×

An almost sure invariance principle for additive functionals of Markov chains. (English) Zbl 1139.60317

Summary: We prove an invariance principle for a vector-valued additive functional of a Markov chain for almost every starting point with respect to an ergodic equilibrium distribution. The hypothesis is a moment bound on the resolvent.

MSC:

60F17 Functional limit theorems; invariance principles
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Billingsley, P., Convergence of Probability Measures (1999), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York · Zbl 0172.21201
[2] Derriennic, Y.; Lin, M., Fractional Poisson equations and ergodic theorems for fractional coboundaries, Israel J. Math., 123, 93-130 (2001) · Zbl 0988.47009
[3] 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
[4] 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
[5] Maxwell, M.; Woodroofe, M., Central limit theorems for additive functionals of Markov chains, Ann. Probab., 28, 2, 713-724 (2000) · Zbl 1044.60014
[6] Rassoul-Agha, F.; Seppäläinen, T., An almost sure invariance principle for random walks in a space-time random environment, Probab. Theory Related Fields, 133, 3, 299-314 (2005) · Zbl 1088.60094
[7] Rassoul-Agha, F.; Seppäläinen, T., Ballistic random walk in a random environment with a forbidden direction, ALEA Lat. Am. J. Probab. Math. Stat., 1, 111-147 (2006), (Electronic) · Zbl 1115.60106
[8] Rassoul-Agha, F.; Seppäläinen, T., Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction, Ann. Probab., 35, 1, 1-31 (2007) · Zbl 1126.60090
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.