There are a variety of forms of a CLT.
One form (see Chung, A course in probability theory, 2nd ed., p. 169, Theorem 6.4.4) is applicable to a sequence of independent random variables with a common distribution. If the mean and variance of that common distribution are finite and, in the case of the variance, positive, there is a CLT for the sum of those random variables.
Chung provides criteria where, suitably scaled, sums of dependent variables also converge to the normal distribution (see p.214 and theorem 7.3.1)
Chung provides additional theorems discussing other ways a CLT holds under various conditions on the distribution(s) of the underlying random variables.
There are other CLTs, such as one used for statistics derived from samples of finite populations.
I highly recommend the book, Counterexamples in Probability and Statistics, Romano and Siegel, Chapman & Hall, 1986 (there might be a newer version). In that version, see examples 5.4.3 -5.4.7 for examples where the classical CLT (the first one mentioned above) fails. Conversely, example 5.48 shows how a Gaussian distribution in the limit is possible when using a different normalizing factor for distributions with infinite variance.
In short, various scaled sums of random variables converge in distribution to a Normal distribution but there are different situations where this can arise.
------------------------------
David Wilson
------------------------------
Original Message:
Sent: 02-07-2018 17:27
From: Manolis Antonoyiannakis
Subject: Central Tendency Theorem
Dear All,
I have one question: Under what conditions does the Central Tendency Theorem fail? In particular, does it fail for the Pareto distribution? Is there a reference on this?
Thank you very much in advance,
Manolis
____________________
Manolis Antonoyiannakis
Associate Editor, Physical Review B
Bibliostatistics Analyst, The American Physical Society