A Singularity Criterion for Countable Gaussian Mixtures Based on the Feldman-Hajek Theorem

We study the mutual singularity of countable Gaussian mixture models (GMMs), with particular emphasis on infinite-dimensional settings.

We first establish that a countable mixture of Gaussian probability measures is itself a well-defined probability measure.

We then prove a general measure-theoretic result showing that if every component of one countable mixture is mutually singular with every component of another, then the two mixtures are mutually singular.

Combining this result with the Feldman--Hájek characterization of equivalence and singularity for Gaussian measures yields a sufficient condition for the mutual singularity of countable Gaussian mixtures.

We also discuss the mixed case, in which the presence of equivalent components prevents mutual singularity and leads naturally to a decomposition into singular and absolutely continuous parts.

To illustrate these theoretical results, we present a series of numerical experiments involving high-dimensional Gaussian mixture models.

The experiments demonstrate the emergence of increasing separability with dimension under different mechanisms, including mean shifts, covariance differences, and independently generated random mixtures.

A complementary experiment with a shared Gaussian component shows that complete asymptotic separation fails when the pairwise singularity condition is violated.

Together, the theoretical and numerical results provide a measure-theoretic framework for understanding asymptotic separability in high-dimensional Gaussian mixture models.