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 $\sqrt{n}$?

  • $\begingroup$ Convergence of the densities is not guaranteed in general. A useful search term is "local limit theorems". $\endgroup$ – cardinal Nov 16 '11 at 20:24
  • $\begingroup$ Well, someone on MathOverflow asked this same question and got a reference for an answer: mathoverflow.net/questions/75829/… I'll try write an actual answer to this later. $\endgroup$ – M.B.M. Nov 16 '11 at 20:24
  • $\begingroup$ Petrov is a good reference. Some modern more-general probability theory books have short introductions to this topic. If I recall, there is a section on this in Durrett. $\endgroup$ – cardinal Nov 16 '11 at 20:26
  • $\begingroup$ @cardinal, Durrett wrote so many textbooks that to provide the title seems mandatory. $\endgroup$ – Did Nov 16 '11 at 21:15
  • $\begingroup$ Unfortunately, Petrov's book is so old, my university's only copy is in the book repository (takes 3 days to get it from there.) Seconding the motion for @cardinal to post specific title of Durrett's book. (I may be able to use the Bounded Convergence Theorem for my problem... not sure yet.) $\endgroup$ – M.B.M. Nov 16 '11 at 21:27

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