Independence Via Uncorrelatedness Under Certain Dependence Structures

Abstract

A characterization of independence via uncorrelatedness is shown to hold for the families satisfying positive and negative dependence conditions. For the associated random variables, the bounds on covariance functions obtained by Lebowitz (Comm. Math. Phys. $\mathbf{28}$ (1972), 313-321) readily yield such a characterization. An elementary proof for the same characterization is also given for a condition weaker than association, labeled as "strong positive (negative) orthant dependence." This condition is compared with the "linear positive dependence," under which Newman and Wright (Ann. Probab. $\mathbf{9}$ (1981), 671-675) obtained the characterization.