distribution and a Gaussian distribution: Berry-Esseen theorem {#BE}
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We use a multidimensional local version of the Berry-Esseen theorem due to Zitikis. More precisely, Theorem 1.2 of [@Zit93] reads:

\[berry-esseen\] Let $V_1, \cdots, V_n$ be a sequence of independent and identically distributed $d$-variate random vectors, let $S_n = \tfrac1n \sum_{i=1}^n V_i$. Let ${\bf \mu}$ and $\Sigma$ be the first two moments of $V_1$, and let $\lambda_{\mathrm{min}}$ be the least eigenvalue of $\Sigma$. Let $G$ be a Gaussian random vector with zero mean and covariance matrix $\mathbbm{1}_d$. Let $\mathcal{C}$ be the class of all convex Borel sets. Then there exists a universal constant $c\geq 0$ such that $$\begin{gathered}
\sup_{C \in \mathcal{C}} \left|P(\sqrt{n} (S_n-\mu)\Sigma^{-1/2}\in \mathcal{C}) - P(G \in \mathcal{C}) \right|\\
\leq c \sqrt{d} \lambda_{\mathrm{min}}^{-3/2}\mathbb{E}\left(||V_1||^3\right)/\sqrt{n},\end{gathered}$$ where $L_n(1)=1/n \sum_i V_i$, $V_i$ is a $d$-dimensional vector and $||.||$ refers to the Euclidean norm.

First, we note that the quantity $$\sup_{C \in \mathcal{C}} \left|P({\bf x} \in \mathcal{C}) - P({\bf y} \in \mathcal{C})  \right|$$ corresponds to the variational distance between the distributions of ${\bf x}$ and ${\bf y}$.

In the case of CVQKD, we need to consider tridimensional random vectors $V_i=[X_i,Y_i,Z_i]$ and one can immediately estimate ${\bf \mu}$ and $\Sigma$ from the experimental data. Using the notations 