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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 $\sqrt{n}$?

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    I got Petrov's book. It's really neat, basically, it methodically goes through a lot of theorems relating to the sums of random variables, including the local limit theorems. For the local limit theorems, it starts with the results requiring the most restrictive condition (bounded) on the distributions for the summands, and then relaxes it (but at the cost of adding more conditions.) I don't think I can answer this question other than tell folks to find Petrov's book.2011-11-19

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