CLT for stationary sequences with infinite variance There is a well-known central limit theorem for as a stationary sequences.
If $( X_n )_n$ is a stationary sequence and $E X_n=0$ then under suitable mixing conditions the sequence $S_n := n^{-1/2}\sum_{i=1}^n X_i$ converges weakly to a normal random variable.
(This is very simplified version of Theorem 7.7.6 of Durrett's Probability Theory ...).
This theorem is very nice but works only when $X_n$ have finite variance (the mixing conditions above require it).
I am almost sure that there must be an analogue of this theorem for variables with infinite variance (of course the sequence will converge to a stable variable). But I couldn't find it in popular textbooks (I check Durrett - "Probability theory...", Kallemberg - "Foundations of probability" and Jacod, Shiryaev - "Limit theorems ..."). Does anybody know any good reference (e.g. a textbook)?
 A: I have found an article "A central limit theorem for independent summands with infinite variances" here:
https://doi.org/10.1007/BF03048130
Also see page 235 of Financial modelling with jump processes more information here:
https://books.google.com/books?id=3X2j2Gjv-oMC&lpg=PP1&pg=PA235#v=onepage&q&f=false
There is a generalization of the central limit theorem involving stable distributions which involves infinite variance see the following:
https://en.wikipedia.org/wiki/Stable_distribution
More on stable distributions:
https://edspace.american.edu/jpnolan/stable/
A: There is a small literature on these topics, mostly from the 90s. The names to look for are A. Jakubowski and M. Kobus (alone and together). For an example see Theorem 1.2 in1 https://www.sciencedirect.com/science/article/pii/S0047259X85710111.
Unfortunately, I am not aware of neither a good general treatment not a textbook treatment.
It is hard to believe that a very general theorem exists with convergence to stable limits because you need to control regular variation of the tails - a problem that does not quite occur in the Gaussian case.
1M. Kobus,
Generalized Poisson Distributions as Limits of Sums for Arrays of Dependent Random Vectors,
Journal of Multivariate Analysis,
Volume 52, Issue 2, 1995, Pages 199-244,
https://doi.org/10.1006/jmva.1995.1011
A: Feller vol 2 Chapter IX should do the trick.  
A more modern reference--which I have not looked at--is
Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Stochastic Modeling) (Hardcover)
~ Gennady Samorodnitsky
Not sure if this covers convergence issues or not.  
A: Just to be more explicit about what PeterR saud. The sum of n Cauchy random varibles (scaled by 1/n) is a cauchy. Maybe it would be helpful if you defined what nice properties you'd like your analog to have. 
A: For IID rv's see Durrett's "Probability: theory & Examples" Section 2.7 Stable Laws
The more general (non-independent) case, is probably in "Stable non-Gaussian random processes: stochastic models with infinite variance" By Gennady Samorodnitsky, Murad S. Taqqu
A: Try to find:


*

*Barbosa, E.G. & Dorea, C.C.Y. "A note on the Lindemberg condition for Convergence to Stable Laws in Mallows Distance", Bernoulli, 2009.


or 


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*Dorea, C.C.Y., Ferreira, D.B. "Conditions for Equivalence Between Mallows
Distance and Convergence to Stable Laws", 2009.


In the first case, I don't remember the exact Vol.
In the second case, I don't know the especific magazine or periodic.
This paper's sources was written in portuguese language (it doesn't help...), but you can find them (the papers, not the sources), using the titles above, at some periodic.
I am suggest them, because they present results similar to CLT, when the variance is infinite. Its enough to remember that convergence in Mallows Distance implies weak convergence. 
A: In reference to ofer zeitouni's answer, you might want to look at this paper:


*

*Davis, R. A., Hsing, T., Point process and partial sum convergence for weakly dependent random variables with infinite variance, Ann. Probab., 1995, 23, 879-917


You might also want to have a look this paper:


*

*Bartkiewicz, K., Jakubowski, A., Mikosch, T., Wintenberger, O., Stable limits for sums of dependent infinite variance random variables, Probab. Theory Related Fields, 2011, 150, 337-372


which gives a good review of what has been done in this field and signposts relevant literature.
Unfortunately, I haven't come across any good textbooks about this topic. As far as I have seen, textbooks usually just treat the i.i.d. case, but the weakly dependent case in quite a limited way.
