Researcher Database

Takahiro Hasebe
Faculty of Science Mathematics Mathematics
Associate Professor

Researcher Profile and Settings

Affiliation

  • Faculty of Science Mathematics Mathematics

Job Title

  • Associate Professor

J-Global ID

Research Interests

  • Complex analysis   Levy processes   Free probability   Infinitely divisible distribution   Combinatorics   

Research Areas

  • Natural sciences / Basic analysis
  • Natural sciences / Basic analysis / Free probability

Academic & Professional Experience

  • 2015 Hokkaido University

Research Activities

Published Papers

  • Hasebe, Takahiro, Simon, Thomas, Wang, Min
    ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES 56 (1) 296 - 325 0246-0203 2020/02 [Refereed][Not invited]
     
    We investigate certain analytical properties of the free alpha-stable densities on the line. We prove that they are all classically infinitely divisible when alpha < 1 and that they belong to the extended Thorin class when alpha < 3/4. The Levy measure is explicitly computed for alpha = 1, showing that free 1-stable distributions are not in the Thorin class except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when alpha > 1. In the one-sided case we prove, refining unimodality, that the densities are whale-shaped, that is their successive derivatives vanish exactly once on their support. We also derive several fine properties of spectrally one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, and several intrinsic features of whale-shaped functions.
  • Hasebe, Takahiro, Szpojankowski, Kamil
    COMPLEX ANALYSIS AND OPERATOR THEORY 13 (7) 3091 - 3116 1661-8254 2019/10 [Refereed][Not invited]
     
    We study here properties of free Generalized Inverse Gaussian distributions (fGIG) in free probability. We show that in many cases the fGIG shares similar properties with the classical GIG distribution. In particular we prove that fGIG is freely infinitely divisible, free regular and unimodal, and moreover we determine which distributions in this class are freely selfdecomposable. In the second part of the paper we prove that for free random variables X, Y where Y has a free Poisson distribution one has X=(d) 1/X+Y if and only if X has fGIG distribution for special choice of parameters. We also point out that the free GIG distribution maximizes the same free entropy functional as the classical GIG does for the classical entropy.
  • Hasebe, Takahiro, Sakuma, Noriyoshi, Thorbjornsen, Steen
    INTERNATIONAL MATHEMATICS RESEARCH NOTICES 2019 (6) 1758 - 1787 1073-7928 2019/03 [Refereed][Not invited]
     
    The class of self-decomposable distributions in free probability theory was introduced by Barndorff-Nielsen and Thorbjornsen. It constitutes a fairly large subclass of the freely infinitely divisible distributions, but so far specific examples have been limited to Wigner's semicircle distributions, the free stable distributions, two kinds of free gamma distributions and a few other examples. In this article, we prove that the (classical) normal distributions are freely self-decomposable. More generally it is established that the Askey-Wimp-Kerov distribution mu(c) is freely self-decomposable for any c in [-1, 0]. The main ingredient in the proof is a general characterization of the freely self-decomposable distributions in terms of the derivative of their free cumulant transform.
  • YAMADA Takayuki, MASAMUNE Jun, TERAMOTO Hiroshi, HASEBE Takahiro, KURODA Hirotoshi
    Transactions of the JSME (in Japanese) 一般社団法人 日本機械学会 85 (877) 19 - 00129-19-00129 2019 [Refereed][Not invited]
     

    This paper aims to develop a scheme for geometrical feature constraints in topology optimization for Additive Manufacturing (AM) without support structures based on the Partial Differential Equation (PDE) of geometrical shape features. To begin with, the basic concept of topology optimization and a level set-based topology optimization method are briefly described. Second, the PDE system for geometrical shape features is formulated. Here, aspects of the distribution of state variables are discussed using an analytical solution of the PDE. Based on the discussion, a function indicating the extended normal vector including geometrical singularity points is formulated. Third, geometrical requirements of product shape in AM without support structures – the so-called overhang constraint – are clarified in two-dimensions. A way of extending of the proposed concept to three-dimensional problems is also clarified. Additionally, geometrical singularities in the overhang constraint are discussed. Based on the PDE system and the clarified geometrical requirements, the overhang constraint including geometrical singularities is formulated. A topology optimization problem of the linear elastic problem is formulated considering the overhang constraint. A level set-based topology optimization algorithm is constructed where the Finite Element Method (FEM) is used to solve the governing equation of the linear elastic problem and the PDE, and to update the level set function. Finally, two-dimensional numerical examples are provided to confirm the validity and utility of the proposed method.

