The Sampling Distribution of the Log-Euclidean Distance Between Sample Correlation Matrices

Abstract: Comparing correlation matrices across time or stress scenarios is critical in quantitative finance and multivariate statistics, yet sample estimation noise often obscures whether an observed distance reflects a true structural shift. We derive the asymptotic sampling distribution of the intrinsic off-log (log-Euclidean) distance between two independently estimated full-rank correlation matrices under the null hypothesis that their population correlation matrices coincide. Under general sampling with finite fourth moments, the scaled squared distance converges to a weighted sum of independent $χ<em>1<sup>2$ variables, with weights determined by the asymptotic covariance of the Generalized Fisher Transformation (GFT) coordinates. Under Gaussian sampling at independence, this simplifies to a parameter-free $4χ_d<sup>2$ law. To calibrate tail probabilities, we provide closed-form cumulant generating functions, Lugannani--Rice saddlepoint quantiles, and an explicit Chernoff envelope requiring no root-finding. The first moment of the limiting law establishes a simple rule of thumb for the baseline expected distance under the null hypothesis ($$\operatorname E[d</em>{\mathrm{LE}}] \lesssim 2\sqrt{d/n}$$ near independence), quantifying the average separation induced strictly by estimation error. We establish plug-in consistency, present an explicit Gaussian covariance factorization, compare the distance statistic with coordinate Wald tests, and characterize its local power.