# Berry Esseen Theorem for Infinite Absolute Third Moment

The simplest form of the Berry-Esseen theorem states that if $Z_n:=(X_1+\cdots+X_n)/\sqrt{n}$, where $X_i$ are iid with mean 0 and variance 1, and $\rho:=E[|X|^3]$ then:

$$F_{Z_n}(y)=F_{N}(y)+O\left(\frac{1}{\sqrt{n}}\rho\right),$$

where the Big-Oh is uniform in $y$, so that the convergence of the CDF of $Z_n$ to the CDF of the standard normal is also uniform. This is where the typical voodoo of "n>=30 is good enough for CLT" probably comes from.

My question is about what happens when $\rho=\infty$, for example in the case of the Pareto distribution $f(x)=c/x^4$ for $x\geq 1$, or even worse for random variables where $E[X^{2+\epsilon}]=\infty$ for $\epsilon>0$, for example $P(X=i)=\frac{c}{i^3\log^2 i}, i\in\mathbb{N}$.

Clearly the convergence is no longer uniform (except perhaps in special cases?). This is also where the non-uniqueness and weirdness of the Hamburger moment problem enters, because the normal distribution has a finite absolute third moment, and the distribution of $X$ must have non-compact support.

But, are there some interesting characterizations of the kind of bad-behavior that you can expect? The extreme (worst-case-scenario) version of this is the Law of the Iterated logorithm, which says that:

$$\limsup_{n\rightarrow\infty} \frac{Z_n}{\sqrt{\log\log(n)}}=_{as}\sqrt{2}.$$

I'm more interested in the distribution of fluctuations, especially far away from the mean. I've tried searching for "Berry Esseen theorem with infinite absolute third moment", but couldn't find anything pertinent.

Let $F_n(x) = P\left(n^{-1/2}\sum_{i=1}^n X_i<x\right)$ . Let $g()$ be a function that is non-negative, even, and non-decreasing in the interval $x>0$, and such that $x/g(x)$ is non-decreasing in the interval $x>0$. If $E[X_1^2g(X_1)]<\infty$. Then
$$\sup_x \left|F_n(x) - \Phi(x)\right| \leq \frac {A}{g(\sqrt{n})}E[X_1^2g(X_1)]$$ for some universal $A>0$.