Advertisement

Journal of Theoretical Probability

, Volume 19, Issue 3, pp 647–689 | Cite as

On the Weak Invariance Principle for Stationary Sequences under Projective Criteria

  • Florence Merlevède
  • Magda Peligrad
Article

In this paper, we study the central limit theorem and its weak invariance principle for sums of a stationary sequence of random variables, via a martingale decomposition. Our conditions involve the conditional expectation of sums of random variables with respect to the distant past. The results contribute to the clarification of the central limit question for stationary sequences.

Keywords

Central limit theorem weak invariance principle projective criteria strong mixing sequences martingale approximation 

Mathematics Subject Classifications 1991

60 F 05 60 F 17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aldous D.J., Eagleson G.K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6, 325–331MathSciNetGoogle Scholar
  2. 2.
    Ango Nzé, P., and Doukhan, P. (2002). Weak dependence: models and applications. In Dehling, H., Mikosch, T., and Sorensen M. (eds.), Empirical process techniques for dependent data, Birkhauser.Google Scholar
  3. 3.
    Billingsley P. (1999). Convergence of Probability Measures, Second Edition. Wiley, New YorkGoogle Scholar
  4. 4.
    Bingham, N. H., Goldie, C. M., and Teugels J. L. (1987). Regular Variation, Cambridge University Press.Google Scholar
  5. 5.
    Bradley R.C. (1997). On quantiles and the central limit question for strongly mixing sequences. J. Theor. Probab.10, 507–555CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bradley, R. C. (2002). Introduction to Strong Mixing Conditions, Volume 1. Technical Report, Department of Mathematics, Indiana University, Bloomington. Custom Publishing of I.U., Bloomington, March 2002.Google Scholar
  7. 7.
    Bradley, R. C. (2003). Introduction to Strong Mixing Conditions, Volume 2. Technical Report, Department of Mathematics, Indiana University, Bloomington. Custom Publishing of I.U., Bloomington, March 2003.Google Scholar
  8. 8.
    Dedecker J., Rio E. (2000). On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Stat. 36: 1–34CrossRefMathSciNetGoogle Scholar
  9. 9.
    Dedecker J., Merlevède F. (2002). Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30, 1044–1081CrossRefMathSciNetGoogle Scholar
  10. 10.
    Dedecker J., Doukhan P. (2003). A new covariance inequality and applications. Stoch. Process Appl. 106, 63–80CrossRefMathSciNetGoogle Scholar
  11. 11.
    Dehling, H., and Philipp, W. (2002). Empirical process techniques for dependent data. In Empirical process techniques for dependent data, Birkhäuser Boston, Boston, MA, 3–113.Google Scholar
  12. 12.
    Dehling H., Denker M., Philipp W. (1986). Central limit theorems for mixing sequences of random variables under minimal conditions. Ann. Probab, 14, 1359–1370MathSciNetGoogle Scholar
  13. 13.
    Denker, M. (1986). Uniform integrability and the central limit theorem for strongly mixing processes. In Eberlain, E., and Taqqu, M. S. (eds.), Dependence in Probability and Statistics. A Survey of Recent Results, Oberwolfach, 1985, Birkhäuser.Google Scholar
  14. 14.
    Gordin M.I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188, 739–741MathSciNetGoogle Scholar
  15. 15.
    Hall, P., and Heyde, C. C. (1980). Martingale Limit Theory and its Application. Academic Press.Google Scholar
  16. 16.
    Herrndorf N. (1983). The invariance principle for φ-mixing sequences. Z. Wahrsch. verw. Gebiete, 63, 97–108MathSciNetGoogle Scholar
  17. 17.
    Ibragimov I.A. (1962). Some limit theorems for stationary processes. Teor. Verojatnost. i Primenen. 7, 361–392MathSciNetGoogle Scholar
  18. 18.
    Ibragimov I.A. (1975). A note on the central limit theorem for dependent random variables. Theory Probab. Appl., 20, 135–141CrossRefGoogle Scholar
  19. 19.
    Ibragimov, I. A., and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Variables, Wolters-Noordholf, Groningen.Google Scholar
  20. 20.
    Maxwell M., Woodroofe M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28, 713–724CrossRefMathSciNetGoogle Scholar
  21. 21.
    McLeish D.L. (1975a). Invariance principles for dependent variables. Z. Wahrsch. verw. Gebiete, 32, 165–178CrossRefMathSciNetGoogle Scholar
  22. 22.
    McLeish D.L. (1975b). A maximal inequality and dependent strong laws. Ann. Probab. 3, 829–839MathSciNetGoogle Scholar
  23. 23.
    Merlevède F. (2003). On the central limit theorem and its weak invariance principle for strongly mixing sequences with values in a Hilbert space via martingale approximation. J. Theoret. Probab. 16, 625–653CrossRefMathSciNetGoogle Scholar
  24. 24.
    Merlevède F., Peligrad M. (2000). The functional central limit theorem for strong mixing sequences of random variables. Ann. Probab. 28(3): 1336–1352CrossRefMathSciNetGoogle Scholar
  25. 25.
    Móricz F.A. (1976). Moment inequalities and the strong laws of large numbers. Z. Wahrscheinlichkeitstheoreie verw. Gebiete. 35(4): 299–314CrossRefGoogle Scholar
  26. 26.
    Peligrad M. (1982). Invariance principles for mixing sequences of random variables. Ann. Probab. 10, 968–981MathSciNetGoogle Scholar
  27. 27.
    Peligrad M. (1986). Recent advances in the central limit theorem and its weak invariance principles for mixing sequences of random variables. In: Eberlein E., Taqqu M.S. (eds). Dependence in Probability and Statistics. A Survey of Recent Results. Birkhäuser, BostonGoogle Scholar
  28. 28.
    Peligrad M. (1990). On Ibragimov–Iosifescu conjecture for φ-mixing sequences. Stoch. Process. Appl. 35, 293–308CrossRefMathSciNetGoogle Scholar
  29. 29.
    Peligrad M., Utev S. (2005). A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33(2): 798–815CrossRefMathSciNetGoogle Scholar
  30. 30.
    Philipp W. (1986). Invariance principles for independent and weakly dependent random variables. In: Eberlein E., Taqqu M.S. (eds). Dependence in Probability and Statistics. A Survey of Recent Results. Birkhäuser, BostonGoogle Scholar
  31. 31.
    Rényi A. (1963). On stable sequences of events. Sankhya Ser. A 25, 293–302MathSciNetGoogle Scholar
  32. 32.
    Rio, E. (2000). Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques et applications de la SMAI. 31 Springer.Google Scholar
  33. 33.
    Rosenblatt M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. 42, 43–47CrossRefMathSciNetGoogle Scholar
  34. 34.
    Shao Q.M. (1993). Almost sure invariance principles for mixing sequences of random variables. Stoch. Process. Appl. 48, 319–334.Google Scholar
  35. 35.
    Stout W. (1974). Almost Sure Convergence. Academic Press, New York, San Francisco, LondonMATHGoogle Scholar
  36. 36.
    Wu, W.B., Woodroofe M. (2004). Martingale approximations for sums of stationary processes. Ann. Probab. 32(2): 1674–1690CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Laboratoire de Probabilités et Modèles AléatoiresUniversité Paris VI, et C.N.R.S UMR 7599ParisFrance
  2. 2.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

Personalised recommendations