# 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)?

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:

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/

• Thank you for the answer. Unfortunately I cannot check the article for next few days (I do not have access from home and I will not be in my library). Could you please tell what the main result of this paper is? Nov 17, 2009 at 18:12
• I couldn't find anything beyond the abstract and the first page. But based on the discussion in the abstract I think that when the terms become identical it is a generalization of the central limit theorem involving stable processes. I have added a couple of references to stable processes one of them talks about the result of Gnedenko and Kolmogorov. I think that when the terms of Govindarajulu's theorem become identical the thereom becomes identical to the result mentioned in the previous sentence. Nov 17, 2009 at 20:43
• Thanks. When I will check this paper I will write something about it here to. Nov 18, 2009 at 9:51

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

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.

• I think that Feller considers only i.i.d. variables. Nov 17, 2009 at 18:07

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.

• More generally, there are "stable distributions" (en.wikipedia.org/wiki/Stable_distribution) that have the property that a sum of n of them, scaled by n^(1/alpha), has the same distribution as the original. Nov 17, 2009 at 18:15
• I not quite understand. What do you mean by "nice properties"? Some properties of r.v. X_n? Or something else? In the first case I would like to assume about X_n as little as possible. Probably something about the tails decay. Nov 17, 2009 at 18:21
• By nice properties I mean properties it would need to have to be considered an analog of the CLT Nov 17, 2009 at 19:48
• I still not quite understand. The assumptions in the CLT are that that X_n are square integrable (loosely speaking) which more or less is eqivalent to the fact that their tails decays like o(t^{-2}). Nov 18, 2009 at 9:54

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

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

• 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.

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.