Advertisement

Journal of Theoretical Probability

, Volume 8, Issue 3, pp 549–570 | Cite as

Uniform CLT for Markov chains and its invariance principle: A martingale approach

  • Jongsig Bae
  • Shlomo Levental
Article

Abstract

The convergence of stochastic processes indexed by parameters which are elements of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for stationary Markov chains. We assume the integrability condition on metric entropy with bracketing. An eventual uniform equicontinuity result is developed which essentially gives the invariance principle of the UCLT. We translate the problem into that of a martingale difference sequence as in Gordin and Lifsic.(7) Then we use the chaining argument with stratification adapted from that of Ossiander.(11) The results of this paper generalize those of Levental(10) and Ossiander.(11)

Key Words

Markov chains invariance principle uniform central limit theorem martingales 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andersen, N. T., and Dobric, V. (1987). The central limit theorem for stochastic processes.Ann. Prob. 15, 164–177.Google Scholar
  2. 2.
    Doukhan, P., Massart, P., and Rio, E. (1994). Invariance principles for absolutely regular empirical processes. Preprint.Google Scholar
  3. 3.
    Dudley, R. M. (1984).A Course on Empirical Processes. Lecture notes in Math., Vol. 1097, Springer-Verlag, New York.Google Scholar
  4. 4.
    Dudley, R. M., and Philipp, W. (1983). Invariance principles for sums of Banach space valued random elements and empirical processes.Z. Wahrsch. verw. Gebiete 62, 509–552.Google Scholar
  5. 5.
    Freedman, D. (1975). On tail probabilities for martingales.Ann. Prob. 3, 100–118.Google Scholar
  6. 6.
    Gordin, M. I. (1969). The central limit theorem for stationary processes.Sov. Math. Dokl. 10, 1174–1176.Google Scholar
  7. 7.
    Gordin, M. I., and Lifsic, B. A. (1978). The central limit theorem for stationary Markov Processes.Sov. Math. Dokl. 19, 392–394.Google Scholar
  8. 8.
    Hoffmann-Jørgensen, J. (1991). Stochastic processes on Polish spaces. Aarhus Universitet. Matematisk Institut.Google Scholar
  9. 9.
    Ibragimov, I. A., and Linnik, J. V. (1971).Independent and Stationary Sequences of Random Variables, “Nauka”, Moscow, 1965; English transl. Noordhoff. Groningen, 1971.Google Scholar
  10. 10.
    Levental, S. (1989). A uniform CLT for uniformly bounded families of martingale differences.J. Theoret. Prob. 2, 271–287.Google Scholar
  11. 11.
    Ossiander, M. (1987). A central limit theorem under metric entropy withL 2 bracketing.Ann. Prob. 15, 897–919.Google Scholar
  12. 12.
    Pollard, D. (1984).Convergence of Stochastic Processes. Springer series in Statistics. Springer-Verlag, New York.Google Scholar
  13. 13.
    Yosida, K. (1965).Functional Analysis. Springer-Verlag, New York.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Jongsig Bae
    • 1
  • Shlomo Levental
    • 1
  1. 1.Department of Statistics and ProbabilityMichigan State UniversityEast Lansing

Personalised recommendations