A finitely additive generalization of Birkhoff’s ergodic theorem
HTML articles powered by AMS MathViewer
- by S. Ramakrishnan PDF
- Proc. Amer. Math. Soc. 96 (1986), 299-305 Request permission
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.References
- Robert Chen, A finitely additive version of Kolmogorov’s law of the iterated logarithm, Israel J. Math. 23 (1976), no. 3-4, 209–220. MR 407947, DOI 10.1007/BF02761801
- 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
- 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
- Lester E. Dubins, On Lebesgue-like extensions of finitely additive measures, Ann. Probability 2 (1974), 456–463. MR 357724, DOI 10.1214/aop/1176996660 N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
- Teturo Kamae, A simple proof of the ergodic theorem using nonstandard analysis, Israel J. Math. 42 (1982), no. 4, 284–290. MR 682311, DOI 10.1007/BF02761408
- Yitzhak Katznelson and Benjamin Weiss, A simple proof of some ergodic theorems, Israel J. Math. 42 (1982), no. 4, 291–296. MR 682312, DOI 10.1007/BF02761409
- Donald Ornstein and Benjamin Weiss, The Shannon-McMillan-Breiman theorem for a class of amenable groups, Israel J. Math. 44 (1983), no. 1, 53–60. MR 693654, DOI 10.1007/BF02763171 R. A. Purves and W. D. Sudderth, Some finitely additive probability, Univ. of Minnesota School of Statistics Tech. Report No. 220, 1973.
- 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
- Roger A. Purves and William D. Sudderth, Finitely additive zero-one laws, Sankhyā Ser. A 45 (1983), no. 1, 32–37. MR 749351
- 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
- S. Ramakrishnan, Potential theory for finitely additive Markov chains, Stochastic Process. Appl. 16 (1984), no. 3, 287–303. MR 723850, DOI 10.1016/0304-4149(84)90026-7
- S. Ramakrishnan, Central limit theorems in a finitely additive setting, Illinois J. Math. 28 (1984), no. 1, 139–161. MR 730717, DOI 10.1215/ijm/1256046159
- S. Ramakrishnan, The tail $\sigma$-field of a finitely additive Markov chain starting from a recurrent state, Proc. Amer. Math. Soc. 89 (1983), no. 3, 493–497. MR 715873, DOI 10.1090/S0002-9939-1983-0715873-2 P. C. Shields, A simple direct proof of Birkhoff’s ergodic theorem, 1982 (unpublished).
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 96 (1986), 299-305
- MSC: Primary 28D05; Secondary 60F15
- DOI: https://doi.org/10.1090/S0002-9939-1986-0818462-7
- MathSciNet review: 818462