The 1971 Wald Memorial Lectures

Distribution Function Inequalities for Martingales
  • Burgess Davis
  • Renming Song
Part of the Selected Works in Probability and Statistics book series (SWPS)


Let uf and Vf be nonnegative random variables associated with a martingale ƒ. In many interesting cases, the inequality
$$P\left( {Vf >\lambda } \right) \leqq cP\left( {Uf >\lambda } \right),$$
which usually does not hold for all λ > 0, does hold for enough λ so that
$$EVf \leqq cEUf$$
and more. The underlying theory, introduced in [6], has also proved fruitful in other probability applications; see [5] and [8]. For an entirely nonprobabilistic application to harmonic functions, see [7].


Maximal Function Sample Function Nonnegative Random Variable Nonnegative Measurable Function Walsh Series 
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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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