Convex majorization with an application to the length of critical paths

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 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 = 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.