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Proceedings of the American Mathematical Society

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Martingale representation in uniformly convex spaces


Authors: D. Landers and L. Rogge
Journal: Proc. Amer. Math. Soc. 75 (1979), 108-110
MSC: Primary 60G48
MathSciNet review: 530761
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Abstract: In this paper we define the concept of a martingale in a uniformly convex Banach space and show that each bounded martingale is convergent and can be represented as a sequence of nearest point projections onto closed convex sets of one element of the Banach space. This result contains classical representation theorems for usual martingales and yields applications in the theory of nearest point projections in $ {L_p}$-spaces.


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  • [1] T. Andô and I. Amemiya, Almost everywhere convergence of prediction sequence in 𝐿_{𝑝}(1<𝑝<∞), Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 113–120 (1965). MR 0189077
  • [2] R. E. Barlow, D. J. Bartholomew, J. M. Bremner and H. D. Brunk, Statistical inference under order restrictions, Wiley, London, New York, 1970.
  • [3] H. D. Brunk, Conditional expectation given a 𝜎-lattice and applications, Ann. Math. Statist. 36 (1965), 1339–1350. MR 0185629
  • [4] H. D. Brunk, Uniform inequalities for conditional 𝑝-means given 𝜎-lattices, Ann. Probability 3 (1975), no. 6, 1025–1030. MR 0385945
  • [5] J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed; North-Holland Mathematical Library, Vol. 10. MR 0402915
  • [6] M. M. Rao, Inference in stochastic processes. III. Nonlinear prediction, filtering, and samlping theorems, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 49–72. MR 0216713

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0530761-1
Keywords: Martingale, uniformly convex Banach space, conditional expectation
Article copyright: © Copyright 1979 American Mathematical Society