  • Hasebe, Takahiro, Huang, Hao-Wei, Wang, Jiun-Chau
    PROBABILITY THEORY AND RELATED FIELDS 172 (3-4) 1081 - 1119 0178-8051 2018/12 [Refereed][Not invited]
     
    In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of self-adjoint commuting random variables in an infinitesimal triangular array are determined. These distributions are characterized by their bi-freely infinite divisibility, and moreover, a transfer principle is established for limit theorems in classical probability theory and Voiculescu's bi-free probability theory. Complete descriptions of bi-free stability are given and fullness of planar probability distributions is studied. All these results reveal one important feature about the theory of bi-free probability that it parallels the classical theory perfectly well. The emphasis in the whole work is not on the tool of bi-free combinatorics but only on the analytic machinery.
  • Benoit Collins, Takahiro Hasebe, Noriyoshi Sakuma
    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 70 (3) 1111 - 1150 0025-5645 2018/07 [Refereed][Not invited]
     
    In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu's sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.
  • Arizmendi, Octavio, Hasebe, Takahiro
    ELECTRONIC JOURNAL OF PROBABILITY 23 1083-6489 2018 [Refereed][Not invited]
     
    We consider different limit theorems for additive and multiplicative free Levy processes. The main results are concerned with positive and unitary multiplicative free Levy processes at small times, showing convergence to log free stable laws for many examples. The additive case is much easier, and we establish the convergence at small or large times to free stable laws. During the investigation we found out that a log free stable law with index 1 coincides with the Dykema-Haagerup distribution. We also consider limit theorems for positive multiplicative Boolean Levy processes at small times, obtaining log Boolean stable laws in the limit.
  • Takahiro Hasebe, Yuki Ueda
    ALEA Lat. Am. J. Probab. Math. Stat. 15 353 - 374 2018 [Refereed][Not invited]
     
    We prove that classical and free Brownian motions with initial distributions
    are unimodal for sufficiently large time, under some assumption on the initial
    distributions. The assumption is almost optimal in some sense. Similar results
    are shown for a symmetric stable process with index 1 and a positive stable
    process with index $1/2$. We also prove that free Brownian motion with initial
    symmetric unimodal distribution is unimodal, and discuss strong unimodality for
    free convolution.
  • Independence and infinite divisibility in non-commutative probability
    Takahiro Hasebe
    Sugaku 70 (3) 296 - 320 2018 [Refereed][Invited]
  • Takahiro Hasebe, Shuhei Tsujie
    JOURNAL OF ALGEBRAIC COMBINATORICS 46 (3-4) 499 - 515 0925-9899 2017/12 [Refereed][Not invited]
     
    Richard P. Stanley conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric functions.
  • Hasebe, Takahiro, Sakuma, Noriyoshi
    ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES 53 (2) 916 - 936 0246-0203 2017/05 [Refereed][Not invited]
     
    We will prove that: (1) A symmetric free Levy process is unimodal if and only if its free Levy measure is unimodal; (2) Every free Levy process with boundedly supported Levy measure is unimodal in sufficiently large time. (2) is completely different property from classical Levy processes. On the other hand, we find a free Levy process such that its marginal distribution is not unimodal for any time s > 0 and its free Levy measure does not have a bounded support. Therefore, we conclude that the boundedness of the support of free Levy 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 Levy process in terms of Levy-Khintchine representation.
  • Bozejko, Marek, Ejsmont, Wiktor, Hasebe, Takahiro
    INTERNATIONAL JOURNAL OF MATHEMATICS 28 (2) 0129-167X 2017/02 [Refereed][Not invited]
     
