Fluctuations of the Diagonal Entries of a Large Sample Precision Matrix

Abstract

For a given $p\times n$ data matrix $\bfX_n$ with i.i.d. centered entries and a population covariance matrix $\bfSigma$, the corresponding sample precision matrix $\hat\bfSigma\inv$ is defined as the inverse of the sample covariance matrix $\hat\bfSigma = (1/n) \bfSigma^{1/2} \bfX_n \bfX_n^\top \bfSigma^{1/2}$. We determine the joint distribution of avector of diagonal entries of the matrix $\hat\bfSigma\inv$ in the situation, where $p_n=p< n$, $p/n \to y \in [0,1)$ for $n\to\infty$ and $\bfSigma$ is a diagonal matrix. . 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 $\hat\bfSigma$ and a submatrix of $\hat\bfSigma$. This difference of spectral statistics fluctuates on a much smaller scale than each single statistic.

Locations

• arXiv (Cornell University) - View - PDF