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Convex Majorization with an Application to the Length of Critical Paths

Isaac Meilijson and Arthur Nádas
Journal of Applied Probability
Vol. 16, No. 3 (Sep., 1979), pp. 671-677
DOI: 10.2307/3213097
Stable URL: http://www.jstor.org/stable/3213097
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Convex Majorization with an Application to the Length of Critical Paths
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Abstract

1. Eφ (X) ≤ Eφ (Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all x, E(X - x)+ ≤ E (Y - x)+. 2. Let H be the Hardy-Littlewood maximal function $H_{Y}(x) = E (Y - x | Y > x)$. Then HY(Y) is the smallest random variable exceeding stochastically all random variables convexly smaller than Y. 3. Let X1X2 ⋯ Xn be random variables with given marginal distributions, let I1, I2, ⋯, Ik be arbitrary non-empty subsets of {1,2, ⋯, n} and let M = max1 ≤ j ≤ k ∑i ∈ Ij Xi (M is the completion time of a PERT network with paths Ij and delay times Xi.) The paper introduces a computation of the convex supremum of M in the class of all joint distributions of the Xi's with specified marginals, and of the 'bottleneck probability' of each path.

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