New limit theorems related to free multiplicative convolution

Noriyoshi Sakuma; Hiroaki Yoshida

Displaying similar documents to “New limit theorems related to free multiplicative convolution”

Semigroups related to additive and multiplicative, free and Boolean convolutions

Octavio Arizmendi, Takahiro Hasebe (2013)

Studia Mathematica

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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...

Symmetrization of probability measures, pushforward of order 2 and the Boolean convolution

Wojciech Młotkowski, Noriyoshi Sakuma (2011)

Banach Center Publications

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We study relations between the Boolean convolution and the symmetrization and the pushforward of order 2. In particular we prove that if μ₁,μ₂ are probability measures on [0,∞) then ${(μ ₁ ⨄ μ ₂)}^{s} = μ ₁^{s} ⨄ μ ₂^{s}$ and if ν₁,ν₂ are symmetric then ${(ν ₁ ⨄ ν ₂)}^{(2)} = ν ₁^{(2)} ⨄ ν ₂^{(2)}$ . Finally we investigate necessary and sufficient conditions under which the latter equality holds.

On some generalization of the t-transformation

Anna Dorota Krystek (2010)

Banach Center Publications

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Using the Nevanlinna representation of the reciprocal of the Cauchy transform of probability measures, we introduce a two-parameter transformation $U^{}$ of probability measures on the real line ℝ, which is another possible generalization of the t-transformation. Using that deformation we define a new convolution by deformation of the free convolution. The central limit measure with respect to the -deformed free convolutions is still a Kesten measure, but the Poisson limit depends on the...

The Lévy-Khintchine formula and Nica-Speicher property for deformations of the free convolution

Łukasz Jan Wojakowski (2007)

Banach Center Publications

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We study deformations of the free convolution arising via invertible transformations of probability measures on the real line T:μ ↦ Tμ. We define new associative convolutions of measures by $μ ⊞_{T} ν = T^{- 1} (T μ ⊞ T ν)$ . We discuss infinite divisibility with respect to these convolutions, and we establish a Lévy-Khintchine formula. We conclude the paper by proving that for any such deformation of free probability all probability measures μ have the Nica-Speicher property, that is, one can find their convolution...

Free powers of the free Poisson measure

Melanie Hinz, Wojciech Młotkowski (2011)

Colloquium Mathematicae

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We compute moments of the measures ${(ϖ^{⊠ p})}^{⊞ t}$ , where ϖ denotes the free Poisson law, and ⊞ and ⊠ are the additive and multiplicative free convolutions. These moments are expressed in terms of the Fuss-Narayana numbers.

Generalized free products

J. D. Monk (2001)

Colloquium Mathematicae

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A subalgebra B of the direct product $\prod_{i \in I} A_{i}$ of Boolean algebras is finitely closed if it contains along with any element f any other member of the product differing at most at finitely many places from f. Given such a B, let B* be the set of all members of B which are nonzero at each coordinate. The generalized free product corresponding to B is the subalgebra of the regular open algebra with the poset topology on B* generated by the natural basic open sets. Properties of this product are...

Probability measures corresponding to Aval numbers

Wojciech Młotkowski (2012)

Colloquium Mathematicae

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We describe the class of probability measures whose moments are given in terms of the Aval numbers. They are expressed as the multiplicative free convolution of measures corresponding to the ballot numbers $(m - k) / (m + k) (\binom{m + k}{m})$ .

A comparison on the commutative neutrix convolution of distributions and the exchange formula

Adem Kiliçman (2001)

Czechoslovak Mathematical Journal

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Let $\tilde{f}$ , $\tilde{g}$ be ultradistributions in $𝒵^{'}$ and let ${\tilde{f}}_{n} = \tilde{f} * δ_{n}$ and ${\tilde{g}}_{n} = \tilde{g} * σ_{n}$ where ${δ_{n}}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $δ$ . Then the neutrix product $\tilde{f} ⋄ \tilde{g}$ is defined on the space of ultradistributions $𝒵^{'}$ as the neutrix limit of the sequence ${\frac{1}{2} ({\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n})}$ provided the limit $\tilde{h}$ exist in the sense that $\underset{n \to \infty}{N - l i m} \frac{1}{2} 〈 {\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n}, ψ 〉 = 〈 \tilde{h}, ψ 〉$ for all $ψ$ in $𝒵$ . We also prove that the neutrix convolution product $f ♦ * g$ exist in $𝒟^{'}$ , if and only if the neutrix product $\tilde{f} ⋄ \tilde{g}$ exist in $𝒵^{'}$ and the exchange formula $F (f ♦ * g) = \tilde{f} ⋄ \tilde{g}$ is then satisfied.

Compact convolution operators between $L_{p} (G)$ -spaces

G. Crombez, W. Govaerts (1978)

Colloquium Mathematicae

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Remarks on the boolean convolution and Kerov's α-transformation

Anna Dorota Krystek (2006)

Banach Center Publications

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This paper consists of two parts. The first part is devoted to the study of continuous diagrams and their connections with the boolean convolution. In the second part we investigate the rectangular Young diagrams and respective discrete measures. We recall the definition of Kerov's α-transformation of diagrams, define the α-transformation of finitely supported discrete measures and generalize the notion of the α-transformation.

Radial Convolutors on Free Groups

T. Pytlik (1982)

Publications du Département de mathématiques (Lyon)

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On the incomplete gamma function and the neutrix convolution

Brian Fisher, Biljana Jolevska-Tuneska, Arpad Takači (2003)

Mathematica Bohemica

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The incomplete Gamma function $γ (α, x)$ and its associated functions $γ (α, x_{+})$ and $γ (α, x_{-})$ are defined as locally summable functions on the real line and some convolutions and neutrix convolutions of these functions and the functions $x^{r}$ and $x_{-}^{r}$ are then found.

The $V_{a}$ -deformation of the classical convolution

Anna Dorota Krystek (2007)

Banach Center Publications

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We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by $μ *_{T} ν = T^{- 1} (T μ * T ν)$ . We deal with the $V_{a}$ -deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_{a}$ -deformed classical convolution and give the orthogonal...

Asymptotic formulae concerning arithmetical functions defined by cross-convolutions. VI: $A$ - $k$ -free integers.

Tóth, László (1997)

Mathematica Pannonica

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Translation-invariant operators on Lorentz spaces L(1,q) with 0 < q < 1

Leonardo Colzani, Peter Sjögren (1999)

Studia Mathematica

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We study convolution operators bounded on the non-normable Lorentz spaces $L^{1, q}$ of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on $L^{1, q}$ . In particular, when the positions of the atoms of a discrete measure are linearly independent over...

Hankel determinant for a class of analytic functions of complex order defined by convolution

S. M. El-Deeb, M. K. Aouf (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we obtain the Fekete-Szego inequalities for the functions of complex order defined by convolution. Also, we find upper bounds for the second Hankel determinant $| a_{2} a_{4} - a_{3}^{2} |$ for functions belonging to the class $S_{γ}^{b} (g (z); A, B)$ .

On the product formula on noncompact Grassmannians

Piotr Graczyk, Patrice Sawyer (2013)

Colloquium Mathematicae

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We study the absolute continuity of the convolution $δ_{e^{X}}^{♮} * δ_{e^{Y}}^{♮}$ of two orbital measures on the symmetric space SO₀(p,q)/SO(p)×SO(q), q > p. We prove sharp conditions on X,Y ∈ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for SO₀(p,q)/SO(p)×SO(q) also serves for the spaces SU(p,q)/S(U(p)×U(q)) and Sp(p,q)/Sp(p)×Sp(q), q > p. We moreover apply our results to...