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Martingales in the limit and amarts


Authors: G. A. Edgar and L. Sucheston
Journal: Proc. Amer. Math. Soc. 67 (1977), 315-320
MSC: Primary 60G45
DOI: https://doi.org/10.1090/S0002-9939-1977-0488272-6
MathSciNet review: 0488272
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Abstract: The notion of amart is compared to that of a martingale in the limit and game fairer with time. Every real-valued amart is a martingale in the limit. More generally, a Banach space E is finite-dimensional if and only if every E-valued amart is a martingale in the limit (or a game fairer with time). Several crucial properties possessed by amarts fail both for martingales in the limit and games fairer with time: the maximal inequality, the optional stopping theorem, the optional sampling theorem, the Riesz decomposition; therefore a general theory analogous to the amart theory cannot be based on the notion of a martingale in the limit. It is also observed that either the optional sampling theorem or a weak form of the Riesz decomposition must fail for any class of sequences of random variables strictly larger than the class of amarts.


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  • [1] D. G. Austin, G. A. Edgar and A. Ionescu Tulcea, Point-wise convergence in terms of expectations, Z. Wahrscheinhchkeitstheorie und Verw. Gebiete 30 (1974), 17-26. MR 0358945 (50:11402)
  • [2] A. Bellow, On vector-valued asymptotic martingales, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 1798-1799. MR 0407966 (53:11733)
  • [3] L. H. Blake, A generalization of martingales and two subsequent convergence theorems, Pacific J. Math. 35 (1970), 279-283. MR 0275505 (43:1259)
  • [4] -, Further results concerning games which become fairer with time, J. London Math. Soc. (2) 6 (1973), 311-316. MR 0312565 (47:1122)
  • [5] D. L. Burkholder, Successive conditional expectations of an integrable function, Ann. Math. Statist. 33 (1962), 887-893. MR 0143246 (26:805)
  • [6] R. V. Chacon, A ``stopped proof of convergence", Advances in Math. 14 (1974), 365-368. MR 0365688 (51:1940)
  • [7] R. V. Chacon and L. Sucheston, On convergence of vector-valued asymptotic martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975), 55-59. MR 0394859 (52:15658)
  • [8] G. A. Edgar and L. Sucheston, Amarts: a class of asymptotic martingales. A. Discrete parameter. B. Continuous parameter, J. Multivariate Analysis 6 (1976), 193-221; 572-591. MR 0413251 (54:1368)
  • [9] -, The Riesz decomposition for vector-valued amarts, Z. Wahrscheinlichkeitstheorie und verw. Gebiete 36 (1976), 85-92. MR 0413252 (54:1369)
  • [10] -, On vector-valued amarts and dimension of Banach spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 39 (1977), 213-216. MR 0448537 (56:6843)
  • [11] U. Krengel and L. Sucheston, On amarts, semiamarts, and processes with finite values, Advances in Prob. (to appear). MR 618785 (82i:60079)
  • [12] B. J. McCabe and L. A. Shepp, On the supremum of $ {S_n}/n$, Ann. Math. Statist. 41 (1970), 2166-2168. MR 0267627 (42:2529)
  • [13] A. G. Mucci, Limits for martingale-like sequences, Pacific J. Math. 48 (1973), 197-202. MR 0358969 (50:11425b)
  • [14] -, Another martingale convergence theorem, Pacific J. Math. 64 (1976), 539-541. MR 0420840 (54:8852)
  • [15] J. Neveu, Discrete parameter martingales, North-Holland, Amsterdam, 1975. MR 0402915 (53:6729)
  • [16] R. Subramanian, On a generalization of martingales due to Blake, Pacific J. Math. 48 (1973), 275-278. MR 0358968 (50:11425a)
  • [17] L. Tzafriri, On Banach spaces with unconditional bases, Israel J. Math. 17 (1974), 84-93. MR 0348456 (50:954)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0488272-6
Keywords: Martingale in the limit, amart, maximal inequality, optional sampling theorem, game fairer with time
Article copyright: © Copyright 1977 American Mathematical Society

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