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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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6 Answers 6

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

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

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

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 in http://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.

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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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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.

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