quantitative CLT template with ideal metrics

Let $X_1,\dots ,X_n$ be independent with mean 0 and variance $\mathbb{V}(X_i)={\sigma }_i^2$. Let ${\rho }_s$ be an ideal metric of order $s$ and set ${\Sigma }_n^2={\sum }_{i\le n}{\sigma }_i^2$. Then,

$$\begin{array}{r}{\rho }_s\left(\frac{1}{{\Sigma }_n}\sum _i{X}_i,N(0,1)\right)={\rho }_s\left(\frac{1}{{\Sigma }_n}\sum _i{X}_i,\frac{1}{{\Sigma }_n}\sum _{i}N(0,{\sigma }_i^2)\right),\end{array}$$

and the definition of an ideal metric gives

$${\rho }_s\left(\frac{1}{{\Sigma }_n}\sum _i{X}_i,\frac{1}{{\Sigma }_n}\sum _{i}N(0,{\sigma }_i^2)\right)\le \sum _{i\le n}{\rho }_s\left(\frac{X_i}{{\Sigma }_n},\frac{N(0,{\sigma }_i^2)}{{\Sigma }_n}\right)\le \frac{1}{{\Sigma }_n^s}\sum _{i\le n}{\rho }_s\left(X_i,N(0,{\sigma }_i^2)\right).$$

Hence, if one shows the right hand side approaches 0 and converge to 0 of the ideal metric implies convergence in distribution (eg with KS distance) you obtain a CLT.