    We construct a deformed Fock space and a Brownian motion coming from Coxeter groups of type D. The construction is analogous to that of the q-Fock space (of type A) and the (alpha, q)-Fock space (of type B).
  • Takahiro Hasebe, Toshinori Miyatani, Masahiko Yoshinaga
    Journal of Singularities 16 212 - 227 1949-2006 2017 [Refereed][Not invited]
     
    The Euler characteristic of a semialgebraic set can be considered as a general- ization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define “negative sets” to be the sets with negative Euler characteristics. Applying this idea to posets, we introduce the notion of semialgebraic posets. Using “negative posets”, we establish Stanley's reciprocity theorems for order polynomials at the level of Euler characteristics. We also formulate the Euler characteristic reciprocities for chromatic and flow polynomials.
  • Takahiro Hasebe, Steen Thorbjornsen
    JOURNAL OF THEORETICAL PROBABILITY 29 (3) 922 - 940 0894-9840 2016/09 [Refereed][Not invited]
     
    We show that any freely selfdecomposable probability law is unimodal. This is the free probabilistic analog of Yamazato's result in (Ann. Probab. 6:523-531, 1978).
  • Octavio Arizmendi, Takahiro Hasebe
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 368 (7) 4873 - 4905 0002-9947 2016/07 [Refereed][Not invited]
     
    We study Boolean stable laws, b(alpha,rho), with stability index b(alpha,rho) and asymmetry parameter rho. We show that the classical scale mixture of b(alpha,rho) coincides with a free mixture and also a monotone mixture of b(alpha,rho). For this purpose we define the multiplicative monotone convolution of probability measures, one supported on the positive real line and the other arbitrary. We prove that any scale mixture of b(alpha,rho) is both classically and freely infinitely divisible for alpha <= 1/2 and also for some alpha > 1/2. Furthermore, we show the multiplicative infinite divisibility of b(alpha), 1 with respect to classical, free and monotone convolutions. Scale mixtures of Boolean stable laws include some generalized beta distributions of the 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 Moller. We use a representation of b(alpha), 1 as the free multiplicative convolution of a free Bessel law and a free stable law to prove a conjecture of Hinz and Mlotkowski regarding the existence of the free Bessel laws as probability measures. The proof depends on the fact that b(alpha), 1 has free divisibility indicator 0 for 1/2 < alpha.
  • Octavio Arizmendi, Takahiro Hasebe
    COMPLEX ANALYSIS AND OPERATOR THEORY 10 (3) 581 - 603 1661-8254 2016/03 [Refereed][Not invited]
     
    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.
  • Nobuhiro Asai, Marek Bozejko, Takahiro Hasebe
    JOURNAL OF MATHEMATICAL PHYSICS 57 (2) 021702  0022-2488 2016/02 [Refereed][Not invited]
     
    Let nu(alpha,q) be the probability and orthogonality measure for the q-Meixner-Pollaczek orthogonal polynomials, which has appeared in the work of Bozejko, Ejsmont, and Hasebe [J. Funct. Anal. 269, 1769-1795 (2015)] as the distribution of the (alpha,q)-Gaussian process (the Gaussian process of type B) over the (alpha,q)-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of nu(alpha,q). Our main results cover not only the representation of q-Gaussian distribution by van Leeuwen and Maassen [J. Math. Phys. 36, 4743-4756 (1995)] but also of q(2)-Gaussian and symmetric free Meixner distributions on R. In addition, non-trivial commutation relations satisfied by (alpha,q)-operators are presented. (C) 2016 AIP Publishing LLC.
  • Free infinite divisibility for powers of random variables
    Takahiro Hasebe
    ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS 13 (1) 309 - 336 1980-0436 2016 [Refereed][Not invited]
     
