Sums of independent random variables and the Burkholder transforms
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- by J.-P. Gabriel PDF
- Proc. Amer. Math. Soc. 66 (1977), 123-127 Request permission
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
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- Kai Lai Chung, A course in probability theory, Harcourt, Brace & World, Inc., New York, 1968. MR 0229268
- J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
- P. Warwick Millar, Martingales with independent increments, Ann. Math. Statist. 40 (1969), 1033–1041. MR 243605, DOI 10.1214/aoms/1177697607
- E. R. van Kampen, Infinite product measures and infinite convolutions, Amer. J. Math. 62 (1940), 417–448. MR 1282, DOI 10.2307/2371464 -P. Gabriel, Loi des grands nombres, séries et martingales indexées par un ensemble filtrant, Thèse de doctorat, EPF-Lausanne, Septembre 1975.
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 66 (1977), 123-127
- MSC: Primary 60G45; Secondary 60G50
- DOI: https://doi.org/10.1090/S0002-9939-1977-0451392-6
- MathSciNet review: 0451392