Fluctuations of the diagonal entries of a large sample precision matrix

Abstract

For a given $p\times n$ data matrix $\textbf{X}_n$ with i.i.d. centered entries and a population covariance matrix $\bf{\Sigma}$, the corresponding sample precision matrix $\hat{\bf\Sigma}^{-1}$ is defined as the inverse of the sample covariance matrix $\hat{\bf{\Sigma}} = (1/n) \bf{\Sigma}^{1/2} \textbf{X}_n\textbf{X}_n^\top \bf{\Sigma}^{1/2}$. We determine the joint distribution of a vector of diagonal entries of the matrix $\hat{\bf\Sigma}^{-1}$ in the situation, where $p_n=p< n$ and $p/n \to y \in [0,1)$ for $n\to\infty$ and $\bf{\Sigma}$ is a diagonal matrix. Remarkably, our results cover both the case where the dimension is negligible in comparison to the sample size and the case where it is of the same magnitude. Our approach is based on a QR-decomposition of the data matrix, yielding a connection to random quadratic forms and allowing the application of a central limit theorem for martingale difference schemes. Moreover, we discuss an interesting connection to linear spectral statistics of the sample covariance matrix. More precisely, the logarithmic diagonal entry of the sample precision matrix can be interpreted as a difference of two highly dependent linear spectral statistics of $\hat{\bf\Sigma}$ and a submatrix of $\hat{\bf\Sigma}$. This difference of spectral statistics fluctuates on a much smaller scale than each single statistic.