My question is how can I use the Berry-Esseen Theorem to know how close to the Gaussian distribution is $L$, where
$$L=nLn(2)+Ln(r_1)+Ln(r_2)+ \cdots + Ln(r_n)$$
$r_i \geq 0$ is a i.i.d. random variable from the Rayleigh distribution.
The mean and variance for $Ln(r_1)$ is
\begin{eqnarray} \mathbb{E}[Ln(r_1)]&=&-\frac{\gamma}{2},\\ \mathbb{E}[(Ln(r_1)-\mathbb{E}[Ln(r_1)])^2]&=&\mathbb{V}\text{ar}[Ln(r_1)]=\frac{\pi^2}{24},\\ \end{eqnarray}
Where $\gamma=0.57721$.
Berry-Esseen Theorem: Let $Z_1,\ldots,Z_n$ be real i.i.d. random variables satisfying \begin{eqnarray*} \mathbb{E}[Z_i]&=&v,\\ \mathbb{E}[(Z_i-v)^2]&=&\sigma^2>0,\\ \mathbb{E}[|Z_i-v|^3]&=&\rho<\infty. \end{eqnarray*} Then let \begin{equation*} Z:=Z_1+\cdots+Z_n, \end{equation*} and let $W \thicksim \mathcal{N}(vn,\sigma^2n)_\mathbb{R}$ be a real gaussian with mean $vn$ and variance $\sigma^2n$. Then for all $x \in \mathbb{R}$,\begin{equation*} \left| Pr[Z\leq x]- Pr[W \leq x] \right|\leq \frac{C\rho}{\sigma^3\sqrt{n}}, \end{equation*} where $C$ is some universal constant.