CONVERGENCE OF THE FOURTH MOMENT AND INFINITE
DIVISIBILITY
Abstract: 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).
2000 AMS Mathematics Subject Classification: Primary: 46L54; Secondary:
60E07.
Keywords and phrases: Bercovici–Pata bijections, Boolean convolution, free
convolution, monotone convolution, infinite divisibility.