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Multi-Dimensional Stochastic Ordering and Associated Random Variables

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Technical rept.

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This paper presents several relationships between the notion of associated random variables and notions of stochastic ordering that have appeared in the literature over the years. More concretely, the discussion centers around the following question Under which conditions does the association of the IR-valued RVs Xsub 1,...,Xsub n imply a possible ordering in some stochastic sense between the IRexp n-valued RV X Xsub 1,...,Xsub n and its independent version X bar X barsub 1,...,X barsub n Some of the results in that direction are as follows i These IRexp n-valued RVs are comparable in either one of the orderings st, ci and cv iff they are identical in law, and ii If the RVs Xsub 1,...,Xsub n are associated, certain comparison properties hold for the stochastic orderings D, K and L defined in D. Stoyan 1983. Strengthening of result i leads to the following results on the stochastic ordering properties of IRexp n-valued RVs X and Y with identical mean j The RVs X and Y are comparable for st iff they are identical in law, and jj If X D Y resp. X K Y, then X and Y are comparable for ci resp. cv iff they are identical in law. These and related results are given direct applications to queueing theory and to the asymptotics of associated random variables. In the process of answering this question, several results were obtained that indicate how multi-dimensional probability distributions are determined by conditions on their one-dimensional marginal distributions in the event of stochastic comparisons. Several interesting consequences of Theorems 1-4 are presented. The first application is given in the context of Fork-Join FJ queue models that arise in many application areas, including flexible manufacturing and parallel processing. Other applications involve bounds on the tail behavior of the maximum of associated RVs and monotone functions.

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  • Statistics and Probability
  • Operations Research

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