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Takahiro Hasebe

Publications and source records attributed to Takahiro Hasebe.

At least 55 records · Page 3Linked to original sources
arXiv2016

Unimodality for free Lévy processes

We will prove that: (1) A symmetric free Lévy process is unimodal if and only if its free Lévy measure is unimodal; (2) Every free Lévy process with boundedly supported Lévy measure is unimodal in sufficiently large time. (2) is completely different property from classical Lévy processes. On the other hand, we find a free Lévy process such that its marginal distribution is not unimodal for any time $s>0$ and its free Lévy measure does not have a bounded support. Therefore, we conclude that the boundedness of the support of free Lévy measure in (2) cannot be dropped. For the proof we will (almost) characterize the existence of atoms and the existence of continuous probability densities of marginal distributions of a free Lévy process in terms of Lévy--Khintchine representation.

math.PR↗
arXiv2016

Radial Bargmann representation for the Fock space of type B

Let $ν_{α,q}$ be the probability and orthogonality measure for the $q$-Meixner-Pollaczek orthogonal polynomials, which has appeared in \cite{BEH15} as the distribution of the $(α,q)$-Gaussian process (the Gaussian process of type B) over the $(α,q)$-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of $ν_{α,q}$. Our main results cover not only the representation of $q$-Gaussian distribution by \cite{LM95}, but also of $q^2$-Gaussian and symmetric free Meixner distributions on $\mathbb R$. In addition, non-trivial commutation relations satisfied by $(α,q)$-operators are presented.

math.FA↗
arXiv2015

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.

math.PR↗
arXiv2014

Fock space associated to Coxeter group of type B

In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an $(α,q)$-Fock space, which satisfy the commutation relation $$ b_{α,q}(x)b_{α,q}^\ast(y)-qb_{α,q}^\ast(y)b_{α,q}(x)=\langle x, y\rangle I+α\langle \overline{x}, y \rangle q^{2N}, $$ where $x,y$ are elements of a complex Hilbert space with a self-adjoint involution $x\mapsto\bar{x}$ and $N$ is the number operator with respect to the grading on the $(α,q)$-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators $b_{α,q}(x)+b_{α,q}^\ast(x)$ with respect to the vacuum expectation becomes a generalized Gaussian distribution, in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions. Our generalized Gaussian distribution associates the orthogonal polynomials called the $q$-Meixner-Pollaczek polynomials, yielding the $q$-Hermite polynomials when $α=0$ and free Meixner polynomials when $q=0$.

math.FA↗
arXiv2014

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.

math.PR↗
arXiv2014

Relations between cumulants in noncommutative probability

We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.

math.CO↗
arXiv2014

Classical Scale Mixtures of Boolean Stable Laws

We study Boolean stable laws, $\mathbf{b}_{α,ρ}$, with stability index $α$ and asymmetry parameter $ρ$. We show that the classical scale mixture of $\mathbf{b}_{α,ρ}$ coincides with a free mixture and also a monotone mixture of $\mathbf{b}_{α,ρ}$. For this purpose we define the multiplicative monotone convolution of probability measures, one is supported on the positive real line and the other is arbitrary. We prove that any scale mixture of $\mathbf{b}_{α,ρ}$ is both classically and freely infinitely divisible for $α\leq1/2$ and also for some $α>1/2$. Furthermore, we show the multiplicative infinite divisibility of $\mathbf{b}_{α,1}$ with respect classical, free and monotone convolutions. Scale mixtures of Boolean stable laws include some generalized beta distributions of second kind, which turn out to be both classically and freely infinitely divisible. One of them appears as a limit distribution in multiplicative free laws of large numbers studied by Tucci, Haagerup and Möller. We use a representation of $\mathbf{b}_{α,1}$ as the free multiplicative convolution of a free Bessel law and a free stable law to prove a conjecture of Hinz and Młotkowski regarding the existence of the free Bessel laws as probability measures. The proof depends on the fact that $\mathbf{b}_{α,1}$ has free divisibility indicator 0 for $1/2<α$.

math.PR↗
arXiv2014

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.

math.OA↗
arXiv2014

On free infinite divisibility for classical Meixner distributions

We prove that symmetric Meixner distributions, whose probability densities are proportional to $|Γ(t+ix)|^2$, are freely infinitely divisible for $0<t\leq\frac{1}{2}$. The case $t=\frac{1}{2}$ corresponds to the law of Lévy's stochastic area whose probability density is $\frac{1}{\cosh(πx)}$. A logistic distribution, whose probability density is proportional to $\frac{1}{\cosh^2(πx)}$, is freely infinitely divisible too.

