On the Rate of Convergence to the Marchenko--Pastur Distribution

2011/10/06 by Götze, F., Tikhomirov, A.
#15B52 #60B20 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1110.1284

Abstract

Let \mathbf X=(X_jk) denote n× p random matrix with entries X_jk, which are independent for 1≤ j≤ n,1≤ k≤ p. We consider the rate of convergence of empirical spectral distribution function of the matrix \mathbf W=\frac1p\mathbf X\mathbf X^* to the Marchenko--Pastur law. We assume that \mathbf E X_jk=0, \mathbf E X_jk²=1 and that the distributions of the matrix elements X_jk have a uniformly sub exponential decay in the sense that there exists a constant \varkappa>0 such that for any 1≤ j ≤ n, 1≤ k≤ p and any t≥ 1 we have Pr\|X_jk|gt;t\≤ \varkappa^-1exp\-t^\varkappa\. By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the sample covariance matrix \mathbf W and the Marchenko--Pastur distribution is of order O(n^-1log^4+\frac4\varkappa n) with high probability.

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