First-order flatness under non-elliptical sampling

Determine whether the first-order flatness property of the GFT coordinate covariance near independence survives for general non-elliptical sampling families.

Background

For elliptical sampling, the paper reports that the GFT covariance at independence is a scalar multiple of the identity and that its dependence on a small departure from independence is flat to first order, with the perturbation beginning at second order. Under non-elliptical sampling, the weighted-chi-squared distance limit remains valid, but the coordinate covariance need not be a scalar rescaling of the Gaussian covariance.

The paper explicitly leaves unresolved whether the first-order cancellation underlying flatness is a general phenomenon beyond elliptical families.

References

Under non-elliptical sampling, the general weighted-\chi2 law still holds, but \mathbf V_C{(r)} need not be a scalar rescaling of the Gaussian covariance. Whether the first-order flatness property survives in general non-elliptical families remains open.

— The Sampling Distribution of the Log-Euclidean Distance Between Sample Correlation Matrices  (2608.25288 - Kuketayev, 26 Aug 2026) in Section 6.5, “Non-Gaussian sampling”