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A Note on Stochastic Dominance and Compactness

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  • Nendel, Max

    (Center for Mathematical Economics, Bielefeld University)

Abstract

In this work, we discuss completeness for the lattice orders of first and second order stochastic dominance. The main results state that, both, first and second order stochastic dominance induce Dedekind super complete lattices, i.e. lattices in which every bounded nonempty subset has a countable subset with identical least upper bound and greatest lower bound. Moreover, we show that, if a suitably bounded set of probability measures is directed (e.g. a lattice), then the supremum and infimum w.r.t. first or second order stochastic dominance can be approximated by sequences in the weak topology or in the Wasserstein-1 topology, respectively. As a consequence, we are able to prove that a sublattice of probability measures is complete w.r.t. first order stochastic dominance or second order stochastic dominance and increasing convex order if and only if it is compact in the weak topology or in the Wasserstein-1 topology, respectively. This complements a set of characterizations of tightness and uniform integrability, which are discussed in a preliminary section.

Suggested Citation

  • Nendel, Max, 2019. "A Note on Stochastic Dominance and Compactness," Center for Mathematical Economics Working Papers 623, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:623
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    File URL: https://pub.uni-bielefeld.de/download/2937261/2937262
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    References listed on IDEAS

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    1. Dianetti, Jodi & Ferrari, Giorgio & Fischer, Markus & Nendel, Max, 2019. "Submodular Mean Field Games. Existence and Approximation of Solutions," Center for Mathematical Economics Working Papers 621, Center for Mathematical Economics, Bielefeld University.
    2. Hu, Tien-Chung & Rosalsky, Andrew, 2011. "A note on the de La Vallée Poussin criterion for uniform integrability," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 169-174, January.
    3. Leskelä, Lasse & Vihola, Matti, 2013. "Stochastic order characterization of uniform integrability and tightness," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 382-389.
    4. Chandra, Tapas Kumar, 2015. "de La Vallée Poussin’s theorem, uniform integrability, tightness and moments," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 136-141.
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    Cited by:

    1. Dianetti, Jodi & Ferrari, Giorgio & Fischer, Markus & Nendel, Max, 2019. "Submodular Mean Field Games. Existence and Approximation of Solutions," Center for Mathematical Economics Working Papers 621, Center for Mathematical Economics, Bielefeld University.

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