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Transactions of the American Mathematical Society

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A new approach to the limit theory of recurrent Markov chains

Authors: K. B. Athreya and P. Ney
Journal: Trans. Amer. Math. Soc. 245 (1978), 493-501
MSC: Primary 60J10; Secondary 60K05
MathSciNet review: 511425
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Abstract: Let $ \{ {X_n};\,n \geqslant 0\} $ be a Harris-recurrent Markov chain on a general state space. It is shown that there is a sequence of random times $ \{ {N_i};\,i \geqslant 1\} $ such that $ \{ {X_{{N_i}}};{\text{ }}i \geqslant 1\} $ are independent and identically distributed. This idea is used to show that $ \{ {X_n}\} $ is equivalent to a process having a recurrence point, and to develop a regenerative scheme which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.

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  • [1] K. B. Athreya, D. McDonald, and P. Ney, Limit theorems for semi-Markov processes and renewal theory for Markov chains, Ann. Probab. 6 (1978), no. 5, 788–797. MR 503952
  • [2] W. Doblin, Éléments d’une théorie générale des chaînes simples constantes de Markoff, Ann. École Norm. (3) 57 (1940), 61–111 (French). MR 0004409
  • [3] J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR 0058896
  • [4] William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
  • [5] David Griffeath, Coupling methods for Markov processes, Studies in probability and ergodic theory, Adv. in Math. Suppl. Stud., vol. 2, Academic Press, New York-London, 1978, pp. 1–43. MR 517252
  • [6] T. E. Harris, The existence of stationary measures for certain Markov processes, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II, University of California Press, Berkeley and Los Angeles, 1956, pp. 113–124. MR 0084889
  • [7] Jacques Neveu, Mathematical foundations of the calculus of probability, Translated by Amiel Feinstein, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1965. MR 0198505
  • [8] Steven Orey, Lecture notes on limit theorems for Markov chain transition probabilities, Van Nostrand Reinhold Co., London-New York-Toronto, Ont., 1971. Van Nostrand Reinhold Mathematical Studies, No. 34. MR 0324774
  • [9] D. Revuz, Markov chains, 2nd ed., North-Holland Mathematical Library, vol. 11, North-Holland Publishing Co., Amsterdam, 1984. MR 758799
  • [10] Charles Stone, On moment generating functions and renewal theory, Ann. Math. Statist. 36 (1965), 1298–1301. MR 0179857,

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Keywords: Markov chains, regeneration, ergodic theorem, invariant measure
Article copyright: © Copyright 1978 American Mathematical Society

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