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Convergence of the fourth moment and infinite divisibility

Arizmendi O.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

201--212

In this note we prove that, for infinitely divisible laws, convergence of the fourth moment to 3 is sufficient to ensure convergence in law to the Gaussian distribution. Our results include infinitely divisible measures with respect to classical, free, Boolean and monotone convolution. A similar criterion is proved for compound Poissons with jump distribution supported on a finite number of atoms. In particular, this generalizes recent results of Nourdin and Poly (2012).