Multi-Dimensional Stochastic Ordering and Associated Random Variables
Abstract
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 RV's {X(sub 1),...,X(sub n)} imply a possible ordering in some stochastic sense between the IR(exp n)-valued RV X := (X(sub 1),...,X(sub n)) and its independent version X bar:= (X bar(sub 1),...,X bar(sub n))? Some of the results in that direction are as follows: (i) These IR(exp n)-valued RV's are comparable in either one of the orderings <=st, <=ci and <=cv iff they are identical in law, and (ii) If the RV's {X(sub 1),...,X(sub 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 IR(exp n)-valued RV's X and Y with identical mean: (j) The RV's 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 RV's and monotone functions.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1987
- Accession Number
- ADA453456
Entities
People
- Armand M. Makowski
- Francois Bacelli
Organizations
- University of Maryland