Annals of Probability

Martingale approximations for sums of stationary processes

Wei Biao Wu and Michael Woodroofe

Full-text: Open access


Approximations to sums of stationary and ergodic sequences by martingales are investigated. Necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained. It is first shown that a martingale approximation is necessary for such normality and then that the sums are asymptotically normal if and only if the approximating martingales satisfy a Lindeberg–Feller condition. Using the explicit construction of the approximating martingales, a central limit theorem is derived for the sample means of linear processes. The conditions are not sufficient for the functional version of the central limit theorem. This is shown by an example, and a slightly stronger sufficient condition is given.

Article information

Ann. Probab., Volume 32, Number 2 (2004), 1674-1690.

First available in Project Euclid: 18 May 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60G42: Martingales with discrete parameter 60F05: Central limit and other weak theorems
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Central limit theorem invariance principle linear process Markov chain martingale Poisson equation stationary process


Wu, Wei Biao; Woodroofe, Michael. Martingale approximations for sums of stationary processes. Ann. Probab. 32 (2004), no. 2, 1674--1690. doi:10.1214/009117904000000351.

Export citation


  • Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Billingsley, P. (1995). Probability and Measure. Wiley, New York.
  • Chow, Y. S. and Teicher, H. (1978). Probability Theory, 2nd ed. Springer, New York.
  • Dedecker, J. and Merlevede, P. (2002). Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30 1044–1081.
  • Doob, J. (1953). Stochastic Processes. Wiley, New York.
  • Doukhan, P. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84 313–342.
  • Feller, W. (1971). {An Introduction to Probability
  • Theory and its Applications}, Wiley, New York.
  • Gänssler, P. and Häeusler, E. (1979). Remarks on the functional central limit theorem for martingales. Z. Wahrsch. Verw. Gebiete 50 237–243.
  • Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 739–741.
  • Gordin, M. I. and Lifsic, B. (1978). The central limit theorem for stationary Markov processes. Soviet. Math. Dokl. 19 392–394.
  • Hall, P. G. and Heyde, C. C. (1980). Martingale Limit Theory and its Applications. Academic Press, New York.
  • Ho, H.-C. and Hsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Probab. 25 1636–1669.
  • Ibragimov, I. A. (1962). Some limit theorems for stationary processes. Theory Probab. Appl. 7 349–382.
  • Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters–Noordhoff, Groningen.
  • Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713–724.
  • Peligrad, M. (1986). Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables (A survey). In Dependence in Probability and Statistics: A Survey of Recent Results (E. Eberlein and M. S. Taqqu, eds.) 193–224. Birkhäuser, Boston.
  • Peligrad, M. (1996). On the asymptotic normality of sequences of weak dependent random variables. J. Theoret. Probab. 9 703–715.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.
  • Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701–1728.
  • Woodroofe, M. (1992). A central limit theorem for functions of a Markov chain with applications to shifts. Stochastic Process. Appl. 41 33–44.
  • Wu, W. B. (2002). Central limit theorems for functionals of linear processes and their applications. Statist. Sinica 12 635–649.\goodbreak
  • Wu, W. B. and Woodroofe, M. (2000). A central limit theorem for iterated random functions. J. Appl. Probab. 37 748–755.