If we use $\Rightarrow$ to represent convergence in distribution and suppose that $X_n \Rightarrow N(0,\sigma_1)$ and $Y_n \Rightarrow N(0,\sigma_2)$, and $X_n$ and $Y_n$ are independent, then we all know that $(X_n-Y_n)/\sqrt{\sigma^2_1+\sigma^2_2} \Rightarrow N(0,1)$. My question is that can we give a concentration bound for the convergence rate of such result?
That is suppose we know that $X_n$ is independent of $Y_n$, $|P(X_n/\sigma_1 \leq x) - P(Z \leq x)| \leq c_1(n)$ and $|P(Y_n/\sigma_2 \leq x) - P(Z \leq x)| \leq c_2(n)$, what will be the convergence rate (written in the form of $c_1(n)$ and $c_2(n)$) for $|P \Big((X_n-Y_n)/\sqrt{\sigma^2_1+\sigma^2_2} \leq x \Big) - P(Z \leq x)|$? Here, $Z \sim N(0,1)$ and for now we can only consider a fixed $x$ for simplicity instead of a uniform bound.
Thank you very much!