Necessary and sufficient conditions for the Marchenko-Pastur theorem

Abstract

We show that a weak concentration property for quadratic forms of isotropic random vectors x is necessary and sufficient for the validity of the Marchenko-Pastur theorem for sample covariance matrices of random vectors having the form Cx, where C is any rectangular matrix with orthonormal rows. We also obtain some general conditions guaranteeing the weak concentration property.

How many samples are needed to reliably approximate the best linear estimator for a linear inverse problem?

“…The proof of Theorem 4.5 in [17, Proposition 2] consists of a combination of two results: the Marchenko-Pastur Law [24] and the Bai-Yin theorem [4]. Since there exist variations of both of these results that hold under assumptions that are weaker than the assumptions in Theorem 4.5-see e.g., [38,Theorem 2.1], and [37, Corollary 3.1] and [10]-we expect that the assumptions in Theorem 4.5 can be weakened. We are currently working on an extension of Theorem 4.5 which uses the typical structure of Z in an inverse problem with additive noise.…”

Limiting spectral distribution for large sample covariance matrices with graph-dependent elements

“…The most general conditions imposed on x p ensure that the quadratic forms x ⊤ p A p x p weakly concentrate around their expectations up to an error term o(p) with probability 1 − o (1), where A p ∈ C p×p is an arbitrary matrix with the spectral norm A p 1. These conditions were studied in [2], [8], [16], [22], [28], [29], and [30]. As shown in [29], the weak concentration property for specific quadratic forms of x p gives necessary and sufficient conditions for the Marchenko-Pastur theorem [18].…”

“…These conditions were studied in [2], [8], [16], [22], [28], [29], and [30]. As shown in [29], the weak concentration property for specific quadratic forms of x p gives necessary and sufficient conditions for the Marchenko-Pastur theorem [18].…”

On spectral distribution of sample covariance matrices from large dimensional and large $k$-fold tensor products

“…components. Later, for the setting τ α ≡ 1, the necessary and sufficient conditions on Y 1 that the Marčenko-Pastur law serves as the limiting spectral distribution (LSD) of M n,1,m were carried out in [11].…”

“…The present paper deals with the case k = O(n) for the model (1.1) and shows that the empirical spectral distribution (ESD) of M n,k,m with τ α ≡ 1 converges to the Marčenko-Pastur law if and only if k = o(n). Therefore, the matrix M n,k,m with k = O(n) can be seen as a new example of bad vectors, for which the necessary and sufficient condition in [11] does not hold.…”