Rate of Convergence of the Expected Spectral Distribution Function to the Marchenko -- Pastur Law

2014/12/19 by Friedrich Götze, Götze, F., А. Н. Тихомиров +1
Mathematics · #15B52 #60B20 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1412.6284

openalex publication_date 2014/12/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let \mathbf X=(X_jk) denote a n× p random matrix with entries X_jk, which are independent for 1≤ j≤ n, 1≤ k≤ p. Let n,p tend to infinity such that \frac np=y+O(n^-1)∈(0,1]. For those values of n,p we investigate the rate of convergence of the expected spectral distribution function of the matrix \mathbf W=\frac1 p\mathbf X\mathbf X^* to the Marchenko-Pastur law with parameter y. Assuming the conditions \mathbf E X_jk=0, \mathbf E X_jk²=1 and sup_n,p≥1sup_{1≤ j≤ n,1≤ k≤ p}\mathbf E |X_jk|⁴=: μ₄

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