# Convergence rate of the inverse covariance matrix [closed]

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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• Every given element of the $X$ appearing in $\hat{\Sigma}$ is to be replaced with the difference of the element and the sample average. – Megadeth May 24 at 15:32
• $X$ 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? – AdamO May 24 at 15:52
• Are you asking about the covariance matrix or about the matrix $\hat \Sigma,$ which almost surely is not the covariance matrix of the samples? – whuber May 24 at 15:54