What characteristics should a distribution have for CLT to work?

Question

If a "distribution" is constant, then CLT is not going to work, obviously. However, even if it is not a constant, but variance is very small, the distribution of the sums is still not normal. For example:

import numpy as np
from collections import Counter

a = np.zeros(1000)
a[0] = 10

samples = [np.sum(np.random.choice(a, 100)) for _ in xrange(100000)]

Result:

>>> Counter(samples)
Counter({0.0: 90339, 10.0: 9141, 20.0: 500, 30.0: 20}) # not normal

Is there any general property or a test that one can do on a distribution to see whether sample measures yield a normal distribution.

Of course I could do a normality test after the fact, but I am more interested about the property of the distribution.

Answer 1

As far as I know, it is easier to actually look for non-gaussianity/normality than gaussianity/normality.

As I explained in this other post, there are several ways of doing this, where the most intuitive is the search for higher-order cummulants in your data, because the gaussian distribution is the only one that has a finite number of non-zero cummulants (this is a theorem known as Marcinkiewicz's theorem). However, this is not recommended because it is computationally expensive and, of course, calculating every cummulant is not possible in real life.

One other way of measuring (and therefore testing for) non-gaussianity that has been particularly useful in Independant Component Analysis (an application where you need to measure the degree of non-gaussianity of samples) is negentropy. For an introduction on these measures, see these notes by Hyvärinen on the subject which is an extract of the paper by Hyvärinen & Oja (2000) on Independant Component Analysis. If you are interested, search for his papers on efficient ways of calculating negentropy. Also, in this post there is an actual derivation of Negentropy if you are looking for it.

Answer 2

The assumptions of the Classical Central Limit Theorem:

Let $X_1,X_2,...,X_n$ be a sequence of $n$ independent identically distributed random variables with probability density function $f$, mean $\mu$, and variance $\sigma^2$. Moreover, you need to assume that $0 < \sigma < \infty$ so that you have finite variance and that your random variables are truly random and not constants.

If you have what I have stated then the Central Limit Theorem will hold.

Note: This says nothing about the speed of convergence. Convergence is an asymptotic property (in the infinite limit). Convergence as I know it depends on the class of tails your density function has. Even for something with finite variance like the t-distribution (for proper exponent), convergence is not necessarily quick as compared to the continuous uniform distribution.