    We prove that X-r follows a free regular distribution, i.e. the law of a nonnegative free Levy process if: (1) X follows a free Poisson distribution without an atom at 0 and r is an element of (-infinity,0] boolean OR [1,infinity); (2) X follows a free Poisson distribution with an atom at 0 and vertical bar r vertical bar >= 1; (3) X follows a mixture of sonic IICM distributions and vertical bar r vertical bar >= 1; (4) X follows some beta distributions and r is taken from some interval. hi particular, if S is a standard semicircular element then vertical bar S vertical bar(r) is freely infinitely divisible for r is an element of (-infinity,0] boolean OR [2,infinity). Also we consider the symmetrization of the above probability measures, and in particular show that S vertical bar(r) sign (S) is freely infinitely divisible for r >= 2. Therefore S-n is freely infinitely divisible for every n is an element of N. The results on free Poisson and semicircular random variables have a good correspondence with classical ID properties of powers of gamma and normal random variables.
  • Takahiro Hasebe, Noriyoshi Sakuma
    Demonstratio Mathematica 48 (3) 424 - 439 2391-4661 2015/09/01 [Refereed][Not invited]
     
    We give a complete list of the LebesgueJordan decomposition of Boolean and monotone stable distributions and a complete list of the mode of them. They are not always unimodal.
  • Octavio Arizmendi, Takahiro Hasebe, Franz Lehner, Carlos Vargas
    ADVANCES IN MATHEMATICS 282 56 - 92 0001-8708 2015/09 [Refereed][Not invited]
     
    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. (C) 2015 Elsevier Inc. All rights reserved.
  • Marek Bozejko, Wiktor Ejsmont, Takahiro Hasebe
    JOURNAL OF FUNCTIONAL ANALYSIS 269 (6) 1769 - 1795 0022-1236 2015/09 [Refereed][Not invited]
     
    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 (alpha, q)-Fock space, which satisfy the commutation relation b(alpha,q)(x)b*(alpha,q)(y) - qb*(alpha,q()y)b(alpha,q)(x) = < x,y > I + alpha <(x) over bar, y)q(2N), where x, y are elements of a complex Hilbert space with a self-adjoint involution x bar right arrow (x) over bar and N is the number operator with respect to the grading on the (alpha, q)-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators b(alpha,q)(x) + b*(alpha,q)(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 is associated to the orthogonal polynomials called the q-Meixner-Pollaczek polynomials, yielding the q-Hermite polynomials when alpha = 0 and free Meixner polynomials when q = 0. (C) 2015 Elsevier Inc. All rights reserved.
  • Takahiro Hasebe
    ELECTRONIC JOURNAL OF PROBABILITY 19 (81) 1 - 33 1083-6489 2014/09 [Refereed][Not invited]
     
    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, many of ultraspherical and Student t-distributions have free divisibility indicator 1.
  • Takahiro Hasebe, Alexey Kuznetsov
    ELECTRONIC COMMUNICATIONS IN PROBABILITY 19 1 - 12 1083-589X 2014/08 [Refereed][Not invited]
     
    We investigate analytical properties of free stable distributions and discover many connections with their classical counterparts. Our main result is an explicit formula for the Mellin transform, which leads to explicit series representations for the characteristic function and for the density of a free stable distribution. All of these formulas bear close resemblance to the corresponding expressions for classical stable distributions. As further applications of our results, we give an alternative proof of the duality law due to Biane and a new factorization of a classical stable random variable into an independent (in the classical sense) product of a free stable random variable and a power of a Gamma(2) random variable.
  • Octavio Arizmendi, Takahiro Hasebe
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 142 (5) 1621 - 1632 0002-9939 2014/05 [Refereed][Not invited]
     
