Spectral gaps and measure decompositions

Abstract: Let $μ$ be a probability measure on $\mathbb{R}^{d}$. In this paper, we introduce a new set of computable quantities in $μ$ that are invariant under orthogonal transformations, namely, the eigenvalues of the 4th moment operator of $μ$. We show how the first and second largest eigenvalues of this operator can determine the extent to which $μ$ can be decomposed as an equal weight mixture of two probability measures with significantly different second order statistics.