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.

Concurrent work:

Shifted Gaussian Correlation (Overleaf, March 2026) https://overleaf.com/read/whbpfsggnzmq

proves c(n)=0 for non-centered GCI via asymptotic methods.

This paper provides complementary explicit quantitative results.

This is the companion paper to "Critical Correlation in Non-Centered Gaussian Vectors: The ρ*(r) Formula" (Paper #2 10.5281/zenodo.20078649).