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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The best constant in Burkholder’s weak-$L^{1}$ inequality for the martingale square function
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by David C. Cox PDF
Proc. Amer. Math. Soc. 85 (1982), 427-433 Request permission

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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • 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