86 research outputs found

### 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 $\mu_{s,r}^{\alpha}$ 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 $\mu_{s,2}^{\alpha}$ as a free compound Poisson law with
L\'{e}vy measure a monotone $\alpha$-stable law. This implies the free infinite
divisibility of $\mu_{s,2}^{\alpha}$. Moreover, when symmetric or positive,
$\mu_{s,2}^{\alpha}$ has a representation as the free multiplication of a free
Poisson law and a monotone $\alpha$-stable law. We also investigate the free
infinite divisibility of $\mu_{s,r}^{\alpha}$ for $r\neq2$. Special cases
include the beta distributions $B(1-\frac{1}{r},1+\frac{1}{r})$ which are
freely infinitely divisible if and only if $1\leq r\leq2$.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

### Free Subordination and Belinschi-Nica Semigroup

We realize the Belinschi-Nica semigroup of homomorphisms as a free
multiplicative subordination. This realization allows to define more general
semigroups of homomorphisms with respect to free multiplicative convolution.
For these semigroups we show that a differential equation holds, generalizing
the complex Burgers equation. We give examples of free multiplicative
subordination and find a relation to the Markov-Krein transform, Boolean stable
laws and monotone stable laws. A similar idea works for additive subordination,
and in particular we study the free additive subordination associated to the
Cauchy distribution and show that it is a homomorphism with respect to
monotone, Boolean and free additive convolutions.Comment: 21 pages, minor corrections, Complex Analysis and Operator Theory,
First online: 16 October 201

### Convergence of the Fourth Moment and Infinite Divisibility: Quantitative estimates

We give an estimate for the Kolmogorov distance between an infinitely
divisible distribution (with mean zero and variance one) and the standard
Gaussian distribution in terms of the difference between the fourth moment and
3. In a similar fashion we give an estimate for the Kolmogorov distance between
a freely infinitely divisible distribution and the Semicircle distribution in
terms of the difference between the fourth moment and 2.Comment: 12 page

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