Convex majorization with an application to the length of critical paths

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Abstract
1. (Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all . 2. Let H be the Hardy–Littlewood maximal function HY (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 X 1 X 2 · ·· Xn be random variables with given marginal distributions, let I 1, I 2, ···, Ik be arbitrary non-empty subsets of {1,2, ···, n} and let M = max (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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