Fluctuations of the diagonal entries of a large sample precision matrix

Abstract

For a p×n data matrix Xn with i.i.d. centered entries and a population covariance matrix Σ, the corresponding sample precision matrix Σˆ−1 is defined as the inverse of the sample covariance matrix Σˆ=(1/n)Σ1/2XnXn⊤Σ1/2. We determine the joint distribution of a vector of diagonal entries of the matrix Σˆ−1 in the situation, where pn=p<n, p/n→y∈[0,1) for n→∞ and Σ is a diagonal matrix.