Let $X_1,...,X_n$ be iid positive random variables and let $Y_i$ represent the truncation of $X_i$ to $(0,b]$, with $b$ a known parameter. I'd like to use the CLT to estimate the distribution of $\sum_i Y_i$ and then find a bound on the error using the Berry-Esseen theorem. If I understand it correctly, the bound to the error in my case is equal to $\frac{C\rho_Y(b)}{\sigma_Y^3(b)\sqrt{n}}$ with $\rho_Y(b)=E[|Y_i-E[Y_i]|^3]$ and $\sigma_Y(b)=E[(Y_i-E[Y_i])^2]$.
While this bound is interesting, I'd prefer to have a bound independent of $b$. I was thinking about using $\frac{C\rho_X}{\sigma_X^3\sqrt{n}}$ since it would be the tightest bound possible for a general $b$. My numerical results seem to show that it is in fact the case but I'm having trouble to prove that $\frac{\rho_Y(b)}{\sigma_Y^3(b)}\leq \frac{\rho_X}{\sigma_X^3}$.
Is there a way to prove that? Or does there exists another bound than Berry-Esseen for my particular case?