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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.

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The Berry -Esseen theorem requires the finiteness of the third moment. A generalization of Berry-Esseen inequality that doesn't, can be found in
Petrov, V. V. (1975). Sums of independent random variables (Vol. 82). Springer Science & Business Media., p. 112 Theorem 5.

For the case of i.i.d. random variables with mean zero and variance 1, the theorem simplifies to the following:

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 $A>0$.

The result can be extended to variables with non-zero and different means, and different variances (see DasGupta, A. (2008). Asymptotic theory of statistics and probability, ch. 11)

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