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Central Limit Theorems for Supercritical Superprocesses with Immigration

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Abstract

In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with immigration with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem extends and generalizes the results obtained by Ren et al. (Stoch Process Appl 125:428–457, 2015). We first give laws of large numbers for supercritical superprocesses with immigration since there are few convergence results on immigration superprocesses, then based on these results, we establish the central limit theorem.

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References

  1. Asmussen, S., Hering, H.: Strong limit theorems for supercritical immigration-branching processes. Math. Scand. 39, 327–342 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adamczak, R., Milós, P.: CLT for Ornstein–Uhlenbeck branching particle system. Electron. J. Probab. 20(42), 1–35 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Chen, Z.Q., Ren, Y.X., Wang, H.: An almost sure limit theorem for Dawson–Watanabe superprocesses. J. Funct. Anal. 254, 1988–2019 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, Z.Q., Shiozawa, Y.: Limit theorems for branching Markov processes. J. Funct. Anal. 250, 374–399 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dynkin, E.B.: Superprocesses and partial differential equations. Ann. Probab. 21, 1185–1262 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dynkin, E.B., Kuznetsov, S.E.: N-measure for branching exit Markov system and their applications to differential equations. Probab. Theory Rel. Fields 130, 135–150 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. El Karoui, N., Roelly, S.: Propriétés de martingales, explosion et représentation de Lévy- Khintchine d’une classe de processus de branchment à valeurs mesures. Stoch. Process. Appl. 38, 239–266 (1991)

    Article  MATH  Google Scholar 

  8. Davies, E.B., Simon, B.: Ultracontractivity and the kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335–395 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  9. Englander, J.: Law of large numbers for superdiffusions: the non-ergodic case. Ann. Inst. Henri Poincaré Probab. Stat. 45, 1–6 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Englander, J., Harris, S.C., Kyprianou, A.E.: Strong law of large numbers for branching diffsions. Ann. Inst. Henri Poincaré Probab. Stat. 46, 279–298 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Englander, J., Winter, A.: Law of large numbers for a class of superdiffusions. Ann. Inst. Henri Poincaré Probab. Stat. 42, 171–185 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, Z.: Measure-valued branching processes with immigration. Stoch. Process. Appl. 43, 249–264 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, Z., Shiga, T.: Measure-valued branching diffusions: immigrations, excursions and limit theorems. J. Math. Kyoto Univ. 35, 233–274 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, Z.: Immigration structures associated with Dawson–Watanabe superprocesses. Stoch. Process. Appl. 62, 73–86 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, Z.: Skew convolution semigroups and related immigration processes. Theory Probab. Appl. 46, 274–296 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, Z.: Measure-Valued Branching Markov Processes. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  17. Liu, R.-L., Ren, Y.-X., Song, R.: Strong law of large number of a class of super-diffusions. Acta Appl. Math. 123(1), 73–97 (2013)

    Article  MathSciNet  Google Scholar 

  18. Milós, P.: Spatial CLT for the supercritical Ornstein–Uhlenbeck superprocess. Preprint, arXiv:1203.6661 (2012)

  19. Ren, Y.-X., Song, R., Zhang, R.: Central limit theorems for super Ornstein–Uhlenbeck processes. Acta Appl. Math. 130, 9–49 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ren, Y.-X., Song, R., Zhang, R.: Central limit theorems for supercritical branching Markov processes. J. Funct. Anal. 266, 1716–1756 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ren, Y.-X., Song, R., Zhang, R.: Central limit theorems for supercritical superprocesses. Stoch. Process. Appl. 125, 428–457 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, L.: An almost sure limit theorem for super-Brownian motion. J. Theoret. Probab. 23, 401–416 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Parts of this paper were written while the author visited Concordia. The author would like to give sincere thanks to Professor Xiaowen Zhou for his encouragement and helpful discussions and hospitality at Concordia. The author also thanks an anonymous referee for useful comments.

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Authors and Affiliations

  1. School of Sciences, Beijing University of Chemical Technology, Beijing, 100029, People’s Republic of China

    Li Wang

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  1. Li Wang
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Correspondence to Li Wang.

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Supported by NSFC 11301020 Grants.

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Wang, L. Central Limit Theorems for Supercritical Superprocesses with Immigration. J Theor Probab 31, 984–1012 (2018). https://doi.org/10.1007/s10959-016-0736-y

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  • DOI: https://doi.org/10.1007/s10959-016-0736-y

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