The tail field of a finitely additive Markov chain starting from a recurrent state
Author:
S. Ramakrishnan
Journal:
Proc. Amer. Math. Soc. 89 (1983), 493497
MSC:
Primary 60F20; Secondary 60J10
MathSciNet review:
715873
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 𝜎field of a Markov chain and a theorem
of Orey, Ann. Math. Statist. 35 (1964),
1291–1295. MR 0164375
(29 #1672)
 [2]
Lester
E. Dubins, On Lebesguelike extensions of finitely additive
measures, Ann. Probability 2 (1974), 456–463.
MR
0357724 (50 #10192)
 [3]
Lester
E. Dubins and Leonard
J. Savage, How to gamble if you must. Inequalities for stochastic
processes, McGrawHill Book Co., New YorkTorontoLondonSydney, 1965.
MR
0236983 (38 #5276)
 [4]
David
Freedman, Markov chains, HoldenDay, San Francisco,
Calif.CambridgeAmsterdam, 1971. MR 0292176
(45 #1263)
 [5]
Jacques
Neveu, Mathematical foundations of the calculus of
probability, Translated by Amiel Feinstein, HoldenDay, Inc., San
Francisco, Calif.LondonAmsterdam, 1965. MR 0198505
(33 #6660)
 [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
(53 #6702)
 [8]
, Finitely additive zero onelaws, Sankhya (to appear).
 [9]
S.
Ramakrishnan, Finitely additive Markov
chains, Trans. Amer. Math. Soc.
265 (1981), no. 1,
247–272. MR
607119 (82i:60121), http://dx.doi.org/10.1090/S00029947198106071193
 [1]
 D. Blackwell and D. A. Freedman, The tail field of a Markov chain and a theorem of Orey, Ann. Math. Statist. 35 (1964), 12911295. MR 0164375 (29:1672)
 [2]
 L. E. Dubins, On Lebesguelike extensions of finitely additive measures, Ann. Probab. 2 (1974), 456463. MR 0357724 (50:10192)
 [3]
 L. E. Dubins and L. J. Savage, How to gamble if you must: inequalities for stochastic processes, McGrawHill, New York, 1965. MR 0236983 (38:5276)
 [4]
 D. A. Freedman, Markov chains, HoldenDay, San Francisco, Calif., 1971. MR 0292176 (45:1263)
 [5]
 J. Neveu, Mathematical foundations of the calculus of probability, HoldenDay, San Francisco, Calif., 1965. MR 0198505 (33:6660)
 [6]
 R. A. Purves and W. D. Sudderth, Some finitely additive probability, Univ. of Minnesota School of Statistics Tech. Report no. 220, 1973.
 [7]
 , Some finitely additive probability, Ann. Probab. 4 (1976), 259276. MR 0402888 (53:6702)
 [8]
 , Finitely additive zero onelaws, Sankhya (to appear).
 [9]
 S. Ramakrishnan, Finitely additive Markov chains, Trans. Amer. Math. Soc. 265 (1981), 247272. MR 607119 (82i:60121)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
60F20,
60J10
Retrieve articles in all journals
with MSC:
60F20,
60J10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198307158732
PII:
S 00029939(1983)07158732
Keywords:
Tail field,
finitely additive probability,
Markov chain,
recurrent state
Article copyright:
© Copyright 1983
American Mathematical Society
