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The Best Constant in the Davis Inequality for the Expectation of the Martingale Square Function

  • Burgess DavisEmail author
  • Renming Song
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Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

A method is introduced for the simultaneous study of the square function and the maximal function of a martingale that can yield sharp norm inequalities between the two. One application is that the expectation of the square function of a martingale is not greater than \(\sqrt {3}\) times the expectation of the maximal function. This gives the best constant for one side of the Davis two-sided inequality. The martingale may take its values in any real or complex Hilbert space. The elementary discrete-time case leads quickly to the analogous results for local martingales M indexed by [0, ∞). Some earlier inequalities are also improved and, closely related, the Lévy martingale is embedded in a large family of submartingales.

Keywords

Maximal Function Complex Hilbert Space Good Constant Local Martingale Pointwise Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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