Abstract.
Let X0,X1,... be a geometrically ergodic Markov chain with state space and stationary distribution π. It is known that if h:→ R satisfies π(|h|2+ɛ)<∞ for some ɛ>0, then the normalized sums of the X i ’s obey a central limit theorem. Here we show, by means of a counterexample, that the condition π(|h|2+ɛ)<∞ cannot be weakened to only assuming a finite second moment, i.e., π(h2)<∞.
Article PDF
Similar content being viewed by others
References
Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation. Oxford University Press, 1992
Cogburn, R.: The central limit theorem for Markov processes. In: Le Cam, L, Neyman, J. Scott, E., (eds.), Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol II. 1972, pp. 485–512
Häggström, O.: A note on disagreement percolation. Random Struct. Algo. 18, 267–278 (2001)
Ibragimov, I.A., Linnik, Y.V.: Independent and Stationary Sequences of RandomVariables. Wolters-Noordhoff, Groningen, 1971
Roberts, G.O., Rosenthal, J.S.: Geometric ergodicity and hybrid Markov chains, Electr. Comm. Probab. 2, 13–25 (1997)
Roberts, G.O., Rosenthal, J.S.: General state space Markov chains and MCMCalgorithms, preprint, 2004, http://xxx.lanl.gov/abs/math.PR/0404033
Rosenthal, J.S.: Faithful coupling of Markov chains: now equals forever. Adv. Appl. Math. 18, 372–381 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Reasearch supported by the Swedish Research Council.
Rights and permissions
About this article
Cite this article
Häggström, O. On the central limit theorem for geometrically ergodic Markov chains. Probab. Theory Relat. Fields 132, 74–82 (2005). https://doi.org/10.1007/s00440-004-0390-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0390-7