the estimator $\bar{V}^m$ converges to the true value and that the error follows a normal distribution.

More precisely, let us note $\Sigma_G$ the true covariance matrix and $\Sigma_{\mathrm{est}}$ the estimated matrix, i.e., $$\Sigma_G = \left[
\begin{smallmatrix}
3\langle x_i^2 \rangle^2 &  \langle x_i^2 \rangle \langle y_i^2 \rangle + 2 \langle x_i y_i \rangle^2 & 3 \langle x_i^2 \rangle \langle x_i y_i \rangle  \\
\langle x_i^2 \rangle \langle y_i^2 \rangle + 2 \langle x_i y_i \rangle^2 & 3\langle y_i^2 \rangle^2& 3 \langle y_i^2 \rangle \langle x_i y_i \rangle\\
3 \langle x_i^2 \rangle \langle x_i y_i \rangle & 3 \langle y_i^2 \rangle \langle x_i y_i \rangle & \langle x_i^2 y_i^2 \rangle\\
\end{smallmatrix}\right],$$ and $$\Sigma_\mathrm{est} = \frac{1}{m} \sum_{i_1, \cdots, i_m}\left[
\begin{smallmatrix}
x_i^4 & x_i^2 y_i^2 & x_i^3 y_i \\
x_i^2 y_i^2 &  y_i^4  &  x_i y_i^3 \\
x_i^3 y_i  & x_i y_i^3  &  x_i^2 y_i^2\\
\end{smallmatrix}\right].$$ Then, the Central Limit Theorem asserts that the random matrix $\sqrt{m} (\Sigma_\mathrm{est}-\Sigma_G)$ converges in distribution to a centered multivariate normal distribution with a covariance matrix depending on moments of order 8 of $(x_k,y_k)$. We do not explicitate this matrix here as it is rather cumbersome, but it is straightforward to compute it.

The result of this analysis is 