Central limit theorems for additive functionals of Markov chains. (English) Zbl 1044.60014

Summary: 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 \(E(S_n\mid X_1)\). 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 \(g\).


60F05 Central limit and other weak theorems
60J55 Local time and additive functionals
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J05 Discrete-time Markov processes on general state spaces
60F17 Functional limit theorems; invariance principles
Full Text: DOI


[1] 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.] · Zbl 0652.60028
[2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[3] 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. · Zbl 0605.60027
[4] R. Durrett, and Resnick, S. (1978). Functional limit theorems for dependent variables. Ann. Probab. 6 829-846. · Zbl 0398.60024
[5] Gordin, M. I. and Lifsic, B. A. (1978). Central limit theorems for stationary Markov processes. Dokl.Akad.Nauk.SSSR 239 766-767. · Zbl 0395.60057
[6] Hall, P. and Heyde, C. (1981). Martingale Limit Theory and Its Applications. Academic Press, New York. · Zbl 0462.60045
[7] 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. · Zbl 0588.60058
[8] Maxwell, M. (1997). Local and global central limit theorems for stationary ergodic sequences. Ph.D. dissertation, Univ. Michigan.
[9] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York. · Zbl 0925.60001
[10] 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. Birkhäuser, Boston. · Zbl 0603.60022
[11] Toth, B. (1986). Persistent random walks in random environment. Probab.Theory Related Fields 71 615-625. · Zbl 0589.60099
[12] Woodroofe, M. (1992). A central limit theorem for functions of a Markov chain with applications to shifts. Stochastic.Process.Appl. 41 31-42. · Zbl 0762.60023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.