Martingale approximations for continuous-time and discrete-time stationary Markov processes. (English) Zbl 1073.60050

Summary: We show that the method of C. Kipnis and S. R. S. Varadhan [Commun. Math. Phys. 104, 1–19 (1986; Zbl 0588.60058)] to construct a martingale approximation to an additive functional of a stationary ergodic Markov process via the resolvent is universal in the sense that a martingale approximation exists if and only if the resolvent representation converges. A sufficient condition for the existence of a martingale approximation is also given. As examples we discuss moving average processes and processes with normal generator.


60G44 Martingales with continuous parameter
60F05 Central limit and other weak theorems
60J25 Continuous-time Markov processes on general state spaces
60J35 Transition functions, generators and resolvents


Zbl 0588.60058
Full Text: DOI


[1] Bhattacharya, R.N., On the functional central limit theorem and the law of the iterated logarithm for Markov processes, Z. wahrsch. verw. gebiete, 60, 2, 185-201, (1982) · Zbl 0468.60034
[2] Billingsley, P., The lindeberg-Lévy theorem for martingales, Proc. amer. math. soc., 12, 788-792, (1961) · Zbl 0129.10701
[3] Birman, M.Sh.; Solomjak, M.Z., Spectral theory of selfadjoint operators in Hilbert space, (1987), D. Reidel Publishing Co. Dordrecht, (Translated from the 1980 Russian original)
[4] Bloom, W.R.; Heyer, H., Harmonic analysis of probability measures on hypergroups, (1995), Walter de Gruyter & Co. Berlin · Zbl 0828.43005
[5] A.N. Borodin, I.A. Ibragimov, Limit theorems for functionals of random walks, Trud. Mat. Inst. Steklov, vol. 195, 1994. (in Russian, English transl. in: Proc. Steklov Inst. Math., 1995). · Zbl 0840.60002
[6] Chikin, D.O., A functional limit theorem for stationary processes: a martingale approach, Theory probab. appl., 34, 4, 668-678, (1989) · Zbl 0703.60031
[7] Cramér, H.; Leadbetter, M.R., Stationary and related processes, (1967), Wiley New York · Zbl 0162.21102
[8] de Masi, A.; Ferrari, P.; Goldstein, S.; Wick, W., An invariance principle for reversible Markov processes, J. statist. phys., 55, 3-4, 787-855, (1989) · Zbl 0713.60041
[9] Derriennic, Y.; Lin, M., Sur le théorème limite central de kipnis et varadhan pour LES chaıˆnes réversibles ou normales, C. R. acad. sci. Paris Sér. I math., 323, 1053-1057, (1996) · Zbl 0869.60019
[10] Derriennic, Y.; Lin, M., The central limit theorem for Markov chains with normal transition operators, started at a point, Probab. theory related fields, 119, 4, 508-528, (2001) · Zbl 0974.60017
[11] Derriennic, Y.; Lin, M., The central limit theorem for Markov chains started at a point, Probab. theory related fields, 125, 1, 73-76, (2003) · Zbl 1012.60028
[12] Gordin, M.I.; Holzmann, H., The central limit theorem for stationary Markov chains under invariant splittings, Stoch. dyn., 4, 1, 15-30, (2004) · Zbl 1077.60023
[13] Gordin, M.I.; Lifšic, B.A., Central limit theorem for stationary Markov processes, Dokl. akad. nauk SSSR, 239, 4, 766-767, (1978)
[14] M.I. Gordin, B.A. Lifšic, A remark about a Markov process with normal transition operator, in: Third Vilnius Conference on Probability and Statistics, vol. 1, Vilnius, 1981, pp. 147-148.
[15] H. Holzmann, The central limit theorem for stationary Markov processes with normal generator and applications to random walks on hypergroups, Working Paper, Göttingen University, 2004.
[16] Ibragimov, I.A., A central limit theorem for a class of dependent random variables, Theory probab. appl., 8, 83-89, (1963) · Zbl 0123.36103
[17] Kipnis, C.; Varadhan, S.R.S., Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm. math. phys., 104, 1, 1-19, (1986) · Zbl 0588.60058
[18] Maxwell, M.; Woodroofe, M., Central limit theorems for additive functionals of Markov chains, Ann. probab., 28, 2, 713-724, (2000) · Zbl 1044.60014
[19] Olla, S., Central limit theorems for tagged particles and for diffusions in random environments, Panor. syntheses, 12, 75-100, (2001) · Zbl 1119.60302
[20] Woodroofe, M., A central limit theorem for functions of a Markov chain with applications to shifts, Stochastic process. appl., 41, 1, 33-44, (1992) · Zbl 0762.60023
[21] Wu, W.B.; Woodroofe, M., Martingale approximations for stationary processes, Ann. probab., 32, 2, 1674-1690, (2004) · Zbl 1057.60022
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