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The 1971 Wald Memorial Lectures

Distribution Function Inequalities for Martingales

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Selected Works of Donald L. Burkholder

Part of the book series: Selected Works in Probability and Statistics ((SWPS))

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Abstract

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].

Received July 6, 1972.

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Authors and Affiliations

  1. Department of Mathematics and Department of Statistics, Purdue University, West Lafayette, IN, 47907-2068, USA

    Burgess Davis

  2. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Renming Song

Authors
  1. Burgess Davis
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  2. Renming Song
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Correspondence to Burgess Davis .

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Editors and Affiliations

  1. , Department of Mathematics and, Purdue University, N. University Street 150, West Lafayette, 47907-2068, Indiana, USA

    Burgess Davis

  2. , Department of Mathematics, University of Illinois, Urbana, W. Green St. 1409, Urbana, 61801, Illinois, USA

    Renming Song

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Davis, B., Song, R. (2011). The 1971 Wald Memorial Lectures. In: Davis, B., Song, R. (eds) Selected Works of Donald L. Burkholder. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7245-3_15

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