Largest magnitude for off-diagonal auto-correlation coefficients in high dimensional framework

Maxime Boucher (), Didier Chauveau () and Marguerite Zani ()
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Maxime Boucher: Université Libre de Bruxelles
Didier Chauveau: Université d’Orléans, Université de Tours, CNRS
Marguerite Zani: Université d’Orléans, Université de Tours, CNRS

Statistical Papers, 2025, vol. 66, issue 4, No 22, 40 pages

Abstract: Abstract This paper studies the coherence of an high-dimensional observations matrix. Specifically, we describe the limiting distribution of the largest magnitude of correlations matrix associated to our data outside a central band which size depends of the sample size. Using the Chen–Stein method, we show the convergence of the normalized coherence towards a Gumbel distribution. We broaden previous results by considering a 3-regime band structure for the off diagonal covariance matrix, where the largest band is composed of asymptotically vanishing coefficients. We provide an hypothesis test on the covariance structure where the alternative shows a clear dichotomy on the vanishing band. Moreover, we provide numerical simulations illustrating the asymptotic behavior of the coherence with Monte-Carlo experiment. We use a splitting strategy computing correlation matrices by blocks in order to avoid the high-dimensional memory issue.

Keywords: Chen–Stein method; Coherence; Gaussian high-dimensional matrices; Banded covariance (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00362-025-01693-y

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