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Generalized Semimodularity: Order Statistics

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High Dimensional Probability VIII

Part of the book series: Progress in Probability ((PRPR,volume 74))

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Abstract

A notion of generalized n-semimodularity is introduced, which extends that of (sub/super)modularity in four ways at once. The main result of this paper, stating that every generalized -semimodular function on the nth Cartesian power of a distributive lattice is generalized n-semimodular, may be considered a multi/infinite-dimensional analogue of the well-known Muirhead lemma in the theory of Schur majorization. This result is also similar to a discretized version of the well-known theorem due to Lorentz, which latter was given only for additive-type functions. Illustrations of our main result are presented for counts of combinations of faces of a polytope; one-sided potentials; multiadditive forms, including multilinear ones—in particular, permanents of rectangular matrices and elementary symmetric functions; and association inequalities for order statistics. Based on an extension of the FKG inequality due to Rinott & Saks and Aharoni & Keich, applications to correlation inequalities for order statistics are given as well.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, USA

    Iosif Pinelis

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  1. Iosif Pinelis
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Correspondence to Iosif Pinelis .

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Editors and Affiliations

  1. MAP 5, Université Paris Descartes, Paris, France

    Nathael Gozlan

  2. Institute of Mathematics, University of Warsaw, Warsaw, Poland

    Rafał Latała

  3. Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France

    Karim Lounici

  4. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    Mokshay Madiman

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Pinelis, I. (2019). Generalized Semimodularity: Order Statistics. In: Gozlan, N., Latała, R., Lounici, K., Madiman, M. (eds) High Dimensional Probability VIII. Progress in Probability, vol 74. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26391-1_8

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