math.PR↗
arXiv2014

Classical and free infinite divisibility for Boolean stable laws

We completely determine the free infinite divisibility for the Boolean stable law which is parametrized by a stability index $α$ and an asymmetry coefficient $ρ$. We prove that the Boolean stable law is freely infinitely divisible if and only if one of the following conditions holds: $0<α\leq\frac{1}{2}$; $\frac{1}{2}<α\leq\frac{2}{3}$ and $2-\frac{1}α\leqρ\leq \frac{1}α-1$; $α=1,~ρ=\frac{1}{2}$. Positive Boolean stable laws corresponding to $ρ=1$ and $α\leq \frac{1}{2}$ have completely monotonic densities and they are both freely and classically infinitely divisible. We also show that continuous Boolean convolutions of positive Boolean stable laws with different stability indices are also freely and classically infinitely divisible. Boolean stable laws, free stable laws and continuous Boolean convolutions of positive Boolean stable laws are non-trivial examples whose free divisibility indicators are infinity. We also find that the free multiplicative convolution of Boolean stable laws is again a Boolean stable law.

math.PR↗
arXiv2013

On a class of explicit Cauchy-Stieltjes transforms related to monotone stable and free Poisson laws

We consider a class of probability measures $μ_{s,r}^α$ 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 $μ_{s,2}^α$ as a free compound Poisson law with Lévy measure a monotone $α$-stable law. This implies the free infinite divisibility of $μ_{s,2}^α$. Moreover, when symmetric or positive, $μ_{s,2}^α$ has a representation as the free multiplication of a free Poisson law and a monotone $α$-stable law. We also investigate the free infinite divisibility of $μ_{s,r}^α$ 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 $1\leq r\leq2$.

math.PR↗
arXiv2013

Semigroups related to additive and multiplicative, free and Boolean convolutions

Belinschi and Nica introduced a composition semigroup on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know whether a probability measure is freely infinitely divisible or not. In this paper we further investigate this indicator, introduce a multiplicative version of it and are able to show many properties. Specifically, on the first half of the paper, we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to infinity; we derive an upper bound of the indicator in terms of Jacobi parameters. This upper bound is achieved only by free Meixner distributions. We also prove Bozejko's conjecture which says the Boolean power of a probability measure mu by 0 < t < 1 is freely infinitely divisible if mu is so. In the other half of this paper, we introduce an analogous composition semigroup for multiplicative convolutions and define free divisibility indicators for these convolutions. Moreover, we prove that a probability measure on the unit circle is freely infinitely divisible concerning the multiplicative free convolution if and only if the indicator is not less than one. We also prove how the multiplicative divisibility indicator changes under free and Boolean powers and then the multiplicative analogue of Bozejko's conjecture. We include an appendix, where the Cauchy distributions and point measures are shown to be the only fixed points of the Boolean-to-free Bercovici-Pata bijection.

math.PR↗
arXiv2013

No Zero Divisor for Wick Product in $(S)^{\ast}$}

In White Noise Analysis (WNA), various random quantities are analyzed as elements of $(S)^{\ast}$, the space of Hida distributions ([1]). Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On $(S)^{\ast}$, the Wick product is defined in terms of the $\mathcal{S}$-transform. We have found such a remarkable property that the Wick product has no zero devisors among Hida distributions. This result is a WNA version of Titchmarsh's theorem and is expected to play fundamental roles in developing the \textquotedblleft operational calculus\textquotedblright in WNA along the line of Mikusiński's version for solving differential equations.

math-ph↗
arXiv2012

On the Law of Free Subordinators

We study the freely infinitely divisible distributions that appear as the laws of free subordinators. This is the free analog of classically infinitely divisible distributions supported on [0,\infty), called the free regular measures. We prove that the class of free regular measures is closed under the free multiplicative convolution, t-th boolean power for $0\leq t\leq 1$, t-th free multiplicative power for $t\geq 1$ and weak convergence. In addition, we show that a symmetric distribution is freely infinitely divisible if and only if its square can be represented as the free multiplicative convolution of a free Poisson and a free regular measure. This gives two new explicit examples of distributions which are infinitely divisible with respect to both classical and free convolutions: χ^2(1) and F(1,1). Another consequence is that the free commutator operation preserves free infinite divisibility.

math.PR↗
arXiv2011

Fourier and Cauchy-Stieltjes transforms of power laws including stable distributions

We introduce a class of probability measures whose densities near infinity are mixtures of Pareto distributions. This class can be characterized by the Fourier transform which has a power series expansion including real powers, not only integer powers. This class includes stable distributions in probability and also non-commutative probability theories. We also characterize the class in terms of the Cauchy-Stieltjes transform and the Voiculescu transform. If the stability index is greater than one, stable distributions in probability theory do not belong to that class, while they do in non-commutative probability.

math.PR↗
arXiv2011

Joint cumulants for natural independence

Many kinds of independence have been defined in non-commutative probability theory. Natural independence is an important class of independence; this class consists of five independences (tensor, free, Boolean, monotone and anti-monotone ones). In the present paper, a unified treatment of joint cumulants is introduced for natural independence. The way we define joint cumulants enables us not only to find the monotone joint cumulants but also to give a new characterization of joint cumulants for other kinds of natural independence, i.e., tensor, free and Boolean independences. We also investigate relations between generating functions of moments and monotone cumulants. We find a natural extension of the Muraki formula, which describes the sum of monotone independent random variables, to the multivariate case.

math.OA↗