A study concerning Blackwell's example in Markov chain theory

Hisao NOMOTO

doi:10.2969/jmsj/01720194

Hisao NOMOTO

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JOURNAL FREE ACCESS

1965 Volume 17 Issue 2 Pages 194-206

DOI https://doi.org/10.2969/jmsj/01720194

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Published: 1965 Received: September 14, 1964 Available on J-STAGE: September 26, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: September 26, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) D. Blackwell, Another countable Markov process with only instantaneous states, Ann. Math. Statist., 29 (1958), 313-316.
2) R. M. Blumenthal, R. K. Getoor, and H. P. McKean, Jr, Markov processes with identical hitting distributions, Illinois, J. Math., 6 (1962), 402-420.
3) N. Bourbaki, Topologie Générale, Hermann, 1940-1949.
4) K. L. Chung, Markov chains with stationary transition probabilities, Springer, 1960.
5) R. L. Dobrusin, An example of a countable homogeneous Markov process all states of which are instantaneous, Theor. Probability Appl., 1 (1956) 481-485.
6) W. Feller and H. P. McKean, A diffusion equivalent to a countable Markov chain, Proc. Nat. Acad. Sci. U. S. A., 42 (1956), 351-354.
7) I. V. Girsanov, Strongly Feller processes I, Theor. Probability Appl., 5 (1960), 7-28.
8) K. Ito, Stochastic processes, Tokyo, 1957. (Japanese)
9) R. Z. Khas'minskii, Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equation, Theor. Probability Appl., 5 (1960), 196-214.
10) H. Kunita and H. Nomoto, A method of compactification concerning Markov processes and its applications, Seminar on Prob., 14 (1962), 1-126. (Japanese)
11) P. Lévy, Systèmes markoviens et stationnaires. Cas dénombrable., Ann. Sci. École Norm. Sup., (3) 68 (1951), 327-381.
12) G. Maruyama and H. Tanaka, Ergodic properties of N-dimensional recurrent Markov processes, Mem. Fac. Sci. Kyushu Univ. Ser. A, 13 (1959), 157-172.
13) M. Nagasawa, The adjoint process of a diffusion with reflecting barrier, Kodai Math. Sem. Rep., 13 (1961), 235-248.
14) D. Ray, Resolvents, transition functions and strongly Markovian processes, Ann. of Math., 70 (1959), 43-72.
15) L. V. Seregin, Continuity conditions for stochastic processes, Theor. Probability Appl., 6 (1961), 3-30.
16) T. Ueno, On recurrent Markov processes, Kodai Math. Sem. Rep., 12 (1960), 109-142.
17) T. Watanabe, Some general properties of Markov processes, J. Inst. Polytech. Osaka City Univ. Ser. A, 9 (1958), 9-29.

Right : [1] D. Blackwell, Another countable Markov process with only instantaneous states, Ann. Math. Statist., 29 (1958), 313-316.
[2] R. M. Blumenthal, R. K. Getoor, and H. P. McKean, Jr, Markov processes with identical hitting distributions, Illinois, J. Math., 6 (1962), 402-420.
[3] N. Bourbaki, Topologie Générale, Hermann, 1940-1949.
[4] K. L. Chung, Markov chains with stationary transition probabilities, Springer, 1960.
[5] R. L. Dobrusin, An example of a countable homogeneous Markov process all states of which are instantaneous, Theor. Probability Appl., 1 (1956) 481-485.
[6] W. Feller and H. P. McKean, A diffusion equivalent to a countable Markov chain, Proc. Nat. Acad. Sci. U. S. A., 42 (1956), 351-354.
[7] I. V. Girsanov, Strongly Feller processes I, Theor. Probability Appl., 5 (1960), 7-28.
[8] K. Ito, Stochastic processes, Tokyo, 1957. (Japanese)
[9] R. Z. Khas'minskii, Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equation, Theor. Probability Appl., 5 (1960), 196-214.
[10] H. Kunita and H. Nomoto, A method of compactification concerning Markov processes and its applications, Seminar on Prob., 14 (1962), 1-126. (Japanese)
[11] P. Lévy, Systèmes markoviens et stationnaires. Cas dénombrable., Ann. Sci. École Norm. Sup., (3) 68 (1951), 327-381.
[12] G. Maruyama and H. Tanaka, Ergodic properties of N-dimensional recurrent Markov processes, Mem. Fac. Sci. Kyushu Univ. Ser. A, 13 (1959), 157-172.
[13] M. Nagasawa, The adjoint process of a diffusion with reflecting barrier, Kodai Math. Sem. Rep., 13 (1961), 235-248.
[14] D. Ray, Resolvents, transition functions and strongly Markovian processes, Ann. of Math., 70 (1959), 43-72.
[15] L. V. Seregin, Continuity conditions for stochastic processes, Theor. Probability Appl., 6 (1961), 3-30.
[16] T. Ueno, On recurrent Markov processes, Kodai Math. Sem. Rep., 12 (1960), 109-142.
[17] T. Watanabe, Some general properties of Markov processes, J. Inst. Polytech. Osaka City Univ. Ser. A, 9 (1958), 9-29.

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