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 Essen 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?

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?

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 $\frac{\sigma Y_n}{\sqrt{n}}$ (resp. of the standard normal distribution). Then, Berry Essen 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?

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 Essen 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?