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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?

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Sure. Look, e.g., at the supplements to chapter VII in the classical book "Sums of independent random variables" by Petrov. – Yvan Velenik Sep 19 '11 at 9:02
ok, thanks! I just need to find a copy of the book... – Anthony Leverrier Sep 19 '11 at 9:16
Hum. If you search for "Sums of independent random variables" Petrov on, you should find what you need on the first page of results ;) ... – Yvan Velenik Sep 19 '11 at 9:24
indeed! Thanks. – Anthony Leverrier Sep 19 '11 at 9:27
up vote 6 down vote accepted

The magic words are "local limit theorem".

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