# CLT for stationary sequences with infinte variance

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

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:

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

More on stable distributions:

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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

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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