    We completely determine the free infinite divisibility for the Boolean stable law which is parametrized by a stability index alpha and an asymmetry coefficient rho. We prove that the Boolean stable law is freely infinitely divisible if and only if one of the following conditions holds: 0 < alpha <= 1/2; 1/2 < alpha <= 2/3 and 2 - 1/alpha <= rho <= 1/alpha -1; alpha = 1, rho = 1/2. Positive Boolean stable laws corresponding to rho = 1 and alpha <= 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.
  • Octavio Arizmendi, Takahiro Hasebe
    BERNOULLI 19 (5B) 2750 - 2767 1350-7265 2013/11 [Refereed][Not invited]
     
    We consider a class of probability measures mu(alpha)(s,r) 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(alpha)(s,2) as a free compound Poisson law with Levy measure a monotone alpha-stable law. This implies the free infinite divisibility of mu(alpha)(s,2). Moreover, when symmetric or positive, mu(alpha)(s,2) 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(alpha)(s,r) for r not equal 2. Special cases include the beta distributions B(1 - 1/r, 1 + 1/r) which are freely infinitely divisible if and only if 1 <= r <= 2.
  • Takahiro Hasebe, Hayato Saigo
    Nagoya Math. J. 215 151 - 167 0027-7630 2013/06/01 [Refereed][Not invited]
     
    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.
  • Takahiro Hasebe
    COMPLEX ANALYSIS AND OPERATOR THEORY 7 (1) 115 - 134 1661-8254 2013/02 [Refereed][Not invited]
     
    We study the multiplicative convolution for c-monotone independence. This convolution unifies the monotone, Boolean and orthogonal multiplicative convolutions. We characterize convolution semigroups for the c-monotone multiplicative convolution on the unit circle. We also prove that an infinitely divisible distribution can always be embedded in a convolution semigroup. We furthermore discuss the (non)-uniqueness of such embeddings including the monotone case. Finally connections to the multiplicative Boolean convolution are discussed.
  • Octavio Arizmendi, Takahiro Hasebe
    Studia Mathematica 215 (2) 157 - 185 0039-3223 2013 [Refereed][Not invited]
     
    Belinschi and Nica introduced a composition semigroup of maps on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know quantitatively if a measure is freely infinitely divisible or not. In the first half of the paper, we further investigate this indicator: 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 that the Boolean powers μt, t [0 1] of a probability measure μ are freely infinitely divisible if the measure μ is freely infinitely divisible. In the other half of the 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 relative to the free multiplicative 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 we establish 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. © Instytut Matematyczny PAN, 2013.
  • Marek Bozejko, Takahiro Hasebe
    PROBABILITY AND MATHEMATICAL STATISTICS-POLAND 33 (2) 363 - 375 0208-4147 2013 [Refereed][Not invited]
     
    We prove that symmetric Meixner distributions, whose probability densities are proportional to vertical bar Gamma(t + ix)vertical bar(2), are freely infinitely divisible for 0 < t <= 1/2. The case t = 1/2 corresponds to the law of Levy's stochastic area whose probability density is 1/cosh(pi x). A logistic distribution, whose probability density is proportional to 1/cosh(2)(pi x), is also freely infinitely divisible.
  • Octavio Arizmendi, Takahiro Hasebe, Noriyoshi Sakuma
    ALEA, Lat. Amer. J. Probab. Math. Stat. 10 (2) 271 - 291 1980-0436 2013 [Refereed][Not invited]
     
    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: \chi^2(1) and F(1,1). Another consequence is that the
    free commutator operation preserves free infinite divisibility.
  • Takahiro Hasebe
    JOURNAL OF THEORETICAL PROBABILITY 25 (3) 756 - 770 0894-9840 2012/09 [Refereed][Not invited]
     
    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.
  • Takahiro Hasebe
    INTERNATIONAL JOURNAL OF MATHEMATICS 23 (3) 1250041 (21 pages)  0129-167X 2012/03 [Refereed][Not invited]
     
    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 noncommutative 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 noncommutative probability.
  • JOINT CUMULANTS FOR NATURAL INDEPENDENCE
    Takahiro Hasebe, Hayato Saigo
    ELECTRONIC COMMUNICATIONS IN PROBABILITY 16 491 - 506 1083-589X 2011/09 [Refereed][Not invited]
     
