of the main text and applying Theorem \[berry-esseen\] gives $$\begin{gathered}
\label{bound}
\iiint_{\mathbb{R}^3} \left|P_v(V)-P_G(V)) \right| \mathrm{d}V\\
\leq c \sqrt{3} \lambda_{\mathrm{min}}^{-3/2}\mathbb{E}\left(||X_1^2+Y_1^2+Z_1^2||^{3/2}\right)/\sqrt{n},\end{gathered}$$ where $P_G$ corresponds to the multivariate Gaussian distribution with the same first two moments as $P_v$.

Error due to the finite estimation sample {#finite}
=========================================

Let us assume that the variables $(x_k,y_k)$ follow a centered bivariate normal distribution, as typical in experimental implementations of CVQKD.

During the parameter estimation, Alice and Bob need to estimate the fourth moments of $(x_k,y_k)$ given by the covariance matrix $\Sigma$: $$\Sigma = \left[
\begin{smallmatrix}
\langle x_i^4 \rangle &  \langle x_i^2 y_i^2 \rangle & \langle x_i^3 y_i \rangle  \\
\langle x_i^2 y_i^2 \rangle & \langle y_i^4 \rangle & \langle x_i y_i^3 \rangle\\
\langle x_i^3 y_i \rangle & \langle x_i y_i^3 \rangle & \langle x_i^2 y_i^2 \rangle\\
\end{smallmatrix}\right]$$ Alternatively, one can describe this matrix by the vector $V_i = [\langle x_i^4 \rangle, \langle x_i^3 y_i \rangle, \langle x_i^2 y_i^2 \rangle, \langle x_i y_i^3 \rangle, \langle y_i^4 \rangle]$. In order to estimate this vector, one uses the estimator $\bar{V}^m$ defined as $$\bar{V}^m:= \frac{1}{\sqrt{m}} \sum_{i_1,\cdots, i_m} V_i,$$ where $i_1,\cdots, i_m$ are $m$ randomly chosen indices among $\{1,\cdots,n\}$. Using the mutlivariate version of the Central Limit Theorem, one obtains that 