Large sample covariance matrices of Gaussian observations with uniform correlation decay

Abstract

We derive the Marchenko–Pastur (MP) law for sample covariance matrices of the form Vn=1nXXT, where X is a p×n data matrix and p/n→y∈(0,∞) as n,p→∞. 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 an/n, where an may grow mildly to infinity. We employ the method of moments tightly: We identify the exact condition on the growth of an 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 Vn under a uniform correlation bound of C/nδ, where C,δ>0 are fixed, and observe a phase transition at δ=1. In particular, convergence of the operator norm to the maximum of the support of the MP distribution can only be guaranteed if δ>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.