Is there a "central distribution" for distributions for which the CLT doesn't apply?

Question

The central limit theorems state roughly that under a certain set of properties of a sampling process, the distribution of a statistic from that sample will converge in distribution to the normal distribution.

As the canonical example, let me take the basic central limit theorem: If we take i.i.d. samples $X_1,X_2...$ from a distribution $f$, then the sample average will converfe in diatribution to the normal distribution if

1. The mean of $f$ exists.

2. The variance of $f$ exists.

My question is:

Assume that $f$ does not have a variance, and that the sample average of an i.i.d.sample from $f$ does not converge in distribution to the normal distribution.

Are there situations where this sample average nevertheless converges in distribution to some other (non-normal) distribution?

In other words, is there a "central limit theorem" for distributions that don't converge to the nornal distribution?

Answer 1

An answer is already found in the wikipedia link from the first comment by Smith.

Q: Are there situations where this sample average nevertheless converges in distribution to some other (non-normal) distribution?

A: Yes, iff a distribution is a stable distribution then it is a limit to sums of the type:

$$\zeta_n = \frac{\xi_1 + \xi_2 + \dots + \xi_n}{B_n} - A_n $$

with the $\xi$ independent and identically distributed random variables, $B_n>0$ and $\vert A_n\vert<\infty$

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The type of distribution laws for $\xi$ that let the above sum converge to a stable distribution (the domain of attraction for that distribution) are described by a theorem in 'Limit distributions for sums of independent random variables' by Gnedenko and Kolmogorov (page 175 in the translated version 1954, link via google)

Theorem 2.* In order that the distribution function F(x) belong to the domain of attraction of a stable law with the characteristic exponent $\alpha$ ($0 \leq \alpha \leq 2$) it is necessary and sufficient that

1) $$\frac{F(-x)}{1-F(x)} \to \frac{c_1}{c_2} \qquad \text{as } k \to \infty$$

2) for every constant $k>0$

$$\frac{1 - F(x) + F(-x)}{1-F(xk) + F(-kx)} \to k^\alpha \qquad \text{as } k \to \infty$$

• *The theorem is attributed to
  • B.V. Gedenko 1939 (no online version available ГНЕДЕНКО, Б. В. К теории областей притяжения устойчивых законов. Ученые записки МГУ, 1939, 2: 30.)
  • and Doeblin 1940, see theorem V in the freely available pdf.
• A slightly more exact description (in comparison to the wikipedia article) is given by https://www.encyclopediaofmath.org/index.php/Attraction_domain_of_a_stable_distribution