Central limit theorem for eigenvalue statistics of sample covariance matrix with random population

2022/11/10 by Ji Oon Lee, Yiting Li, Lee, Ji Oon +1
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2211.05546

openalex publication_date 2022/11/10 · openalex created_date 2022/11/16 · openalex updated_date 2026/07/28

Abstract

Consider the sample covariance matrix Σ^1/2XX^TΣ^1/2 where X is an M× N random matrix with independent entries and Σ is an M× M diagonal matrix. It is known that if Σ is deterministic, then the fluctuation of ∑_if(λ_i) converges in distribution to a Gaussian distribution. Here \λ_i\ are eigenvalues of Σ^1/2XX^TΣ^1/2 and f is a good enough test function. In this paper we consider the case that Σ is random and show that the fluctuation of (1)/(√ N)∑_if(λ_i) converges in distribution to a Gaussian distribution. This phenomenon implies that the randomness of Σ decreases the correlation among \λ_i\.

Related

• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• The Central Limit Theorem for Linear Eigenvalue Statistics of the Sum of Independent Matrices of Rank One
• Fluctuations for linear eigenvalue statistics of sample covariance matrices
• Central Limit Theorem for Linear Eigenvalue Statistics of non-Hermitian\n Random Matrices
• Partial linear eigenvalue statistics for non-Hermitian random matrices