The central limit theorem for additive functionals of Markov processes and the weak convergence to Wiener measure

Masuyuki HITSUDA; Akinobu SHIMIZU

doi:10.2969/jmsj/02240551

Masuyuki HITSUDA, Akinobu SHIMIZU

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1970 Volume 22 Issue 4 Pages 551-566

DOI https://doi.org/10.2969/jmsj/02240551

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Published: 1970 Received: December 15, 1969 Available on J-STAGE: September 29, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: September 29, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) Yu. A. Davydov, I. A. Ibragimov, M. I. Gordin and V. N. Solev, Stationary processes. Limit theorems. Regularity conditions, Proceedings of USSR-Japan Symposium on Probability, Khabarovsk, August (1969), 72-99.
2) E. B. Dynkin, Markov processes, Vol. I, Springer-Verlag, 1965.
3) M. Fukushima and M. Hitsuda, On a class of Markov processes taking values on lines and the central limit theorem, Nagoya Math. J., 30 (1967), 47-56.
4) E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Collq. Publ., 31, 1957.
5) J. Keilson and D. M. G. Wishart, A central limit theorem for processes defined on a finite Markov chain, Proc. Cambridge Philos. Soc., 60 (1964), 547-567.
6) L. D. Meshalkin, Limit theorem for Markov chain with a finite state, Theor. Probability Appl., 6 (1961), 257-275.
7) S. V. Nagaev, Some limit theorems for stationary Markov chains, Theor. Probability Appl., 2 (1957), 379-406.
8) Yu. V. Prokhorov, Convergence of stochastic processes and limit theorem of the probability, Theor. Probability Appl., 1 (1956), 177-238.
9) F. Riesz and B. Sz-Nagy, Functional analysis, Frederick Unger Publ. Co., 1965.
10) V. N. Tutubalin, On limit theorems for the product of random matrices, Theor. Probability Appl., 10 (1965), 15-27.
11) I. S. Volkov, On the distribution of sums of random variables defined on a homogeneous Markov chain with finite number of states, Theor. Probability Appl., 3 (1958), 413-429.

Right : [1] Yu. A. Davydov, I. A. Ibragimov, M. I. Gordin and V. N. Solev, Stationary processes. Limit theorems. Regularity conditions, Proceedings of USSR-Japan Symposium on Probability, Khabarovsk, August (1969), 72-99.
[2] E. B. Dynkin, Markov processes, Vol. I, Springer-Verlag, 1965.
[3] M. Fukushima and M. Hitsuda, On a class of Markov processes taking values on lines and the central limit theorem, Nagoya Math. J., 30 (1967), 47-56.
[4] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Collq. Publ., 31, 1957.
[5] J. Keilson and D. M. G. Wishart, A central limit theorem for processes defined on a finite Markov chain, Proc. Cambridge Philos. Soc., 60 (1964), 547-567.
[6] L. D. Meshalkin, Limit theorem for Markov chain with a finite state, Theor. Probability Appl., 6 (1961), 257-275.
[7] S. V. Nagaev, Some limit theorems for stationary Markov chains, Theor. Probability Appl., 2 (1957), 379-406.
[8] Yu. V. Prokhorov, Convergence of stochastic processes and limit theorem of the probability, Theor. Probability Appl., 1 (1956), 177-238.
[9]F. Riesz and B. Sz-Nagy, Functional analysis, Frederick Unger Publ. Co., 1965.
[10] V. N. Tutubalin, On limit theorems for the product of random matrices, Theor. Probability Appl., 10 (1965), 15-27.
[11] I. S. Volkov, On the distribution of sums of random variables defined on a homogeneous Markov chain with finite number of states, Theor. Probability Appl., 3 (1958), 413-429.

Date of correction: September 29, 2006 Reason for correction: - Correction: PDF FILE Details: -

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