Spectral analysis of the Moore–Penrose inverse of a large dimensional sample covariance matrix

For a sample of n independent identically distributed p-dimensional centered random vectors with covariance matrix Σn let S̃n denote the usual sample covariance (centered by the mean) and Sn the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where p>n. In this paper, we provide the limiting spectral distribution and central limit theorem for linear spectral statistics of the Moore–Penrose inverse of Sn and S̃n. We consider the large dimensional asymptotics when the number of variables p→∞ and the sample size n→∞ such that p/n→c∈(1,+∞). We present a Marchenko–Pastur law for both types of matrices, which shows that the limiting spectral distributions for both sample covariance matrices are the same. On the other hand, we demonstrate that the asymptotic distribution of linear spectral statistics of the Moore–Penrose inverse of S̃n differs in the mean from that of Sn.