongli
  • Ren, Yan-Xia
  • Song, Renming

Abstract

Suppose X={Xt,t≥0} is a supercritical superprocess. Let ϕ be the non-negative eigenfunction of the mean semigroup of X corresponding to the principal eigenvalue λ>0. Then Mt(ϕ)=e−λt〈ϕ,Xt〉,t≥0, is a non-negative martingale with almost sure limit M∞(ϕ). In this paper we study the rate at which Mt(ϕ)−M∞(ϕ) converges to 0 as t→∞ when the process may not have finite variance. Under some conditions on the mean semigroup, we provide sufficient and necessary conditions for the rate in the almost sure sense. Some results on the convergence rate in Lp with p∈(1,2) are also obtained.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:154:y:2022:i:c:p:286-327
    DOI: 10.1016/j.spa.2022.09.009
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    References listed on IDEAS

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    1. Li Wang, 2010. "An Almost Sure Limit Theorem for Super-Brownian Motion," Journal of Theoretical Probability, Springer, vol. 23(2), pages 401-416, June.
    2. 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.
    3. 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.
    4. 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.
    5. Kim, Panki & Song, Renming & Vondraček, Zoran, 2013. "Potential theory of subordinate Brownian motions with Gaussian components," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 764-795.
    6. Cohn, H. & Hering, H., 1983. "Inhomogeneous Markov branching processes: Supercritical case," Stochastic Processes and their Applications, Elsevier, vol. 14(1), pages 79-91, January.
    7. Kouritzin, Michael A. & Ren, Yan-Xia, 2014. "A strong law of large numbers for super-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 505-521.
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