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Stochastic order characterization of uniform integrability and tightness

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  • Leskelä, Lasse
  • Vihola, Matti

Abstract

We show that a family of random variables is uniformly integrable if and only if it is stochastically bounded in the increasing convex order by an integrable random variable. This result is complemented by proving analogous statements for the strong stochastic order and for power-integrable dominating random variables. In particular, we show that, whenever a family of random variables is stochastically bounded by a p-integrable random variable for some p>1, there is no distinction between the strong order and the increasing convex order. These results also yield new characterizations of relative compactness in Wasserstein and Prohorov metrics.

Suggested Citation

  • Leskelä, Lasse & Vihola, Matti, 2013. "Stochastic order characterization of uniform integrability and tightness," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 382-389.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:1:p:382-389
    DOI: 10.1016/j.spl.2012.09.023
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    References listed on IDEAS

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    1. Alison L. Gibbs & Francis Edward Su, 2002. "On Choosing and Bounding Probability Metrics," International Statistical Review, International Statistical Institute, vol. 70(3), pages 419-435, December.
    2. Marco Scarsini & Alfred Muller, 2006. "Stochastic order relations and lattices of probability measures," Post-Print hal-00539119, HAL.
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    Cited by:

    1. Lasse Leskelä, 2022. "Ross’s second conjecture and supermodular stochastic ordering," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 213-215, April.
    2. 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.
    3. Nendel, Max, 2019. "A Note on Stochastic Dominance and Compactness," Center for Mathematical Economics Working Papers 623, Center for Mathematical Economics, Bielefeld University.

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