From micro-correlations to macro-correlations

Random vectors with a symmetric correlation structure share a common value of pair-wise correlation between their different components. The symmetric correlation structure appears in a multitude of settings, e.g. mixture models. In a mixture model the components of the random vector are drawn independently from a general probability distribution that is determined by an underlying parameter, and the parameter itself is randomized. In this paper we study the overall correlation of high-dimensional random vectors with a symmetric correlation structure. Considering such a random vector, and terming its pair-wise correlation "micro-correlation", we use an asymptotic analysis to derive the random vector's "macro-correlation" : a score that takes values in the unit interval, and that quantifies the random vector's overall correlation. The method of obtaining macro-correlations from micro-correlations is then applied to a diverse collection of frameworks that demonstrate the method's wide applicability.