We prove some invariance principles for processes which generalize FARIMAprocesses, when the innovations are in the domain of attraction of anonGaussian stable distribution. The limiting processes are extensions of thefractional L\'evy processes. The technique used is interesting in itself; itextends an older idea of splitting a sample into a central part and an extremeone, analyzing each part with different techniques, and then combining theresults. This technique seems to have the potential to be useful in otherproblems in the domain of nonGaussian stable distributions.
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