On the Failure of the Gaussian Correlation Inequality for Non-Centered Distributions

The Gaussian Correlation Inequality (GCI), proved by Royen for centered Gaussian measures, asserts that Pr(X ∈ A ∩ B) ≥ Pr(X ∈ A) Pr(X ∈ B) for symmetric convex sets A, B. We show this inequality fails for non-centered Gaussian vectors. We exhibit an explicit counterexample in dimension two with GCI ratio R ≈ 7.24 × 10−18, verified to 60 decimal digits, and establish a local sign theorem: at the identity covariance, the sign of ∂R/∂ρij equals sign(mi · mj ), where m is the mean vector. When the means have opposite signs, the GCI ratio R is locally decreasing in ρ, producing violations at positive correlation. We further prove that this sign structure is universal: it holds for all symmetric unimodal densities, including Laplace, logistic, and Student-t, with heavier tails producing larger GCI ratios. The counterexample and sign theorem have implications for the composition analysis of differential privacy mechanisms that assume or exploit positive Gaussian correlation.