Large Sample Covariance Matrices of Gaussian Observations with Uniform Correlation Decay

TL;DR

This paper establishes the Marchenko-Pastur law for large sample covariance matrices derived from Gaussian data with uniformly decaying correlations, identifying conditions for spectral convergence and phase transitions in operator norm behavior.

Contribution

It extends the Marchenko-Pastur law to correlated Gaussian data with uniform correlation decay, providing precise growth conditions and analyzing operator norm phase transitions.

Findings

Spectral distribution converges to MP law under specific correlation decay conditions.

Operator norm converges to MP support only if correlation decay exponent exceeds 1.

Constructs examples where spectral convergence occurs but operator norm remains stochastic.

Abstract

We derive the Marchenko-Pastur (MP) law for sample covariance matrices of the form $V_n=\frac{1}{n}XX^T$, where $X$ is a $p\times n$ data matrix and $p/n\to y\in(0,\infty)$ as $n,p \to \infty$. We assume the data in $X$ stems from a correlated joint normal distribution. In particular, the correlation acts both across rows and across columns of $X$, and we do not assume a specific correlation structure, such as separable dependencies. Instead, we assume that correlations converge uniformly to zero at a speed of $a_n/n$, where $a_n$ may grow mildly to infinity. We employ the method of moments tightly: We identify the exact condition on the growth of $a_n$ which will guarantee that the moments of the empirical spectral distributions (ESDs) converge to the MP moments. If the condition is not met, we can construct an ensemble for which all but finitely many moments of the ESDs diverge. We also investigate the operator norm of $V_n$ under a uniform correlation bound of $C/n^{\delta}$, where $C,\delta>0$ are fixed, and observe a phase transition at $\delta=1$. In particular, convergence of the operator norm to the maximum of the support of the MP distribution can only be guaranteed if $\delta>1$. The analysis leads to an example for which the MP law holds almost surely, but the operator norm remains stochastic in the limit, and we provide its exact limiting distribution.