Convex majorization with an application to the length of critical paths
- 1 September 1979
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 16 (3) , 671-677
- https://doi.org/10.2307/3213097
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.Keywords
This publication has 4 references indexed in Scilit:
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- The Completion Time of PERT NetworksOperations Research, 1977
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- A maximal theorem with function-theoretic applicationsActa Mathematica, 1930