A finitely additive generalization of Birkhoff's ergodic theorem
Author:
S. Ramakrishnan
Journal:
Proc. Amer. Math. Soc. 96 (1986), 299305
MSC:
Primary 28D05; Secondary 60F15
MathSciNet review:
818462
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Abstract 
References 
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Additional Information
Abstract: A finitely additive generalization of Birkhoff s ergodic theorem is obtained which yields, in particular, strong laws of large numbers in the i.i.d. setting as well as for positive recurrent Markov chains.
 [1]
Robert
Chen, A finitely additive version of Kolmogorov’s law of the
iterated logarithm, Israel J. Math. 23 (1976),
no. 34, 209–220. MR 0407947
(53 #11714)
 [2]
Robert
Chen, Some finitely additive versions of the strong law of large
numbers, Israel J. Math. 24 (1976), no. 34,
244–259. MR 0418203
(54 #6244)
 [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]
Lester
E. Dubins, On Lebesguelike extensions of finitely additive
measures, Ann. Probability 2 (1974), 456–463.
MR
0357724 (50 #10192)
 [5]
N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
 [6]
Teturo
Kamae, A simple proof of the ergodic theorem using nonstandard
analysis, Israel J. Math. 42 (1982), no. 4,
284–290. MR
682311 (84i:28019), http://dx.doi.org/10.1007/BF02761408
 [7]
Yitzhak
Katznelson and Benjamin
Weiss, A simple proof of some ergodic theorems, Israel J.
Math. 42 (1982), no. 4, 291–296. MR 682312
(84i:28020), http://dx.doi.org/10.1007/BF02761409
 [8]
Donald
Ornstein and Benjamin
Weiss, The ShannonMcMillanBreiman theorem for a class of amenable
groups, Israel J. Math. 44 (1983), no. 1,
53–60. MR
693654 (85f:28018), http://dx.doi.org/10.1007/BF02763171
 [9]
R. A. Purves and W. D. Sudderth, Some finitely additive probability, Univ. of Minnesota School of Statistics Tech. Report No. 220, 1973.
 [10]
Roger
A. Purves and William
D. Sudderth, Some finitely additive probability, Ann.
Probability 4 (1976), no. 2, 259–276. MR 0402888
(53 #6702)
 [11]
Roger
A. Purves and William
D. Sudderth, Finitely additive zeroone laws, Sankhyā
Ser. A 45 (1983), no. 1, 32–37. MR 749351
(85j:60057)
 [12]
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
 [13]
S.
Ramakrishnan, Potential theory for finitely additive Markov
chains, Stochastic Process. Appl. 16 (1984),
no. 3, 287–303. MR 723850
(86c:60110), http://dx.doi.org/10.1016/03044149(84)900267
 [14]
S.
Ramakrishnan, Central limit theorems in a finitely additive
setting, Illinois J. Math. 28 (1984), no. 1,
139–161. MR
730717 (85j:60004)
 [15]
S.
Ramakrishnan, The tail 𝜎field of a finitely
additive Markov chain starting from a recurrent state, Proc. Amer. Math. Soc. 89 (1983), no. 3, 493–497. MR 715873
(85a:60039), http://dx.doi.org/10.1090/S00029939198307158732
 [16]
P. C. Shields, A simple direct proof of Birkhoff's ergodic theorem, 1982 (unpublished).
 [1]
 R. Chen, A finitely additive version of Kolmogorov's law of iterated logarithm, Israel J. Math. 23 (1976), 209220. MR 0407947 (53:11714)
 [2]
 , Some finitely additive versions of the strong law of large numbers, Israel J. Math. 24 (1976), 244259. MR 0418203 (54:6244)
 [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]
 L. E. Dubins, On Lebesguelike extensions of finitely additive measures, Ann. Probab. 2 (1974), 456463. MR 0357724 (50:10192)
 [5]
 N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
 [6]
 T. Kamae, A simple proof of the ergodic theorem using nonstandard analysis, Israel J. Math. 42 (1982), 284290. MR 682311 (84i:28019)
 [7]
 Y. Katznelson and B. Weiss, A simple proof of some ergodic theorems, Israel J. Math. 42 (1982), 291296. MR 682312 (84i:28020)
 [8]
 D. Ornstein and B. Weiss, The ShannonMcMillanBreiman theorem for a class of amenable groups, Israel J. Math. 44 (1983), 5360. MR 693654 (85f:28018)
 [9]
 R. A. Purves and W. D. Sudderth, Some finitely additive probability, Univ. of Minnesota School of Statistics Tech. Report No. 220, 1973.
 [10]
 , Some finitely additive probability, Ann. Probab. 4 (1976), 259276. MR 0402888 (53:6702)
 [11]
 , Finitely additive zeroone laws, Sankhya 45A (1983), 3237. MR 749351 (85j:60057)
 [12]
 S. Ramakrishnan, Finitely additive Markov chains, Trans. Amer. Math. Soc. 265 (1981), 247272. MR 607119 (82i:60121)
 [13]
 , Potential theory for finitely additive Markov chains, Stochastic Process. Appl. 16 (1984), 287303. MR 723850 (86c:60110)
 [14]
 , Central limits theorems in a finitely additive setting, Illinois J. Math. 28 (1984), 139161. MR 730717 (85j:60004)
 [15]
 , The tail field of a finitely additive Markov chain starting from a recurrent state, Proc. Amer. Math. Soc. 89 (1983), 493497. MR 715873 (85a:60039)
 [16]
 P. C. Shields, A simple direct proof of Birkhoff's ergodic theorem, 1982 (unpublished).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198608184627
PII:
S 00029939(1986)08184627
Keywords:
Ergodic theorem,
finitely additive probability,
i.i.d. measure,
Markov chain
Article copyright:
© Copyright 1986
American Mathematical Society
