Berry-Esseen Theorem states that the rate of convergence of the probability distribution of normalized sample mean converges to Gaussian at rate $O(1/\sqrt{n})$ (given that certain conditions are met, s.t. independence and finite absolute third moment.)
The theorem is given in terms of maximum discrepancy between the cdf's. For example, the simple version involving a sequence of i.i.d. random variables $\{X_i\}$ with mean zero, variance $\sigma^2$, absolute third moment $\rho$, and $F_n$ being the cdf of $\frac{\sum_{i=1}^nX_i}{\sqrt{n}\sigma}$:
$\sup_x|F_n(x)-\Phi(x)|\leq \frac{C\rho}{\sigma^3\sqrt{n}}$
where $C$ is a constant.
I am wondering if there is a convergence result for the probability density function of $\frac{\sum_{i=1}^nX_i}{\sqrt{n}\sigma}$ (assuming it exists). Does the discrepancy with the pdf of the Gaussian decrease as $\sqrt{n}$?