math.PR — Probability

Beyond Small Balls: A Determinant Inequality for Functions of Matrix Gaussian Series

Contributed by Suvrit Sra

We prove a functional extension of Gaussian operator-norm small-ball inequalities. Let $X=\sum_{i=1}^d g_iB_i$, where $g\sim N(0,I_d)$, the matrices $B_i\in\mathbb{R}^{d\times d}$ are symmetric, and $\sum_iB_i^2\preceq I_d$. For every even log-concave $f\colon\mathbb{R}\to[0,\infty)$, we obtain $\mathbb{E}\det f(X)\ge(\mathbb{E} f(\kappa G))^d$ at dilation $\kappa=13/5$, where $G$ is standard Gaussian. The proof reduces each function to one small-ball estimate, obtained from matrix-weighted Poincaré inequalities, and controls the rest by averages over the spectrum. We give an equivalent entropy formulation. An explicit counterexample shows that no universal dilation below $\sqrt{\pi/2}$ can work; we conjecture that this value is optimal.