On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations

2021/03/04 by Dembczak-Kołodziejczyk, Alicja, Lytova, Anna
#47N30 #60B20 #60F05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2103.03204

Abstract

Given n,m∈ ℕ, we study two classes of large random matrices of the form L_n =∑_α=1^mξ_αy_αy_α^T\quadand A_n =∑_α=1^mξ_α(y_αx_α^T+x_αy_α^T), where for every n, (ξ_α)_α⊂ ℝ are iid random variables independent of (x_α,y_α)_α, and (x_α)_α, (y_α)_α⊂ ℝⁿ are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as n,m(n)→ ∞: a standard one, where m/n→ c, and a slightly modified one, where m/n→∞ and Eξ→ 0 while mEξ/n→ c for some c≥ 0. Assuming that vectors (x_α)_α and (y_α)_α are normalized and isotropic "in average", we prove the convergence in probability of the empirical spectral distributions of L_n and A_n to a version of the Marchenko-Pastur law and so called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as (ξ_α)_α, in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [9, 21].

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