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Acta Applicandae Mathematicae

, Volume 147, Issue 1, pp 137–175 | Cite as

Functional Central Limit Theorems for Supercritical Superprocesses

  • Yan-Xia Ren
  • Renming Song
  • Rui Zhang
Article
  • 151 Downloads

Abstract

In this paper, we establish some functional central limit theorems for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. In the particular case when the state \(E\) is a finite set and the underlying motion is an irreducible Markov chain on \(E\), our results are superprocess analogs of the functional central limit theorems of Janson (Stoch. Process. Appl. 110:177–245, 2004) for supercritical multitype branching processes. The results of this paper are refinements of the central limit theorems in Ren et al. (Stoch. Process. Appl. 125:428–457, 2015).

Keywords

Functional central limit theorem Supercritical superprocess Excursion measures of superprocesses 

Mathematics Subject Classification

60J68 60F05 60G57 60J45 

Notes

Acknowledgements

We thank the two referees for their helpful comments on the first version of this paper.

References

  1. 1.
    Adamczak, R., Miłoś, P.: CLT for Ornstein-Uhlenbeck branching particle system. Electron. J. Probab. 20(42), 1–35 (2015) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Adamczak, R., Miłoś, P.: \(U\)-statistics of Ornstein-Uhlenbeck branching particle system. J. Theor. Probab. 27(4), 1071–1111 (2014) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Asmussen, S., Hering, H.: Branching Processes. Birkhäuser, Boston (1983) CrossRefzbMATHGoogle Scholar
  4. 4.
    Asmussen, S., Keiding, N.: Martingale central limit theorems and asymptotic estimation theory for multitype branching processes. Adv. Appl. Probab. 10, 109–129 (1978) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Athreya, K.B.: Limit theorems for multitype continuous time Markov branching processes I: the case of an eigenvector linear functional. Z. Wahrscheinlichkeitstheor. Verw. Geb. 12, 320–332 (1969) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Athreya, K.B.: Limit theorems for multitype continuous time Markov branching processes II: the case of an arbitrary linear functional. Z. Wahrscheinlichkeitstheor. Verw. Geb. 13, 204–214 (1969) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Athreya, K.B.: Some refinements in the theory of supercritical multitype Markov branching processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 20, 47–57 (1971) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bojdecki, T., Gorostiza, L.G., Talarczyk, A.: Limit theorems for occupation time fluctuations of branching systems I: long-range dependence. Stoch. Process. Appl. 116, 1–18 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bojdecki, T., Gorostiza, L.G., Talarczyk, A.: Limit theorems for occupation time fluctuations of branching systems II: critical and lage dimensions. Stoch. Process. Appl. 116, 19–35 (2006) CrossRefzbMATHGoogle Scholar
  10. 10.
    Bojdecki, T., Gorostiza, L.G., Talarczyk, A.: Occupation time limits of inhomogeneous Poisson systems of independent particles. Stoch. Process. Appl. 118, 28–52 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Bojdecki, T., Gorostiza, L.G., Talarczyk, A.: Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses. Electron. J. Probab. 14(46), 1328–1371 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Chung, K.L., Walsh, J.B.: Markov Processes, Brownian Motion, and Time Symmetry. Springer, Berlin (2005) CrossRefzbMATHGoogle Scholar
  13. 13.
    Davies, E.B., Simon, B.: Ultracontractivity and the kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335–395 (1984) CrossRefzbMATHGoogle Scholar
  14. 14.
    Dawson, D.A.: Measure-Valued Markov Processes. Springer, Berlin (1993) CrossRefzbMATHGoogle Scholar
  15. 15.
    Dynkin, E.B.: Superprocesses and partial differential equations. Ann. Probab. 21, 1185–1262 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Dynkin, E.B., Kuznetsov, S.E.: ℕ-measure for branching exit Markov system and their applications to differential equations. Probab. Theory Relat. Fields 130, 135–150 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    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) CrossRefzbMATHGoogle Scholar
  18. 18.
    Hong, W.: Functional central limit theorem for super \(\alpha \)-stable processes. Sci. China Ser. A 47, 874–881 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Iscoe, I.: A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Relat. Fields 71, 85–116 (1986) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, Berlin (2003) CrossRefzbMATHGoogle Scholar
  21. 21.
    Janson, S.: Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110, 177–245 (2004) CrossRefzbMATHGoogle Scholar
  22. 22.
    Kesten, H., Stigum, B.P.: A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Stat. 37, 1211–1223 (1966) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Kesten, H., Stigum, B.P.: Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Ann. Math. Stat. 37, 1463–1481 (1966) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Li, Z.: Skew convolution semigroups and related immigration processes. Theory Probab. Appl. 46, 274–296 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Li, Z.: Measure-Valued Branching Markov Processes. Springer, Heidelberg (2011) CrossRefzbMATHGoogle Scholar
  26. 26.
    Li, Z.H., Shiga, T.: Measure-valued branching diffusions: immigrations, excursions and limit theorems. J. Math. Kyoto Univ. 35, 233–274 (1995) zbMATHMathSciNetGoogle Scholar
  27. 27.
    Miłoś, P.: Occupation time fluctuations of Poisson and equilibrium finite variance branching systems. Probab. Math. Stat. 27, 181–203 (2007) zbMATHMathSciNetGoogle Scholar
  28. 28.
    Miłoś, P.: Occupation time fluctuations of Poisson and equilibrium branching systems in critical and large dimensions. Probab. Math. Stat. 28, 235–256 (2008) zbMATHMathSciNetGoogle Scholar
  29. 29.
    Miłoś, P.: Occupation times of subcritical branching immigration systems with Markov motions. Stoch. Process. Appl. 119, 3211–3237 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Miłoś, P.: Occupation times of subcritical branching immigration systems with Markov motion, CLT and deviation principles. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15, 1250002 (2012). 28 pp. CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Miłoś, P.: Spatial CLT for the supercritical Ornstein-Uhlenbeck superprocess (2012). Preprint. arXiv:1203.6661
  32. 32.
    Ren, Y.-X., Song, R., Zhang, R.: Central limit theorems for super Ornstein-Uhlenbeck processes. Acta Appl. Math. 130, 9–49 (2014) CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Ren, Y.-X., Song, R., Zhang, R.: Central limit theorems for supercritical branching Markov processes. J. Funct. Anal. 266, 1716–1756 (2014) CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Ren, Y.-X., Song, R., Zhang, R.: Central limit theorems for supercritical superprocesses. Stoch. Process. Appl. 125, 428–457 (2015) CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Ren, Y.-X., Song, R., Zhang, R.: Central limit theorems for supercritical branching nonsymmetric Markov processes. Ann. Probab. (2015, to appear). Available at: arXiv:1404.0116
  36. 36.
    Zhang, M.: Functional central limit theorem for the super-Brownian motion with super-Brownian immigration. J. Theor. Probab. 18, 665–685 (2005) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.LMAM School of Mathematical Sciences & Center for Statistical SciencePeking UniversityBeijingP.R. China
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA
  3. 3.School of Mathematical SciencesCapital Normal UniversityBeijingP.R. China

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