pr.probability - When does a pointwise CLT hold?

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 $n$ $$|F_n(x)-N(x)| leq frac{3rho}{sigma^3 sqrt{n}}.$$

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
$$2frac{3rho}{epsilon sigma^3 sqrt{n}}.$$
So, if $epsilonsqrt{n}$ 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.