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A Stochastic Comparison for Arrangement Increasing Functions

Published online by Cambridge University Press:  12 September 2008

Abba M. Krieger  and
Paul R. Rosenbaum
Abba M. Krieger
Affiliation:
Department of Statistics, University of Pennsylvania, Philadelphia PA 19104–6302, USA
Paul R. Rosenbaum
Affiliation:
Department of Statistics, University of Pennsylvania, Philadelphia PA 19104–6302, USA
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Abstract

Let h(·) be an arrangement increasing function, let X have an arrangement increasing density, and let XE be a random permutation of the coordinates of X. We prove E{h(XE)} ≤ E{h(X)}. This comparison is delicate in that similar results are sometimes true and sometimes false. In a finite distributive lattice, a similar comparison follows from Holley's inequality, but the set of permutations with the arrangement order is not a lattice. On the other hand, the set of permutations is a lattice, though not a distributive lattice, if it is endowed with a different partial order, but in this case the comparison does not hold.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 3 , September 1994 , pp. 345 - 348
Copyright
Copyright © Cambridge University Press 1994

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