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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A short proof of the martingale convergence theorem
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by Charles W. Lamb PDF
Proc. Amer. Math. Soc. 38 (1973), 215-217 Request permission


The martingale convergence theorem is first proved for uniformly integrable martingales by a standard application of Doob’s maximal inequality. A simple truncation argument is then given which reduces the proof of the ${L^1}$-bounded martingale theorem to the uniformly integrable case. A similar method is used to prove Burkholder’s martingale transform convergence theorem.
  • D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504. MR 208647, DOI 10.1214/aoms/1177699141
  • J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
  • Adriano M. Garsia, Topics in almost everywhere convergence, Lectures in Advanced Mathematics, No. 4, Markham Publishing Co., Chicago, Ill., 1970. MR 0261253
  • Luis Báez-Duarte, On the convergence of martingale transforms, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19 (1971), 319–322. MR 300328, DOI 10.1007/BF00535836
  • Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288
  • Murali Rao, Doob’s decomposition and Burkholder’s inequalities, Séminaire de Probabilités, VI (Univ. Strasbourg, année universitaire 1970–1971), Lecture Notes in Math., Vol. 258, Springer, Berlin, 1972, pp. 198–201. MR 0375457
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 38 (1973), 215-217
  • MSC: Primary 60G45
  • DOI:
  • MathSciNet review: 0324770