Multidimensional Stochastic Ordering and Associated Random Variables

Abstract

This paper presents several relationships between the concept of associated random variables (RVs) and notions of stochastic ordering. The question that provides the impetus for this work is whether the association of the ℝ-valued RVs {X₁, …, X_n} implies a possible stochastic ordering between the ℝⁿ-valued RV X ≔ (X₁, …, X_n) and its independent version X̄ ≔ (X̄₁, …, X̄_n). This leads to results on how multidimensional probability distributions are determined by conditions on their one-dimensional marginal distributions in the event of comparison under the stochastic orderings ≤_st, ≤_c1, ≤_cv, ≤_D and ≤_K. Such results have direct implications for the comparison of bounds for Fork-Join queues and for the structure of monotone functions of several variables.