    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.
  • Takahiro Hasebe
    INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS 14 (3) 465 - 516 0219-0257 2011/09 [Refereed][Not invited]
     
    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 noncommutative 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.
  • Takahiro Hasebe
    COLLOQUIUM MATHEMATICUM 124 (1) 79 - 94 0010-1354 2011 [Refereed][Not invited]
     
    We generalize the infinitesimal independence appearing in free probability of type B in two directions: to higher order derivatives and other natural independences: tensor, monotone and Boolean. Such generalized infinitesimal independences can be defined by using associative products of infinitely many linear functionals, and therefore the associated cumulants can be defined. These products can be seen as the usual natural products of linear maps with values in formal power series.
  • INDEPENDENCE GENERALIZING MONOTONE AND BOOLEAN INDEPENDENCES
    Takahiro Hasebe
    QUANTUM PROBABILITY AND RELATED TOPICS 27 190 - 201 1793-5121 2011 [Refereed][Not invited]
     
    We define conditionally monotone independence in two states which interpolates monotone and Boolean ones. This independence is associative, and therefore leads to a natural probability theory in a non-commutative algebra.
  • Takahiro Hasebe
    INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS 13 (4) 619 - 627 0219-0257 2010/12 [Refereed][Not invited]
     
    The energy representation of a gauge group on a Riemannian manifold has been discussed by several authors. Y. Shimada has shown the irreducibility with the use of white noise analysis for compact Riemannian manifolds. In this paper we extend its technique to the noncompact Riemannian manifolds which have differential operators satisfying some conditions.
  • Takahiro Hasebe
    INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS 13 (1) 111 - 131 0219-0257 2010/03 [Refereed][Not invited]
  • Takahiro Hasebe
    STUDIA MATHEMATICA 200 (2) 175 - 199 0039-3223 2010 [Refereed][Not invited]
     
    We study how a property of a monotone convolution semigroup changes with respect to the time parameter. Especially we focus on "time-independent properties": in the classical case, there are many properties of convolution semigroups (or Levy processes) which are determined at an instant, and moreover, such properties are often characterized by the drift term and Levy measure. In this paper we exhibit such properties of monotone convolution semigroups; an example is the concentration of the support of a probability measure on the positive real line. Most of them are characterized by the same conditions on drift terms and Levy measures as known in probability theory. These kinds of properties are mapped bijectively by a monotone analogue of the Bercovici-Pata bijection. Finally we compare such properties with classical, free, and Boolean cases, which will be important in an approach to unify these notions of independence.
  • Takahiro Hasebe, Hayato Saigo
    Ann. Inst. Henri Poincare Probab. Stat. 47 (4) 1160 - 1170 0246-0203 2009/07/28 [Refereed][Not invited]
     
    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''.
  • Takahiro Hasebe, Izumi Ojima, Hayato Saigo
    Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 11 (2) 307 - 311 0219-0257 2007/12/23 [Refereed][Not invited]
     
    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 Mikusi\'{n}ski's version for solving differential equations.

MISC

Educational Activities

Teaching Experience

  • Basic Analysis A
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 初等関数,指数関数, 対数関数, 三角関数,整級数,複素積分,原始関数,不定積分,正則関数, コーシーの積分定理, コーシーの積分公式, 調和関数 テーラーの定理,留数定理
  • Exercises on Basic Analysis A
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 初等関数, 指数関数, 対数関数, 三角関数, 複素積分,正則関数, コーシーの積分定理, コーシーの積分公式, 調和関数,テーラーの定理, 留数定理
  • Basic Mathematics D
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 多変数関数,偏微分,テイラーの定理,逆写像定理,陰関数定理,重積分,広義積分
  • The World of Science and Technology
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 全学教育
    キーワード : 対称性,マルコフ連鎖,複素関数


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