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On the global limit behaviour of Markov chains and of general nonsingular Markov processes. (English) Zbl 0218.60060


MSC:

60J27 Continuous-time Markov processes on discrete state spaces
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J25 Continuous-time Markov processes on general state spaces
60F99 Limit theorems in probability theory
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[1] Blackwell, D.; Freedman, D., The tail σ-field of a Markov chain and a theorem of Orey, Ann. math. Statistics, 35, 1291-1295 (1964) · Zbl 0127.35204
[2] Chacon, R. V.: Convergence of operator averages. Symposium on Ergodic Theory, Tulane Univ., New Orleans (1961), ed. Wright (1963). Corrected proofs in [14].
[3] Chacon, R. V.; Ornstein, D. S., A general ergodic theorem, Illinois J. Math., 4, 153-160 (1960) · Zbl 0134.12102
[4] Chung, K. L., Markov chains with stationary transition probabilities (1960), Berlin-Göttingen-Heidelberg: Springer, Berlin-Göttingen-Heidelberg · Zbl 0092.34304
[5] Derman, C., A solution to a set of fundamental equations in Markov chains, Proc. Amer. math. Soc., 5, 332-334 (1954) · Zbl 0058.34504
[6] Doeblin, W., Sur deux problèmes de M. Kolmogoroff concernant les chaÎnes dénombrables, Bull. Soc. math. France, 66, 210-220 (1938) · JFM 64.0538.02
[7] Dowker, Y. N., On measurable transformations in finite measure spaces, Ann. of Math., II. Ser., 62, 504-516 (1955) · Zbl 0065.28701
[8] Dowker, Y. N.; Erdös, P., Some examples in ergodic theory, Proc. London math. Soc., III. Ser., 9, 227-241 (1959) · Zbl 0084.34102
[9] Dunford, N.; Schwartz, J. T., Convergence almost everywhere of operator averages, J. Math. Mech., 5, 129-178 (1956) · Zbl 0075.12102
[10] Feldman, J., Subinvariant measures for Markov operators, Duke math. J., 29, 71-98 (1962) · Zbl 0106.33401
[11] Harris, T. E.; Robbins, H. E., Ergodic theory of Markov chains admitting an infinite invariant measure, Proc. nat. Acad. Sci. U. S. A., 39, 860-864 (1953) · Zbl 0051.10503
[12] Hopf, E., The general temporally discrete Markov process, J. Math. Mech., 3, 13-43 (1954) · Zbl 0055.36705
[13] Ionescu-Tulcea, C., Mesures dans les espaces produits, Atti Accad. naz. Lincei, Rend., Cl. Sci. fis. mat. nat. VIII. Ser., 7, 208-211 (1949) · Zbl 0035.15203
[14] Jacobs, K.: Lecture notes on ergodic theory. Universitet Aarhus, Matematisk Institut, 1962/63. · Zbl 0196.31301
[15] Jain, N. C., Some limit theorems for a general Markov process, Z. Wahrscheinlichkeits-theorie verw. Geb., 6, 206-223 (1966) · Zbl 0234.60086
[16] Krengel, U.: Classification of states for operators. Proc. Fifth Berkeley Sympos. math. Statist. Probability, 1966. · Zbl 0236.60051
[17] Moy, S. T. C., λ-continuous Markov chains, Trans. Amer. math. Soc., 117, 68-91 (1965) · Zbl 0137.11901
[18] -λ-continuous Markov chains II. (to appear). · Zbl 0219.60050
[19] Neveu, J., Bases mathématiques du calcul des probabilitées (1964), Paris: Masson, Paris · Zbl 0137.11203
[20] Neveu, J., Sur l’existence de mesures invariantes en théorie ergodique, C. r. Acad. Sci. Paris, 260, 393-396 (1965) · Zbl 0127.09301
[21] Orey, S., An ergodic theorem for Markov chains, Z. Wahrscheinlichkeitstheorie verw. Geb., 1, 174-176 (1962) · Zbl 0109.36302
[22] Chung, K. L.; Erdös, P., Probability limit theorems assuming only the first moment, Mem. Amer. math. Soc., 6, 1-19 (1951) · Zbl 0042.37601
[23] Kingman, J. F. C.; Orey, S., Ratio limit theorems for Markov chains, Proc. Amer. math. Soc., 15, 907-910 (1964) · Zbl 0131.16701
[24] Krickeberg, K.: Strong mixing properties of Markov chains with infinite invariant measure. Proc. Fifth Berkeley Sympos. math. Statist. Probability (to appear 1966). · Zbl 0211.48503
[25] Orey, S., Strong ratio limit property, Bull. Amer. math. Soc., 67, 571-574 (1961) · Zbl 0109.11801
[26] Pruitt, W. E., Strong ratio limit property for ℛ-recurrent Markov chains, Proc. Amer. math. Soe., 16, 196-200 (1965) · Zbl 0131.16603
[27] Chacon, R. V.; Krengel, U., Linear modulus of a linear operator, Proc. Amer. math. Soc., 15, 553-559 (1964) · Zbl 0168.11701
[28] Chacon, R. V., A class of linear transformations, Proc. Amer. math. Soc., 15, 560-564 (1964) · Zbl 0168.11702
[29] Ionescu-Tulcea, A., On the category of certain classes of transformations in ergodic theory, Trans. Amer. math. Soc., 114, 261-279 (1965) · Zbl 0178.38501
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