2026/08/04 by Peter Reinhard Hansen
Mathematics · Economics, Econometrics and Finance · #math.PR #econ.EM #math.ST #stat.TH
arxiv created 2026/08/04 · arxiv updated 2026/08/06
The set of n× n correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate exp\-\tfrac14 n2log n\. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular: its maximum absolute correlation is of order √(log n/n), its Frobenius distance is asymptotic to √ n, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact Beta(1,d) distribution, where d=n(n-1)/2, and is therefore of order n-2. More generally, distinct off-diagonal entries are exactly pairwise independent under every LKJ(η) law. For the uniform law, this yields a Chen-Stein proof of the extreme-correlation point-process limit and an O(n-1) total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-n means. We also identify two distinct scales: ηn\asymp n alters the limiting spectrum, whereas ηn\asymp n2 is needed to keep the Frobenius distance bounded. Finally, for a bounded, centered i.i.d. off-diagonal specification, projection to the nearest correlation matrix incurs a squared repair cost asymptotically at least one-half of the squared Frobenius norm of its off-diagonal part.