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Convergence of the Fourth Moment and Infinite Divisibility
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.Comment: 10 page
On a class of explicit Cauchy-Stieltjes transforms related to monotone stable and free Poisson laws
We consider a class of probability measures which have
explicit Cauchy-Stieltjes transforms. This class includes a symmetric beta
distribution, a free Poisson law and some beta distributions as special cases.
Also, we identify as a free compound Poisson law with
L\'{e}vy measure a monotone -stable law. This implies the free infinite
divisibility of . Moreover, when symmetric or positive,
has a representation as the free multiplication of a free
Poisson law and a monotone -stable law. We also investigate the free
infinite divisibility of for . Special cases
include the beta distributions which are
freely infinitely divisible if and only if .Comment: Published in at http://dx.doi.org/10.3150/12-BEJ473 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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