A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

Abstract

In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form Q = T X(T X) * , where the sample X is an M2 × N random matrix with i.i.d. entries with mean zero and variance N −1 , and T is an M1 × M2 deterministic matrix satisfying T * T is diagonal. We study the asymptotic behavior of the largest eigenvalues of Q when M := min{M1, M2} and N tends to infinity with limN→∞ N/M = d ∈ (0, ∞). Under mild assumptions of T , we prove that the Tracy-Widom law holds for the largest eigenvalue of Q if and only if lims→∞ s 4 P(| √ N xij| ≥ s) = 0. This condition was first proposed for Wigner matrices by Lee and Yin [24]. *

Edge universality of separable covariance matrices

“…So far, this is the strongest edge universality result for sample covariance matrices with correlated data (i.e. non-diagonal A) and heavy tails, which improves the previous results in [6,37] (assuming high moments and diagonal A), [35] (assuming high moments) and [14] (assuming diagonal A).…”

“…non-diagonal A) with heavy tails as in (1.1). So far, this is the strongest edge universality for sample covariance matrices compared with [6,37] (assuming high moments and diagonal A), [35] (assuming high moments) and [14] (assuming diagonal A). The sample covariance matrices are widely used in various applied fields: multivariate statistics, empirical finance, signal processing, population genetics, and machine learning, to name a few.…”

“…A is not a multiple of I). For general diagonal A, the problem beyond Gaussian was considered in [6,37] under the assumption that the entries of X have arbitrarily high moments, and in [14] under the condition (1.1). Our result shows that, for heavy-tailed correlated data satisfying (1.1), one can still use the largest eigenvalue as our test static in the above high-dimensional statistical inference problems.…”

“…When A is a non-scalar diagonal matrix, the Tracy-Widom distribution was first proved for Wishart matrix X in [15] (non-singular A case) and [44] (singular A case). Later the edge universality with general diagonal A was proved in [6,37] for X with entries having arbitrarily high moments, and in [14] for X with entries satisfying the tail condition (1.1). The most general case with non-diagonal A is considered in [35], where the edge universality was proved under the arbitrarily high moments assumption.…”

Tracy–Widom limit for Kendall’s tau

“…Therefore, mathematically, independence test based on the spectral statistics such as the largest eigenvalue of the sample covariance matrix or correlation matrix is also doubtful under general distribution assumption. On the other hand, although the TW law was shown to be universal for the sample covariance matrices, assumptions on the distribution of the matrix entries is still required to certain extent, see for instance the minimal moment condition in [11]. This moment requirement certainly excludes all heavy-tailed data sets.…”