Journal of Theoretical Probability

, Volume 26, Issue 2, pp 489–513 | Cite as

A Nonconventional Invariance Principle for Random Fields

Article
First Online:
Received:
Revised:
  • 135 Downloads

Abstract

In Kifer and Varadhan (Ann Probab, to appear), we obtained a nonconventional invariance principle (functional central limit theorem) for sufficiently fast mixing stochastic processes with discrete and continuous time. In this article, we derive a nonconventional invariance principle for sufficiently well-mixing random fields.

Keywords

Random fields Limit theorems Mixing 

Mathematics Subject Classification (2000)

60F17 60G60 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Alexander, K.S.: Mixing properties and exponential decay for lattice systems in finite volumes. Ann. Probab. 32, 441–487 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Austin, T.: On the norm convergence of non-conventional ergodic averages. Ergod. Theory Dyn. Syst. 30, 321–338 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bergelson, V.: Weakly mixing PET. Ergod. Theory Dyn. Syst. 7, 337–349 (1987)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bogachev, V.I.: Gaussian Measures. American Mathematical Society, Providence (1998)MATHGoogle Scholar
  5. 5.
    Bolthausen, E.: On the central limit theorem for stationary mixing random fields. Ann. Probab. 10, 1047–1050 (1982)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bradley, R.C.: Introduction to Strong Mixing Conditions. Kendrick Press, Heber City (2007)Google Scholar
  7. 7.
    Bulinski, A., Shashkin, A.: Strong invariance principle for dependent random fields. IMS Lecture Notes. Dyn. Stoch. 48, 128–143 (2006)Google Scholar
  8. 8.
    Bulinski, A., Shashkin, A.: Limit Theorems for Associated Random Fields and Related Systems. World Scientific, Singapore (2007)Google Scholar
  9. 9.
    Bickel, P.J., Wichura, M.J.: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42, 1656–1670 (1971)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    de Jong, R.M.: Limit theorems for dependent heterogeneous random variables. Econ Theory 13, 353–367 (1997)CrossRefGoogle Scholar
  11. 11.
    Dobruschin, R.L.: The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13, 197–224 (1968)CrossRefGoogle Scholar
  12. 12.
    Doukhan, P.: Mixing: Properties and Examples. Lecture Notes in Mathematics, vol. 85. Springer, New York (1994).Google Scholar
  13. 13.
    Dunford, N., Schwartz, J.T.: Linear Operators Part I. Wiley, New York (1958)Google Scholar
  14. 14.
    Furstenberg, H.: Nonconventional ergodic averages. Proc. Symp. Pure Math. 50, 43–56 (1990)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Garling, D.J.H.: Inequalities: A Journey into Linear Analysis. Cambridge University Press, Cambridge (2007)MATHCrossRefGoogle Scholar
  16. 16.
    Georgii, H.-O.: Gibbs Measures and Phase Transitions. De Gruyter, Berlin (1988)MATHCrossRefGoogle Scholar
  17. 17.
    Kifer, Yu.: Nonconventional limit theorems. Probab. Theory Rel. Fields 148, 71–106 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kifer, Yu., Varadhan, S.R.S.: Nonconventional limit theorems in discrete and continuous time via martingales. Ann. Probab. (accepted)Google Scholar
  19. 19.
    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge (1990)Google Scholar
  20. 20.
    Leibman, A.: Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. 146, 303–315 (2005)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    McLeish, D.L.: Invariance principles for dependent variables. Z. Wahrsch. Verw. Geb. 32, 165–178 (1975)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    McLeish, D.L.: On the invariance principle for nonstationary mixingales. Ann. Probab. 5, 616–621 (1977)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Sunklodas, J.: On normal approximation for strongly mixing random fields. Theory Probab. Appl. 52, 125–132 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations