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There is a well-known central limit theorem for as a stationary sequences.

If $( X_n )_n$ is a sationary 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 - "Fundations of probability" and Jacod, Shiryaev - "Limit theorems ..."). Does anybody know any good reference (e.g. a textbook)?

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4 Answers

I have found an article "A central limit theorem for independent summands with infinite variances" here:

http://www.springerlink.com/content/n462221853449467/

Also see page 235 of Financial modelling with jump processes more information here:

http://books.google.com/books?id=3X2j2Gjv-oMC&dq=central+limit+theorem+%22infinite+variance%22&lr=&source=gbs_navlinks_s

There is a generalization of the central limit theorem involving stable distributions which involves infinite variance see the following:

http://en.wikipedia.org/wiki/Stable_distribution

More on stable distributions:

http://academic2.american.edu/~jpnolan/stable/stable.html

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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? –  Piotr Miłoś Nov 17 '09 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. –  Kristal Cantwell Nov 17 '09 at 20:43
    
Thanks. When I will check this paper I will write something about it here to. –  Piotr Miłoś Nov 18 '09 at 9:51
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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.

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I think that Feller considers only i.i.d. variables. –  Piotr Miłoś Nov 17 '09 at 18:07
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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

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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.

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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. –  Michael Lugo Nov 17 '09 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. –  Piotr Miłoś Nov 17 '09 at 18:21
    
By nice properties I mean properties it would need to have to be considered an analog of the CLT –  Jonathan Kariv Nov 17 '09 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}). –  Piotr Miłoś Nov 18 '09 at 9:54
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