Finitely additive Markov chains

Ramakrishnan, S.

doi:10.1090/S0002-9947-1981-0607119-3

Finitely additive Markov chains
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Trans. Amer. Math. Soc. 265 (1981), 247-272 Request permission

Abstract:

In this paper we develop the theory of Markov chains with stationary transition probabilities, where the transition probabilities and the initial distribution are assumed only to be finitely additive. We prove a strong law of large numbers for recurrent chains. The problem of existence and uniqueness of finitely additive stationary initial distributions is studied and the ergodicity of recurrent chains under a stationary initial distribution is proved.

References

Leo Breiman, Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
Robert Chen, On almost sure convergence in a finitely additive setting, Z. Wahrsch. Verw. Gebiete 37 (1976/77), no. 4, 341–356. MR 571674, DOI 10.1007/BF00533425
Robert Chen, Some finitely additive versions of the strong law of large numbers, Israel J. Math. 24 (1976), no. 3-4, 244–259. MR 418203, DOI 10.1007/BF02834755
Kai Lai Chung, A course in probability theory, 2nd ed., Probability and Mathematical Statistics, Vol. 21, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0346858
Kai Lai Chung, Markov chains with stationary transition probabilities, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 104, Springer-Verlag New York, Inc., New York, 1967. MR 0217872
C. Derman, A solution to a set of fundamental equations in Markov chains, Proc. Amer. Math. Soc. 5 (1954), 332–334. MR 60757, DOI 10.1090/S0002-9939-1954-0060757-0
Lester E. Dubins, On Lebesgue-like extensions of finitely additive measures, Ann. Probability 2 (1974), 456–463. MR 357724, DOI 10.1214/aop/1176996660
Lester E. Dubins and Leonard J. Savage, How to gamble if you must. Inequalities for stochastic processes, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. MR 0236983

Linear operators

David Freedman, Markov chains, Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1971. MR 0292176
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-Los Angeles, Calif., 1956, pp. 113–124. MR 0084889
T. E. Harris, Transient Markov chains with stationary measures, Proc. Amer. Math. Soc. 8 (1957), 937–942. MR 91564, DOI 10.1090/S0002-9939-1957-0091564-3
Roger A. Purves and William D. Sudderth, Some finitely additive probability, Ann. Probability 4 (1976), no. 2, 259–276. MR 402888, DOI 10.1214/aop/1176996133

Some finitely additive probability

S. Ramakrishnan, Finitely additive Markov chains, Trans. Amer. Math. Soc. 265 (1981), no. 1, 247–272. MR 607119, DOI 10.1090/S0002-9947-1981-0607119-3