Independence via uncorrelatedness under certain dependence structures. (English) Zbl 0542.62046

Joag-Dev, Kumar

Ann. Probab. 11, 1037-1041 (1983).

Summary: 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 J. L. Lebowitz [Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems. Commun. Math. Phys. 28, 313-321 (1972)] 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 C. M. Newman and A. L. Wright [Ann. Probab. 9, 671-675 (1981; Zbl 0465.60009)] obtained the characterization.

MSC:

62H20	Measures of association (correlation, canonical correlation, etc.)
62H05	Characterization and structure theory for multivariate probability distributions; copulas

Keywords:

characterization of independence; uncorrelatedness; positive and negative dependence conditions; bounds on covariance functions; orthant dependence; linear positive dependence

Citations:

Zbl 0465.60009