Eigenvectors of some large sample covariance matrix ensembles

2009/11/16 by Olivier Ledoit, Sandrine Péché, Ledoit, Olivier +1 · 6 citations
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0911.3010

openalex publication_date 2009/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider sample covariance matrices S_N=(1)/(p)Σ_N^1/2X_NX_N^* Σ_N^1/2 where X_N is a N × p real or complex matrix with i.i.d. entries with finite 12^{\rm th} moment and Σ_N is a N × N positive definite matrix. In addition we assume that the spectral measure of Σ_N almost surely converges to some limiting probability distribution as N → ∞ and p/N → γ>0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type (1)/(N) Tr (g(Σ_N) (S_N-zI)^-1)), where I is the identity matrix, g is a bounded function and z is a complex number. This is then used to compute the asymptotically optimal bias correction for sample eigenvalues, paving the way for a new generation of improved estimators of the covariance matrix and its inverse.

Citations

• On the distribution of the largest eigenvalue in principal components analysis
• Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices

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