If, $\Sigma$ is the population covariance matrix and $S$ is the sample covariance matrix, $p$ is the number of variables, $\frac{p}{n} \rightarrow c$ as $n \rightarrow 0$,
$\frac{1}{p}|S|_{F}^{2} \rightarrow |\Sigma|_{F}^{2} + \frac{c}{p}(trace(\Sigma))^2$ almost surely.
If we consider $S_{D}$ as the $diag(S)$ and its population covariance matrix is, $\Sigma_{D}$, can we say,
$\frac{1}{p}|S_{D}|_{F}^{2} \rightarrow |\Sigma_{D}|_{F}^{2} + \frac{c}{p}(trace(\Sigma_{D}))^2$ almost surely?
Any help regarding this is greatly appreciated.
Thanks