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:


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.


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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