# An improved estimate for the condition number anomaly of univariate Gaussian correlation matrices

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## Abstract

In this short note, it is proved that the derivatives of the parametrized univariate Gaussian correlation matrix R_g (Î¸) = (exp(âÎ¸(x_i â x_j )^2_{i,j} â R^{nÃn} are rank-deficient in the limit Î¸ = 0 up to any order m < (n â 1)/2. This result generalizes the rank deficiency theorem for Euclidean distance matrices, which appear as the first-order derivatives of the Gaussian correlation matrices in the limit Î¸ = 0. As a consequence, it is shown that the condition number of R_g(Î¸) grows at least as fast as 1(/Î¸^(mË +1) for Î¸ â 0, where mË is the largest integer such that mË < (n â 1)/2. This considerably improves the previously known growth rate estimate of 1/Î¸^22 for the so-called Gaussian condition number anomaly.

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