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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

The best constant in Burkholder's weak-$ L\sp{1}$ inequality for the martingale square function


Author: David C. Cox
Journal: Proc. Amer. Math. Soc. 85 (1982), 427-433
MSC: Primary 60G42; Secondary 42B25
MathSciNet review: 656117
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Abstract: Let $ {Y_1},{Y_2}, \ldots $ be a martingale with difference sequence $ {X_1} = {Y_1},{X_i} = {Y_i} - {Y_{i - 1}},i \geqslant 2$. We give a new proof of the inequality

$\displaystyle P\left( {\sum\limits_{i \geqslant 1} {X_i^2 \geqslant {\lambda ^2... ...{ - 1}}C\mathop {\sup }\limits_{i \geqslant 1} E\left\vert {{Y_i}} \right\vert,$

for all $ y > 0$, and show that the best constant is $ C = {e^{1/2}}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0656117-9
PII: S 0002-9939(1982)0656117-9
Article copyright: © Copyright 1982 American Mathematical Society