My problem appears to be pretty simple. I've obtained the speed of convergence — Berry-Esseen bound — for the expression $$ \underset{C \in \mathcal{C}}{\mathrm{sup}} | Q_{X, n} (C) - \Phi (C)| \leqslant \frac{\widetilde{C}}{\sqrt{n}} $$ — where $Q_{X, n}$ is the CDF of the sum $S_n = \sum_{i=1}^n X_i$ of iid RVs with zero mean, the covariance of $S_n$ is finite and is equal to $\sigma^2$; $\Phi$ is the CDF of Normal RV with expectation zero and $C$ is some set in the class of measurable convex sets in $\mathbb{R}$. Now, I can't handle the problem of deriving the same bound for the squares: that is, I have no idea how fast the sum of the squares of $X_i$'s is supposed to converge to the $\chi^2$-distribution.
Could you please help me? Is there anything we can say about the situation without thinking much about the distribution of $X_i$'s? I've tried looking for some results concerning functions of polynomials of random variables, and haven't succeeded much.