Analytic aspects of the bi-free partial R-transform

Huang, Hao-Wei; Wang, Jiun-Chau

doi:10.1016/j.jfa.2016.04.026

Cited by 17 publications

(55 citation statements)

References 8 publications

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“…In the present paper, we continue the previous work [13] and contribute to the research of bifree harmonic analysis without any emphasis on Voiculescu's original motivation. To accommodate objects like planar probability distributions or integral representations, it is natural to constraint ourselves to commuting and self-adjoint pairs (a, b) in a certain C * -probability space, i.e.…”

Section: Introductionmentioning

confidence: 61%

“…Since the introduction of bi-free probability by Voiculescu in 2013, combinatorial and analytical approaches have been the main research focuses so far [7][8] [11] [13][17] [18]. Given a two-faced pair (a, b) in a C * -probability space (A, ϕ), its bi-free partial R-transform is defined through its Cauchy transform G (a,b) (z, w) = ϕ((z − a) −1 (w − b) −1 ) as the left faces a, c and the freeness of the right ones b, d. The reader is referred to [2] [12] for some recent developments.…”

Section: Introductionmentioning

confidence: 99%

“…Distributions of this kind are determined by their characteristic functions or free R-transforms, the so-called Lévy-Hinčin type representations. Measures in P R 2 which are ⊞⊞-infinitely divisible were first studied in [11] in the case when they are compactly supported and considered in the general case in [13].…”

Section: Introductionmentioning

confidence: 99%

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Limit theorems in bi-free probability theory

Huang

Wang

2018

Probab. Theory Relat. Fields

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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 selfadjoint 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 and fullness of planar probability distributions are also set down. 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.for complex values z and w in a neighborhood of zero, where R a and R b are the usual R-transforms of a and b, respectively [20]. This transform plays the same role as the usual R-transform does in the free case. As in the classical situation, the bi-freeness of (a, b) and (c, d) yields the freeness of

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Section: Introductionmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

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Limit theorems in bi-free probability theory

Huang

Wang

2018

Probab. Theory Relat. Fields

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“…We conclude this section with an example, which views as an analog of bi-free compound Poisson distribution [9]. It will be shown in the next section that virtually all bi-freely max-i.d.…”

Section: Limit Theorems and Bi-freely Max-infinite Divisibilitymentioning

confidence: 88%

“…We have seen that γ c is non-full if and only if c = ±1. When |c| < 1, γ c is absolutely continuous with respect to the Lebesgue measure, and its density is given by the formula [9]:…”

Section: Examplementioning

confidence: 99%

Bi-free extreme values

Huang

Wang

2020

Journal of Functional Analysis

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Bi-Boolean Independence for Pairs of Algebras

Skoufranis

2017

Complex Anal. Oper. Theory

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In this paper, the notion of bi-Boolean independence for non-unital pairs of algebras is introduced thereby extending the notion of Boolean independence to pairs of algebras. The notion of B-(ℓ, r)cumulants is defined via a bi-Boolean moment-cumulant formula over the lattice of bi-interval partitions, and it is demonstrated that bi-Boolean independence is equivalent to the vanishing of mixed B-(ℓ, r)-cumulants. Furthermore, some of the simplest bi-Boolean convolutions are considered, and a bi-Boolean partial ηtransform is constructed for the study of limit theorems and infinite divisibility with respect to the additive bi-Boolean convolution. In particular, a bi-Boolean Lévy-Hinčin formula is derived in perfect analogy with the bi-free case, and some Bercovici-Pata type bijections are provided. Additional topics considered include the additive bi-Fermi convolution, some relations between the (ℓ, r)-and B-(ℓ, r)-cumulants, and bi-Boolean independence in an amalgamated setting.The last section of this paper also includes an errata that will be published with this copy of the paper. Date

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Analytic aspects of the bi-free partial R-transform

Cited by 17 publications

References 8 publications

Limit theorems in bi-free probability theory

Limit theorems in bi-free probability theory

Bi-free extreme values

Bi-Boolean Independence for Pairs of Algebras

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