185 research outputs found

### Free infinite divisibility for beta distributions and related ones

We prove that many of beta, beta prime, gamma, inverse gamma, Student t- and
ultraspherical distributions are freely infinitely divisible, but some of them
are not. The latter negative result follows from a local property of
probability density functions. Moreover, we show that the Gaussian,
ultraspherical and many of Student t-distributions have free divisibility
indicator 1.Comment: 37 pages, 6 figures, slightly different from the published versio

### Conditionally monotone independence I: Independence, additive convolutions and related convolutions

We define a product of algebraic probability spaces equipped with two states.
This product is called a conditionally monotone product. This product is a new
example of independence in non-commutative probability theory and unifies the
monotone and Boolean products, and moreover, the orthogonal product. Then we
define the associated cumulants and calculate the limit distributions in
central limit theorem and Poisson's law of small numbers. We also prove a
combinatorial moment-cumulant formula using monotone partitions. We investigate
some other topics such as infinite divisibility for the additive convolution
and deformations of the monotone convolution. We define cumulants for a general
convolution to analyze the deformed convolutions.Comment: 41 pages; small mistakes revised; to appear in Infin. Dimens. Anal.
Quantum Probab. Relat. To

### Analytic continuations of Fourier and Stieltjes transforms and generalized moments of probability measures

We consider analytic continuations of Fourier transforms and Stieltjes
transforms. This enables us to define what we call complex moments for some
class of probability measures which do not have moments in the usual sense.
There are two ways to generalize moments accordingly to Fourier and Stieltjes
transforms; however these two turn out to coincide. As applications, we give
short proofs of the convergence of probability measures to Cauchy distributions
with respect to tensor, free, Boolean and monotone convolutions.Comment: 13 pages; to appear in Journal of Theoretical Probabilit

### Unimodality for free multiplicative convolution with free normal distributions on the unit circle

We study unimodality for free multiplicative convolution with free normal
distributions $\{\lambda_t\}_{t>0}$ on the unit circle. We give four results on
unimodality for $\mu\boxtimes\lambda_t$: (1) if $\mu$ is a symmetric unimodal
distribution on the unit circle then so is $\mu\boxtimes \lambda_t$ at any time
$t>0$; (2) if $\mu$ is a symmetric distribution on $\mathbb{T}$ supported on
$\{e^{i\theta}: \theta \in [-\varphi,\varphi]\}$ for some $\varphi \in
(0,\pi/2)$, then $\mu \boxtimes \lambda_t$ is unimodal for sufficiently large
$t>0$; (3) ${\bf b} \boxtimes \lambda_t$ is not unimodal at any time $t>0$,
where ${\bf b}$ is the equally weighted Bernoulli distribution on $\{1,-1\}$;
(4) $\lambda_t$ is not freely strongly unimodal for sufficiently small $t>0$.
Moreover, we study unimodality for classical multiplicative convolution (with
Poisson kernels), which is useful in proving the above four results.Comment: 19 pages, 4 figure

### The Monotone Cumulants

In the present paper we define the notion of generalized cumulants which
gives a universal framework for commutative, free, Boolean, and especially,
monotone probability theories. The uniqueness of generalized cumulants holds
for each independence, and hence, generalized cumulants are equal to the usual
cumulants in the commutative, free and Boolean cases. The way we define
(generalized) cumulants needs neither partition lattices nor generating
functions and then will give a new viewpoint to cumulants. We define ``monotone
cumulants'' in the sense of generalized cumulants and we obtain quite simple
proofs of central limit theorem and Poisson's law of small numbers in monotone
probability theory. Moreover, we clarify a combinatorial structure of
moment-cumulant formula with the use of ``monotone partitions''.Comment: 13 pages; minor changes and correction

### On operator-valued monotone independence

We investigate operator-valued monotone independence, a noncommutative
version of independence for conditional expectation. First we introduce
operator-valued monotone cumulants to clarify the whole theory and show the
moment-cumulant formula. As an application, one can obtain an easy proof of
Central Limit Theorem for operator-valued case. Moreover, we prove a
generalization of Muraki's formula for the sum of independent random variables
and a relation between generating functions of moments and cumulants.Comment: Proof of Theorem 3.4 is explaine

### 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

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