Fluctuations of the diagonal entries of a large sample precision matrix

TL;DR

This paper analyzes the fluctuations of the diagonal entries of a large sample precision matrix, deriving their joint distribution in high-dimensional settings where the dimension and sample size grow proportionally.

Contribution

It provides a novel joint distribution result for diagonal entries of the inverse sample covariance matrix in high-dimensional regimes, covering both negligible and comparable dimension scenarios.

Findings

Derived the joint distribution of diagonal entries in high-dimensional limit

Connected fluctuations to linear spectral statistics of the sample covariance matrix

Identified that differences of spectral statistics fluctuate on smaller scales

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.