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High Dimensional Probability VIIIGeneralized Semimodularity: Order Statistics

High Dimensional Probability VIII: Generalized Semimodularity: Order Statistics [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

High Dimensional Probability VIIIGeneralized Semimodularity: Order Statistics

Part of the Progress in Probability Book Series (volume 74)
Editors: Gozlan, Nathael; Latała, Rafał; Lounici, Karim; Madiman, Mokshay
Pinelis, Iosif
High Dimensional Probability VIII — Nov 27, 2019
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References (27)

Publisher
Springer International Publishing
Copyright
© Springer Nature Switzerland AG 2019
ISBN
978-3-030-26390-4
Pages
99 –119
DOI
10.1007/978-3-030-26391-1_8
Publisher site
See Chapter on Publisher Site

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.]

Published: Nov 27, 2019

Keywords: Semimodularity; Submodularity; Supermodularity; FKG-type inequalities; Association inequalities; Correlation inequalities; Primary 06D99; 26D15; 26D20; 60E15; Secondary 05A20; 05B35; 06A07; 60C05; 62H05; 62H10; 82D99; 90C27

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