ac{1}{k} \sum_{j=1}^k {\mathbb{E}}\left({\left\Vert X_j \right\Vert}^2
    {\mathbbm{1}_{{\left\Vert X_j \right\Vert} > {\varepsilon}\sqrt{k}}} \right).$$ Then $$d_{BL} \left(\frac{1}{\sqrt{k}} \sum_{j=1}^k X_j, Z\right) \le
    C_d \inf_{0 \le {\varepsilon}\le 1} ({\varepsilon}+ \theta({\varepsilon})),$$ where $Z$ is a standard Gaussian random vector in ${\mathbb{R}}^d$, and $C_d >
  0$ depends only on $d$.

Let $f:{\mathbb{C}}\to {\mathbb{R}}$ with ${\left\Vert f \right\Vert}_{BL} \le 1$. Observe that $$\label{E:mean-int}
    {\mathbb{E}}\mu(f) = 
    \frac{1}{{\left\vert G \right\vert}} \sum_{\chi \in \widehat{G}} {\mathbb{E}}f(\lambda_\chi)
    = \frac{1}{{\left\vert G \right\vert}} \sum_{\chi \in \widehat{G}} 
    {\mathbb{E}}f\left(\frac{1}{\sqrt{{\left\vert G \right\vert}}}\sum_{a \in G} \chi(a) Y_a\right),$$ where $(n)$ superscripts have been omitted for simplicity. We consider $\lambda_\chi$ as a sum of independent random vectors in ${\mathbb{R}}^2 \cong {\mathbb{C}}$. The relevant covariances are $$\operatorname{Cov}(\chi(a) Y_a) = \begin{bmatrix}
    {\mathbb{E}}({\operatorname{Re}}\chi(a) Y_a)^2 & {\mathbb{E}}({\operatorname{Re}}\chi(a) Y_a) ({\operatorname{Im}}\chi(a) Y_a) \\
    {\mathbb{E}}({\operatorname{Re}}\chi(a) Y_a) ({\operatorname{Im}}\chi(a) Y_a) & {\mathbb{E}}({\operatorname{Im}}\chi(a) Y_a)^2
  \end{bmatrix}.$$ The identities $$\begin{split}\label{E:Re-Im}
      ({\operatorname{Re}}w)({\operatorname{Re}}z)
      &= \tfrac{1}{2} {\operatorname{Re}}\bigl[(w + \overline{w}) z \bigr], \\
      ({\operatorname{Im}}w)({\operatorname{Im}}z)
      &= \tfrac{1}{2} {\operatorname{Re}}\bigl[(\overline{w} - w) z \bigr], \\
      ({\operatorname{Re}}w)({\operatorname{Im}}z) &= \tfrac{1}{2} {\operatorname{Im}}\bigl[(w - \overline{w}) z
      \bigr],
  \end{split}$$ will be useful.

Setting $w = z = \chi(a) Y_a$ for a fixed $\chi \in \widehat{G}$, $$\begin{aligned}
    \sum_{a \in G} {\mathbb{E}}({\operatorname{Re}}\chi(a) Y_a)^2 &=
    \sum_{a \in G} \left[\frac{1}{2} {\operatorname{Re}}{\mathbb{E}}\le