The Annals of Probability

Central limit theorems for additive functionals of Markov chains

Michael Maxwell and Michael Woodroofe

Full-text: Open access

Abstract

Central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic Markov chain, say $S_n = g(X_1)+ \cdots + g(X_n)$ where $E[g(X_1)]= 0$ and $E[g(X_1)^2]<\infty$. The conditions imposed restrict the moments of $g$ and the growth of the conditional means $E(S_n|X_1)$. No other restrictions on the dependence structure of the chain are required. When specialized to shift processes,the conditions are implied by simple integral tests involving $g$.

Article information

Source
Ann. Probab. Volume 28, Number 2 (2000), 713-724.

Dates
First available in Project Euclid: 18 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1019160258

Mathematical Reviews number (MathSciNet)
MR1782272

Digital Object Identifier
doi:10.1214/aop/1019160258

Zentralblatt MATH identifier
01906373

Subjects
Primary: 60F05: Central limit and other weak theorems

Keywords
Asymptotic normality ergodic theorem functional central limit theorem Hilbert space martingale maximal inequality one-sided shifts Poisson’s equation

Citation

Maxwell, Michael; Woodroofe, Michael. Central limit theorems for additive functionals of Markov chains. The Annals of Probability 28 (2000), no. 2, 713--724. doi:10.1214/aop/1019160258. http://projecteuclid.org/euclid.aop/1019160258.


Export citation

References

  • Bhattacharrya, R. and Lee, O. (1988). Asymptotics for a class of Markov processes that are not in general irreducible. Ann.Probab. 16 1333-1347. [Correction (1997) Ann.Probab. 25 1541-1543.]
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Dehling, H., Denker, M. and Phillipp, W. (1986). Central limit theorems for mixing sequences of random variables under minimal conditions. Ann.Probab. 14 1359-1370.
  • R. Durrett, and Resnick, S. (1978). Functional limit theorems for dependent variables. Ann. Probab. 6 829-846.
  • Gordin, M. I. and Lifsic, B. A. (1978). Central limit theorems for stationary Markov processes. Dokl.Akad.Nauk.SSSR 239 766-767.
  • Hall, P. and Heyde, C. (1981). Martingale Limit Theory and Its Applications. Academic Press, New York.
  • Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorems for additive functionals of reversible Markov processes and applications to simple excursions. Comm.Math.Phys. 104 1-19.
  • Maxwell, M. (1997). Local and global central limit theorems for stationary ergodic sequences. Ph.D. dissertation, Univ. Michigan.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.
  • Peligrad, M. (1986). Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables. In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 193-224. Birkh¨auser, Boston.
  • Toth, B. (1986). Persistent random walks in random environment. Probab.Theory Related Fields 71 615-625.
  • Woodroofe, M. (1992). A central limit theorem for functions of a Markov chain with applications to shifts. Stochastic.Process.Appl. 41 31-42.