Skip to main content
SpringerLink
Log in

On the Weak Invariance Principle for Stationary Sequences under Projective Criteria

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aldous D.J., Eagleson G.K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6, 325–331

    MathSciNet  Google Scholar 

  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.

  3. Billingsley P. (1999). Convergence of Probability Measures, Second Edition. Wiley, New York

    Google Scholar 

  4. Bingham, N. H., Goldie, C. M., and Teugels J. L. (1987). Regular Variation, Cambridge University Press.

  5. Bradley R.C. (1997). On quantiles and the central limit question for strongly mixing sequences. J. Theor. Probab.10, 507–555

    Article  MathSciNet  Google Scholar 

  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.

  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.

  8. Dedecker J., Rio E. (2000). On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Stat. 36: 1–34

    Article  MathSciNet  Google Scholar 

  9. Dedecker J., Merlevède F. (2002). Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30, 1044–1081

    Article  MathSciNet  Google Scholar 

  10. Dedecker J., Doukhan P. (2003). A new covariance inequality and applications. Stoch. Process Appl. 106, 63–80

    Article  MathSciNet  Google Scholar 

  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.

  12. Dehling H., Denker M., Philipp W. (1986). Central limit theorems for mixing sequences of random variables under minimal conditions. Ann. Probab, 14, 1359–1370

    MathSciNet  Google Scholar 

  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.

  14. Gordin M.I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188, 739–741

    MathSciNet  Google Scholar 

  15. Hall, P., and Heyde, C. C. (1980). Martingale Limit Theory and its Application. Academic Press.

  16. Herrndorf N. (1983). The invariance principle for φ-mixing sequences. Z. Wahrsch. verw. Gebiete, 63, 97–108

    MathSciNet  Google Scholar 

  17. Ibragimov I.A. (1962). Some limit theorems for stationary processes. Teor. Verojatnost. i Primenen. 7, 361–392

    MathSciNet  Google Scholar 

  18. Ibragimov I.A. (1975). A note on the central limit theorem for dependent random variables. Theory Probab. Appl., 20, 135–141

    Article  Google Scholar 

  19. Ibragimov, I. A., and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Variables, Wolters-Noordholf, Groningen.

  20. Maxwell M., Woodroofe M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28, 713–724

    Article  MathSciNet  Google Scholar 

  21. McLeish D.L. (1975a). Invariance principles for dependent variables. Z. Wahrsch. verw. Gebiete, 32, 165–178

    Article  MathSciNet  Google Scholar 

  22. McLeish D.L. (1975b). A maximal inequality and dependent strong laws. Ann. Probab. 3, 829–839

    MathSciNet  Google Scholar 

  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, 625–653

    Article  MathSciNet  Google Scholar 

  24. Merlevède F., Peligrad M. (2000). The functional central limit theorem for strong mixing sequences of random variables. Ann. Probab. 28(3): 1336–1352

    Article  MathSciNet  Google Scholar 

  25. Móricz F.A. (1976). Moment inequalities and the strong laws of large numbers. Z. Wahrscheinlichkeitstheoreie verw. Gebiete. 35(4): 299–314

    Article  Google Scholar 

  26. Peligrad M. (1982). Invariance principles for mixing sequences of random variables. Ann. Probab. 10, 968–981

    MathSciNet  Google Scholar 

  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, Boston

    Google Scholar 

  28. Peligrad M. (1990). On Ibragimov–Iosifescu conjecture for φ-mixing sequences. Stoch. Process. Appl. 35, 293–308

    Article  MathSciNet  Google Scholar 

  29. Peligrad M., Utev S. (2005). A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33(2): 798–815

    Article  MathSciNet  Google Scholar 

  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, Boston

    Google Scholar 

  31. Rényi A. (1963). On stable sequences of events. Sankhya Ser. A 25, 293–302

    MathSciNet  Google Scholar 

  32. Rio, E. (2000). Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques et applications de la SMAI. 31 Springer.

  33. Rosenblatt M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. 42, 43–47

    Article  MathSciNet  Google Scholar 

  34. Shao Q.M. (1993). Almost sure invariance principles for mixing sequences of random variables. Stoch. Process. Appl. 48, 319–334.

    Google Scholar 

  35. Stout W. (1974). Almost Sure Convergence. Academic Press, New York, San Francisco, London

    MATH  Google Scholar 

  36. Wu, W.B., Woodroofe M. (2004). Martingale approximations for sums of stationary processes. Ann. Probab. 32(2): 1674–1690

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, et C.N.R.S UMR 7599, Boîte 188, 175 rue du Chevaleret, 75 013, Paris, France

    Florence Merlevède

  2. Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH, 45221-0025, USA

    Magda Peligrad

Authors
  1. Florence Merlevède
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Magda Peligrad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Florence Merlevède.

Additional information

Magda Peligrad is supported in part by a Charles Phelps Taft research support grant at the Univeristy of Cincinnati and the NSA grant H98230-05-1-0066.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Merlevède, F., Peligrad, M. On the Weak Invariance Principle for Stationary Sequences under Projective Criteria. J Theor Probab 19, 647–689 (2006). https://doi.org/10.1007/s10959-006-0029-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-006-0029-y

Keywords

Mathematics Subject Classifications 1991

Search

Navigation