Summary
The error bound O(1/√n) is derived in the central limit theorem for partial sums \(\sum\limits_{j = 1}^n {f(\xi _j )} \) where ξj is a recurrent discrete Markov chain and f is a real valued function on the state space. In particular it is shown that for bounded f and starting distribution dominated by some multiple of the stationary one, it is sufficient for the chain to have recurrence times with third moments on order to get this bound.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Battacharga, R.N., Rao, R.R.: Normal approximation and asymptotic expansions. New York: Wiley 1976
Bolthausen, E.: On rates of convergence in a random central limit theorem and in the central limit theorem for Markov chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 38, 279–286 (1977)
Chung, K.L.: Markov chains with stationary transition probabilities. Berlin, Heidelberg, New York: Springer 1967
Feller, W.: An introduction to probability theory and its applications, Vol. II. New York: Wiley 1966
Landers, D., Rogge, L.: The exact approximation order in the central limit theorem for random summation. Z. Wahrscheinlichkeitstheorie verw. Gebiete 36, 169–183 (1976)
Landers, D., Rogge, L.: On the rate of convergence in the central limit theorem for Markov chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 35, 169–183 (1976)
Lifshits, B.A.: On the central limit theorem for Markov chains. Theory Probab. Appl. 23, 279–295 (1978)
Pitman, J.W.: Uniform rates of convergence for Markov chain transition probabilities. Z. Wahrscheinlichkeitstheorie verw. Gebiete 29, 193–227 (1974)
Pitman, J.W.: Occupation measures for Markov chains. Adv. in Appl. Probab. 9, 69–86 (1977)
Rosenblatt, M.: Markov processes. Structure and asymptotic behaviour. Berlin, Heidelberg, New York: Springer 1971
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bolthausen, E. The Berry-Esseen theorem for functionals of discrete Markov chains. Z. Wahrscheinlichkeitstheorie verw Gebiete 54, 59–73 (1980). https://doi.org/10.1007/BF00535354
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00535354