Question

I have found an article "A central limit theorem for independent summands with infinite variances" here:http://academic2.american.edu/~jpnolan/stable/stable.html

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