185 research outputs found

    Free infinite divisibility for beta distributions and related ones

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    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

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    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

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    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

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    We study unimodality for free multiplicative convolution with free normal distributions {λt}t>0\{\lambda_t\}_{t>0} on the unit circle. We give four results on unimodality for μλt\mu\boxtimes\lambda_t: (1) if μ\mu is a symmetric unimodal distribution on the unit circle then so is μλt\mu\boxtimes \lambda_t at any time t>0t>0; (2) if μ\mu is a symmetric distribution on T\mathbb{T} supported on {eiθ:θ[φ,φ]}\{e^{i\theta}: \theta \in [-\varphi,\varphi]\} for some φ(0,π/2)\varphi \in (0,\pi/2), then μλt\mu \boxtimes \lambda_t is unimodal for sufficiently large t>0t>0; (3) bλt{\bf b} \boxtimes \lambda_t is not unimodal at any time t>0t>0, where b{\bf b} is the equally weighted Bernoulli distribution on {1,1}\{1,-1\}; (4) λt\lambda_t is not freely strongly unimodal for sufficiently small t>0t>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

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    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

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    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

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    We consider a class of probability measures μs,rα\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 μs,2α\mu_{s,2}^{\alpha} as a free compound Poisson law with L\'{e}vy measure a monotone α\alpha-stable law. This implies the free infinite divisibility of μs,2α\mu_{s,2}^{\alpha}. Moreover, when symmetric or positive, μs,2α\mu_{s,2}^{\alpha} has a representation as the free multiplication of a free Poisson law and a monotone α\alpha-stable law. We also investigate the free infinite divisibility of μs,rα\mu_{s,r}^{\alpha} for r2r\neq2. Special cases include the beta distributions B(11r,1+1r)B(1-\frac{1}{r},1+\frac{1}{r}) which are freely infinitely divisible if and only if 1r21\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