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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.

MSC:

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

Citations:

Zbl 0588.60058
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References:

[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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