  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.
