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Sums of independent random variables and the Burkholder transforms

Author: J.-P. Gabriel
Journal: Proc. Amer. Math. Soc. 66 (1977), 123-127
MSC: Primary 60G45; Secondary 60G50
MathSciNet review: 0451392
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Abstract: This note shows a connection between the unconditionally a.e. convergence of series with independent increments and the a.e. convergence of their Burkholder transforms. Using this result, it is then proved that the $ {L_1}$-bounded condition of Burkholder is the best one in the class of martingales, which assures the a.e. convergence of their transforms.

References [Enhancements On Off] (What's this?)

  • [D] L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494-1504. MR 0208647 (34:8456)
  • [K] L. Chung, A course in probability theory, Harcourt, Brace and World, New York, 1968. MR 0229268 (37:4842)
  • [J] L. Doob, Stochastic processes, Wiley, New York, 1953. MR 0058896 (15:445b)
  • [P] W. Millar, Martingales with independent increments, Ann. Math. Statist. 40 (1969), 1033-1041. MR 0243605 (39:4926)
  • [E] R. van Kampen, Infinite product measure and infinite convolutions, Amer. J. Math. 62 (1940). MR 0001282 (1:209d)
  • [J] -P. Gabriel, Loi des grands nombres, séries et martingales indexées par un ensemble filtrant, Thèse de doctorat, EPF-Lausanne, Septembre 1975.

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Keywords: Independent random variables, unconditionally a.e. convergence, Burkholder transform, martingales
Article copyright: © Copyright 1977 American Mathematical Society

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