[Paper Review] Rate of Convergence of the Expected Spectral Distribution Function to the Marchenko -- Pastur Law

Friedrich Götze, А. Н. Тихомиров|arXiv (Cornell University)|Dec 19, 2014

Random Matrices and Applications17 references6 citations

TL;DR

This paper establishes the rate of convergence of the expected spectral distribution function of sample covariance matrices to the Marchenko-Pastur law under moment and boundedness conditions on the matrix entries. Under assumptions of zero mean, unit variance, finite fourth moments, and entrywise boundedness by $ Dn^{1/4} $, the Kolmogorov distance between the expected spectral distribution and the Marchenko-Pastur law is shown to be $ O(n^{-1}) $, providing a non-asymptotic convergence rate.

ABSTRACT

Let $\mathbf X=(X_{jk})$ denote a $n imes p$ random matrix with entries $X_{jk}$, which are independent for $1\le j\le n, 1\le k\le p$. Let $n,p$ tend to infinity such that $\frac np=y+O(n^{-1})\in(0,1]$. 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 $\mathbf E X_{jk}=0$, $\mathbf E X_{jk}^2=1$ and $ \quad \quad \quad \quad \quad \quad \quad \sup_{n,p\ge1}\sup_{1\le j\le n,1\le k\le p}\mathbf E |X_{jk}|^4=: μ_4

Motivation & Objective

• To establish the non-asymptotic rate of convergence of the expected spectral distribution function of sample covariance matrices to the Marchenko-Pastur law.
• To extend previous results on Wigner matrices to sample covariance matrices by analyzing the Kolmogorov distance between the expected spectral distribution and the limiting Marchenko-Pastur law.
• To derive a sharp $ O(n^{-1}) $ bound on the Kolmogorov distance under minimal moment and entrywise boundedness assumptions.
• To provide a quantitative convergence rate that holds uniformly over the parameter range $ y \in (0,1] $, with explicit dependence on $ D $, $ \mu_4 $, $ y $, and $ c_y $.

Proposed method

• Utilizes the Stieltjes transform of the spectral distribution and compares it to the Stieltjes transform of the Marchenko-Pastur law to analyze convergence.
• Employs resolvent techniques and the Hermitian matrix $ \mathbf{V} $ constructed from $ \mathbf{X} $ and $ \mathbf{X}^* $ to relate eigenvalue behavior to the resolvent entries.
• Applies concentration inequalities and moment bounds to control the error terms in the resolvent expansion, particularly focusing on the difference between $ m_n(z) $ and $ s_y(z) $.
• Uses conditional moment estimates and decoupling techniques to bound the fourth moments of error terms in the resolvent identity, leveraging assumptions on $ \mu_4 $ and entrywise boundedness.
• Implements a multistep error decomposition involving $ \beta_{j1}, \beta_{j2}, \beta_{j3} $ to isolate and control contributions from different parts of the matrix structure.
• Applies a priori bounds on imaginary parts of resolvent entries and uses the Green's function comparison method in a controlled manner to derive uniform estimates over the spectral parameter $ z $.

Experimental results

Research questions

• RQ1 What is the non-asymptotic rate of convergence of the expected spectral distribution function of a sample covariance matrix to the Marchenko-Pastur law?
• RQ2 How does the convergence rate depend on the fourth moment and entrywise boundedness of the matrix entries?
• RQ3 Can a sharp $ O(n^{-1}) $ rate be established under minimal moment and boundedness conditions?
• RQ4 How does the Kolmogorov distance behave when the ratio $ n/p \to y \in (0,1] $, with $ |n/p - y| = O(n^{-1}) $?

Key findings

• Under the conditions $ \mathbb{E}[X_{jk}] = 0 $, $ \mathbb{E}[X_{jk}^2] = 1 $, $ \sup \mathbb{E}[|X_{jk}|^4] = \mu_4 < \infty $, and $ \sup |X_{jk}| \leq Dn^{1/4} $, the Kolmogorov distance $ \Delta_n = \sup_x |F_n(x) - G_y(x)| $ is bounded by $ C n^{-1} $.
• The constant $ C $ in the bound depends only on $ D $, $ \mu_4 $, $ y $, and $ c_y $, and is independent of $ n $ and $ p $.
• The result holds uniformly for $ y \in (0,1] $, and the convergence rate is sharp under the given moment and boundedness assumptions.
• The $ O(n^{-1}) $ rate is established via a detailed analysis of the Stieltjes transform and resolvent error terms, using moment bounds and conditional expectation techniques.
• The paper shows that the difference between the expected spectral distribution and the Marchenko-Pastur law decays at least as fast as $ n^{-1} $, even when the matrix entries are not identically distributed, as long as the fourth moment is uniformly bounded.