The Feldman-H\'ajek Dichotomy for Countable Gaussian Mixtures and their Asymptotic Separability in High Dimensions

Abstract

This paper establishes the theoretical foundations for the asymptotic separability of Gaussian Mixture Models (GMMs) in high dimensions by extending the classical Feldman-H\'ajek theorem. We first prove that a countable mixture of Gaussian measures is a well-defined probability measure. Our primary result, the Gaussian Mixture Dichotomy Theorem, demonstrates that the mutual singularity of individual Gaussian components is a sufficient condition for the mutual singularity of the resulting mixtures. We provide a rigorous proof and further discuss the ``Mixed Case,'' where the presence of even a single equivalent pair of components leads to partial absolute continuity via the Lebesgue decomposition, thereby defining the theoretical limits of perfect classification in infinite-dimensional spaces.