TITLE: When does a pointwise CLT hold?
QUESTION [9 upvotes]: Let $X$ be a random variable with mean $0$ and variance $1$, and let $X_1, X_2, X_3, \dots$ be iid copies of $X$.  Under what conditions can we say that the density of $\frac{X_1+\dots+X_n}{\sqrt{n}}$ converges pointwise to $N(0,1)$?
In particular, when can I say that for any sequence $\epsilon_n \rightarrow 0$ we have 
$$\frac{P(|\frac{X_1+\dots+X_n}{\sqrt{n}}|<\epsilon_n)-P(|N(0,1)|<\epsilon_n)}{\epsilon_n} \rightarrow 0?$$
In flavor this is somewhat similar to what I've seen termed as "local limit theorems", except a little bit stronger; for example if $X$ is a Bernoulli variable the above would not hold (take $\epsilon_n=2^{-n}$).  My guess would be that a sufficient condition would be for the usual CLT to hold and $X$ to have bounded density functions, though I haven't seen this cited anywhere.

REPLY [2 votes]: Feller states the Berry-Esseen theorem in the following way. Let the $X_k$ be independent variables with a common distribution $F$ such that $$E[X_k]=0, E[X_k^2]=\sigma^2>0, E[|X_k|^3]=\rho<\infty,$$ and let $F_n$ stand for the distribution of the normalized sum $$(X_1+ \dots X_n)/(\sigma \sqrt{n}).$$ Then for all $x$ and $n$ $$|F_n(x)-N(x)| \leq \frac{3\rho}{\sigma^3 \sqrt{n}}.$$
The expression you are interested in is 
  $$\left|\frac{F_n(\epsilon)-F_n(-\epsilon)-N(\epsilon)+N(-\epsilon)}{\epsilon}\right|,$$ 
which is less than
  $$\left| \frac{F_n(\epsilon)-N(\epsilon)}{\epsilon} \right| + \left| \frac{F_n(-\epsilon)-N(-\epsilon)}{\epsilon} \right|,$$
which by Berry-Esseen is bounded by
  $$2\frac{3\rho}{\epsilon \sigma^3 \sqrt{n}}.$$
So, if $\epsilon\sqrt{n}$ goes to infinity, then you are good.
I realize this isn't what you asked, in that you wanted conditions on $X$, and this instead gives you conditions on $\epsilon_n$. Still, perhaps it'll help.
Reference: Feller, An Introduction to Probability Theory and Its Applications, Volume II, Chapter XVI.5.