Asymptotic covariance of the bivariate Gaussian covariance estimator
= Asymptotic covariance of the bivariate Gaussian covariance estimator
Writing $a=\Sigma_{11}$, $c=\Sigma_{12}$, and $b=\Sigma_{22}$, the vector $(\widehat\Sigma_{11},\widehat\Sigma_{12},\widehat\Sigma_{22})$ has asymptotic covariance
$$
\begin{pmatrix}
2a^2&2ac&2c^2\\
2ac&ab+c^2&2bc\\
2c^2&2bc&2b^2
\end{pmatrix}.
$$