Tracy-Widom distribution for the edge eigenvalues of elliptical model

04/16/2023

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by   Xiucai Ding, et al.

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In this paper, we study the largest eigenvalues of sample covariance matrices with elliptically distributed data. We consider the sample covariance matrix Q=YY^*, where the data matrix Y ∈ℝ^p × n contains i.i.d. p-dimensional observations 𝐲_i=ξ_iT𝐮_i, i=1,…,n. Here 𝐮_i is distributed on the unit sphere, ξ_i ∼ξ is independent of 𝐮_i and T^*T=Σ is some deterministic matrix. Under some mild regularity assumptions of Σ, assuming ξ^2 has bounded support and certain proper behavior near its edge so that the limiting spectral distribution (LSD) of Q has a square decay behavior near the spectral edge, we prove that the Tracy-Widom law holds for the largest eigenvalues of Q when p and n are comparably large.