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

Abstract

Let $\mathbf{X}=(X_{jk})$ denote $n\times p$ random matrix with entries $X_{jk}$, which are independent for $1\le j\le n,1\le k\le p$. We consider the rate of convergence of empirical spectral distribution function of the matrix $\mathbf{W}=\frac{1}{p}\mathbf{X}{\mathbf{X}}^{\ast }$ to the Marchenko--Pastur law. We assume that $\mathbf{E}{X}_{jk}=0$, $\mathbf{E}{X}_{jk}^2=1$ 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 $1\le j\le n,1\le k\le p$ and any $t\ge 1$ we have $$\mathbf{Pr}\{|X_{jk}|>t\}\le {\varkappa }^{-1}\exp \{-t^{\varkappa }\}.$$ 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 $O(n^{-1}{\log }^{4+\frac{4}{\varkappa }}n)$ with high probability.

A much shortened version of the proof rewriting Section 6 by using transfer principles to Gaussian variables for moments of convex functions in Section 4 and the Appendix

Cited by in corpus (10)

• The Church of the Symmetric Subspace
• Spectral Graph Matching and Regularized Quadratic Relaxations I: The Gaussian Model
• Random Subsets of Structured Deterministic Frames have MANOVA Spectra
• Eigenvalue variance bounds for covariance matrices
• Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups
• Detecting Rare and Weak Spikes in Large Covariance Matrices
• Rate of Convergence of the Expected Spectral Distribution Function to the Marchenko -- Pastur Law
• Normal Approximation and Confidence Region of Singular Subspaces
• Adaptive Estimation of Noise Variance and Matrix Estimation via USVT Algorithm
• Analog Coding Frame-work