The tail -field of a finitely additive Markov chain starting from a recurrent state

Author:
S. Ramakrishnan

Journal:
Proc. Amer. Math. Soc. **89** (1983), 493-497

MSC:
Primary 60F20; Secondary 60J10

MathSciNet review:
715873

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Abstract: For a Markov chain with an arbitrary nonempty state space, with stationary finitely additive transition probabilities and with initial distribution concentrated on a recurrent state, it is shown that the probability of every tail set is either zero or one. This generalizes and in particular gives an alternative proof of the result due to Blackwell and Freedman [**1**] in case the state space is countable and all transition probabilities are countably additive.

**[1]**David Blackwell and David Freedman,*The tail 𝜎-field of a Markov chain and a theorem of Orey*, Ann. Math. Statist.**35**(1964), 1291–1295. MR**0164375****[2]**Lester E. Dubins,*On Lebesgue-like extensions of finitely additive measures*, Ann. Probability**2**(1974), 456–463. MR**0357724****[3]**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****[4]**David Freedman,*Markov chains*, Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1971. MR**0292176****[5]**Jacques Neveu,*Mathematical foundations of the calculus of probability*, Translated by Amiel Feinstein, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1965. MR**0198505****[6]**R. A. Purves and W. D. Sudderth,*Some finitely additive probability*, Univ. of Minnesota School of Statistics Tech. Report no. 220, 1973.**[7]**Roger A. Purves and William D. Sudderth,*Some finitely additive probability*, Ann. Probability**4**(1976), no. 2, 259–276. MR**0402888****[8]**-,*Finitely additive zero one-laws*, Sankhya (to appear).**[9]**S. Ramakrishnan,*Finitely additive Markov chains*, Trans. Amer. Math. Soc.**265**(1981), no. 1, 247–272. MR**607119**, 10.1090/S0002-9947-1981-0607119-3

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DOI:
https://doi.org/10.1090/S0002-9939-1983-0715873-2

Keywords:
Tail -field,
finitely additive probability,
Markov chain,
recurrent state

Article copyright:
© Copyright 1983
American Mathematical Society