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Proceedings of the American Mathematical Society

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

 
 

 

The best constant in Burkholder’s weak-$L^{1}$ inequality for the martingale square function


Author: David C. Cox
Journal: Proc. Amer. Math. Soc. 85 (1982), 427-433
MSC: Primary 60G42; Secondary 42B25
DOI: https://doi.org/10.1090/S0002-9939-1982-0656117-9
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 \[ P\left ( {\sum \limits _{i \geqslant 1} {X_i^2 \geqslant {\lambda ^2}} } \right ) \leqslant {\lambda ^{ - 1}}C\sup \limits _{i \geqslant 1} E\left | {{Y_i}} \right |,\] for all $y > 0$, and show that the best constant is $C = {e^{1/2}}$.


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Article copyright: © Copyright 1982 American Mathematical Society