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A divergent, two-parameter, bounded martingale


Authors: Lester E. Dubins and Jim Pitman
Journal: Proc. Amer. Math. Soc. 78 (1980), 414-416
MSC: Primary 60G42; Secondary 60G44
DOI: https://doi.org/10.1090/S0002-9939-1980-0553386-9
MathSciNet review: 553386
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Abstract | References | Similar Articles | Additional Information

Abstract: An example is given of a divergent, uniformly bounded martingale $ X = \{ {X_t}:t \in T\} $ where the index t ranges over the set T of pairs of positive integers with the usual coordinatewise ordering.


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  • [1] R. Cairoli, Une inégalité pour martingales à indices multiples et ses applications, Séminaire de Probabilités IV, Lecture Notes in Math., vol. 124, Springer-Verlag, Berlin, 1970, pp. 1-28. MR 0270424 (42:5312)
  • [2] R. Cairoli and J. B. Walsh, Stochastic integrals in the plane, Acta Math. 134 (1975), 111-183. MR 0420845 (54:8857)
  • [3] J. Dieudonné, Sur un théorème de Jessen, Fund. Math. 37 (1950), 242-248. MR 0043176 (13:218d)
  • [4] L. E. Dubins and J. W. Pitman, A pointwise ergodic theorem for the group of rational rotations, Trans. Amer. Math. Soc. 251 (1979), 299-308. MR 531981 (80i:60062)
  • [5] A. Gut, Convergence of reversed martingales with multidimensional indices, Duke Math. J. 43 (1976), 269-275. MR 0407969 (53:11736)
  • [6] K. Krickeberg, Convergence of martingales with a directed index set, Trans. Amer. Math. Soc. 83 (1976), 313-337. MR 0091328 (19:947e)
  • [7] J. Neveu, Discrete parameter martingales, North-Holland, Amsterdam; American Elsevier, New York, 1975. MR 0402915 (53:6729)
  • [8] R. T. Smythe, Sums of independent random variables on partially ordered sets, Ann. Probability 2 (1974), 906-917. MR 0358973 (50:11429)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0553386-9
Keywords: Multi-parameter martingales
Article copyright: © Copyright 1980 American Mathematical Society

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