In the following version of the Multivariate Lindeberg-Feller CLT, what does $\overline{\mathbf{V}}_n^{-1/2}$ mean? It is not mentioned anywhere in the text and I have never seen the square root of a matrix before.
$\begingroup$
$\endgroup$
1
-
$\begingroup$ Could you point me to your reference if you still have it / remember it? I'm looking for exactly something like this. $\endgroup$– bobCommented May 18, 2022 at 21:23
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
$\bar{V_n}^{-1}$ is a positive-semidefinite average of covariance matrix, then by Spectral theorem we can decompose it as $\bar{V_n}^{-1} = Q \Lambda Q^T$ and $\bar{V_n}^{-1/2} = Q \Lambda^{1/2} Q^T$, where $Q$ is orthonormal and $\Lambda$ is the diagonal matrix containing eigenvalues of $\bar{V_n}^{-1}$.
