Geodesic
Regular Variation and the Functional Central Limit Theorem for Heavy Tailed Random Vectors
Publications de l'Institut Mathématique, _N_S_71 (2002) no. 85, p. 55 .

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Multivariable regular variation is used, along with the martingale central limit theorem, to give a very simple proof that the partial sum process for a sequence of independent, identically distributed random vectors converges to a Brownian motion whenever the summands belong to the generalized domain of attraction of a normal law. This includes the heavy tailed case, where the covariance matrix is undefined because some of the marginals have infinite variance.
DOI : 10.2298/PIM0271055M
Classification : 60F17 62E20
Keywords: martingale, invariance principle, Donsker's Theorem, partial sum process, generalized domain of attraction, operator normalization 
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Mark M. Meerschaert; Steven J. Sepanski. Regular Variation and the Functional Central Limit Theorem for Heavy Tailed Random Vectors. Publications de l'Institut Mathématique, _N_S_71 (2002) no. 85, p. 55 . doi : 10.2298/PIM0271055M. http://geodesic.mathdoc.fr/articles/10.2298/PIM0271055M/

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