setup

r.v.$$\begin{pmatrix}X_1\\Y_1\end{pmatrix},\ldots,\begin{pmatrix}X_n\\Y_n\end{pmatrix}\overset{i.i.d.}{\sim} N_2\left(\begin{pmatrix}\xi\\\eta\end{pmatrix},\begin{pmatrix}\sigma^2 & \rho\sigma\tau\\ \rho\sigma\tau & \tau^2\end{pmatrix}\right).$$Let $\gamma = \rho\sigma\tau.$

estimators:$\hat{\xi} = \bar{X}, \hat{\eta} = \bar{Y}, \hat{\sigma}^2=\frac{1}{n}\sum (X_i-\bar{X})^2, \hat{\tau}^2=\frac{1}{n}\sum (Y_i - \bar{Y})^2, \hat{\gamma}=\frac{1}{n}\sum(X_i-\bar{X})(Y_i-\bar{Y}).$

Problem

By the multivariate CLT, we can show that the joint limit distribution of$$\sqrt{n}(\hat{\xi}-\xi), \sqrt{n}(\hat{\eta}-\eta), \sqrt{n}(\hat{\sigma}^2-\sigma^2), \sqrt{n}(\hat{\tau}^2-\tau^2), \sqrt{n}(\hat{\gamma}-\gamma)$$is the 5-variate normal distribution with zero means and covariance matrix$$\Sigma = \begin{pmatrix} \Sigma_1 & \color{red}{0} \\ \color{red}{0} & \Sigma_2\end{pmatrix},$$where$$\Sigma_1 = \begin{pmatrix}\sigma^2 & \rho\sigma\tau\\ \rho\sigma\tau & \tau^2\end{pmatrix}$$and $\Sigma_2$ is the $3\times 3$ matrix.

I would like to show in red.

What I know

For example, we can show that the covariance of $(X_i - \xi)$ and $(Y_i - \eta)^2$ is zero:$$\mathrm{Cov}((X_i - \xi), (Y_i - \eta)^2) = \mathbb{E}[(X_i - \xi)(Y_i - \eta)^2] - \mathbb{E}[X_i - \xi]\mathbb{E}[(Y_i - \eta)^2]$$second term of right side is $0$, but how to calculate the first term...?

I thought it would be a bit complicated to do the integral calculation from the definition of expectation.