I am trying to find results regarding the convergence rate of the inverse covariance matrix in the case where the number of observations $n$ is larger than the number of dimensions $p$.

Assume that $X \in R^p$ and $Var(X) = \Sigma$.

Then $\hat{\Sigma} = \frac{X'X}{n-1}$, $n$ being the number of observations. Are there results regarding the convergence rate of $\hat{\Sigma}^{-1}$ to $\Sigma^{-1}$ as $n$ grows?


closed as unclear what you're asking by Michael Chernick, mdewey, Frans Rodenburg, Sycorax, whuber May 31 at 18:28

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  • $\begingroup$ Every given element of the $X$ appearing in $\hat{\Sigma}$ is to be replaced with the difference of the element and the sample average. $\endgroup$ – Megadeth May 24 at 15:32
  • $\begingroup$ $X$ is usually a sample. Why would $X \in \mathbb{R}^p$? Do you mean to say that $X$ is a $n \times p$ matrix of real values? $\endgroup$ – AdamO May 24 at 15:52
  • $\begingroup$ Are you asking about the covariance matrix or about the matrix $\hat \Sigma,$ which almost surely is not the covariance matrix of the samples? $\endgroup$ – whuber May 24 at 15:54