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A stochastic comparison for arrangement increasing functions. (English) Zbl 0806.60003

Summary: Let \(h(\cdot)\) be an arrangement increasing function, let \({\mathbf X}\) have an arrangement increasing density, and let \({\mathbf X}_ E\) be a random permutation of the coordinates of \({\mathbf X}\). We prove \(E \{h({\mathbf X}_ E)\} \leq E \{h({\mathbf X})\}\). 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.

MSC:

60C05 Combinatorial probability
05A05 Permutations, words, matrices
06A07 Combinatorics of partially ordered sets
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[1] DOI: 10.1214/aos/1176345688 · Zbl 0492.62052 · doi:10.1214/aos/1176345688
[2] DOI: 10.1214/lnms/1215465624 · doi:10.1214/lnms/1215465624
[3] Bollobás, Combinatorics (1986)
[4] Berge, Principles of Combinatorics (1971)
[5] DOI: 10.2307/1401973 · Zbl 0134.36403 · doi:10.2307/1401973
[6] Hajek, Nonparametric Techniques in Statistical Inference pp 3– (1970)
[7] DOI: 10.1214/aoms/1177706797 · Zbl 0086.35001 · doi:10.1214/aoms/1177706797
[8] Marshall, Inequalities: Theory of Majorization and Its Applications (1979) · Zbl 0437.26007
[9] DOI: 10.1007/BF01645980 · doi:10.1007/BF01645980
[10] DOI: 10.1214/aos/1176343895 · Zbl 0356.62043 · doi:10.1214/aos/1176343895
[11] DOI: 10.1214/aos/1176347131 · Zbl 0684.62038 · doi:10.1214/aos/1176347131
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