Let $X\_1, X\_2, \cdots$ be i.i.d. random variables with $E(X\_1) = 0, E(X\_1^2) = \sigma^2 >0, E(|X\_1|^3) = \rho < \infty$.
Let $Y\_n = \frac{1}{n} \sum\_{i=1}^n X\_i$ and let us note $F\_n$ (resp. $\Phi$) the *cumulative distribution function* of $\frac{Y\_n \sqrt{n}}{\sigma}$ (resp. of the standard normal distribution).
Then, Berry Esseen theorem states that there exists a positive constant $C$ such that for all $x$ and $n$,
$$|F\_n(x)-\Phi(x)| \leq \frac{C \rho}{\sigma^3 \sqrt{n}}.$$


Are there known conditions on the distribution of $X\_1$ that allow to derive a similar statement for *probability density functions* instead of cumulative distribution functions?



Berry Esseen type result for probability density functions
The magic words are "local limit theorem".


