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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  -bounded condition of Burkholder is the best one in the class of martingales, which assures the a.e. convergence of their transforms.
- [D]
D.
L. Burkholder, Martingale transforms, Ann. Math. Statist.
37 (1966), 1494–1504. MR 0208647
(34 #8456)
- [K]
Kai
Lai Chung, A course in probability theory, Harcourt, Brace
& World, Inc., New York, 1968. MR 0229268
(37 #4842)
- [J]
J.
L. Doob, Stochastic processes, John Wiley & Sons Inc., New
York, 1953. MR
0058896 (15,445b)
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P.
Warwick Millar, Martingales with independent increments, Ann.
Math. Statist. 40 (1969), 1033–1041. MR 0243605
(39 #4926)
- [E]
E.
R. van Kampen, Infinite product measures and infinite
convolutions, Amer. J. Math. 62 (1940),
417–448. 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.
- [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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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1977-0451392-6
PII:
S 0002-9939(1977)0451392-6
Keywords:
Independent random variables,
unconditionally a.e. convergence,
Burkholder transform,
martingales
Article copyright:
© Copyright 1977 American Mathematical Society
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