Independences and partial $R$-transforms in bi-free probability

Skoufranis, Paul

doi:10.1214/15-aihp691

Cited by 25 publications

(42 citation statements)

References 22 publications

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“…Obviously our result here complements the recent work on operations on bi-free bi-partite hermitian two-faced pairs ( [10], [7], [5], [11], [8]). …”

Section: Introductionsupporting

confidence: 84%

Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

Voiculescu

2015

J Theor Probab

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Abstract. We compute the bi-free max-convolution which is the operation on bi-variate distribution functions corresponding to the max-operation with respect to the spectral order on bi-free bipartite two-faced pairs of hermitian non-commutative random variables. With the corresponding definitions of bi-free max-stable and max-infinitely-divisible laws their determination becomes in this way a classical analysis question. IntroductionThe definition and classification of free max-stable laws in [2] had been an unexpected addition to the list of free probability analogues to classical probability items. Here we take the first step in a similar direction in bi-free probability [9]. We show that there is a simple formula for computing the bi-free extremal convolution of probability measures in the plane. This corresponds to computing the distribution of (a ∨ c, b ∨ d), where (a, b) and (c, d) are two bi-free two-faced pairs of commuting hermitian operators. Like the free extremal convolution on R defined in [2], the bi-free extremal convolution in the plane reduces the question of bi-free max-stable laws in the plane to an analysis problem in the classical context which we won't pursue here.

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“…Obviously our result here complements the recent work on operations on bi-free bi-partite hermitian two-faced pairs ( [10], [7], [5], [11], [8]). …”

Section: Introductionsupporting

confidence: 84%

Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

Voiculescu

2015

J Theor Probab

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“…Due to the recent work ( [10], [7], [1], [2], [8], [3], [5]) we already have a much better understanding of bi-free probability.…”

Section: Introductionmentioning

confidence: 99%

“…The additive and multiplicative cases being closely related in [6], we will be able to make also substantial use here of our work in [10] on the bi-free partial R-transform. Since a combinatorial proof for the partial R-transform was found in [8] and there is also combinatorial work for handling multiplication of bi-free systems of variables using additive bi-free cumulants [1] it is to be expected that a combinatorial approach to the 2-variables partial Sand T -transforms we consider here is possible. On the other hand, the study of infinite divisibility for additive bi-free convolution of probability measures on R 2 in [5] may not have an immediate analogue for the operations considered here, since the result of the operations ⊠⊠ and ⊞⊠ on probability measures seems to produce signed measures only in general.…”

Section: Introductionmentioning

confidence: 99%

Free probability for pairs of faces III: 2-Variables bi-free partial S- and T-transforms

Voiculescu

2016

Journal of Functional Analysis

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Abstract. We introduce two 2-variables transforms: the partial bi-free S-transform and the partial bi-free T -transform. These transforms are the analogues for the bi-multiplicative and, respectively, for the additive-multiplicative bi-free convolution of the 2-variables partial bi-free R-transform in our previous paper in this series. IntroductionIn the first paper in this series [9] we proposed an extension of free probability to systems with left and right variables, based on the notion of bi-freeness. Due to the recent work ([10]we already have a much better understanding of bi-free probability.Here, we complement our paper [10] on the 2-variables bi-free Rtransform with two further 2-variables partial transforms adapted to the bi-multiplicative operation ⊠⊠ and to the additive-multiplicative operation ⊞⊠. In view of free-probability it was obvious to use the letter S for the first transform, while for the second which corresponds to passing from bi-free pairs (a 1 , b 1 ), (a 2 , b 2 ) to (a 1 + a 2 , b 1 b 2 ) we went for T the next letter in the alphabet.Like in the case of addition [10] we found that also for multiplication instead of our original approach to the one-variable S-transform

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“…A two-faced family a = ((a i ) i∈I , (a j ) j∈J ) in a non-commutative probability space (A, ϕ) has a bi-Boolean central limit distribution (or centred bi-Boolean Gaussian distribution) with covariance matrix C if there exists a complex matrix C = (C k,ℓ ) k,ℓ∈I⊔J such that The algebraic bi-Boolean central limit theorem immediately follows from [11, Theorem 6.2] with the given sequence being bi-Boolean independent and the limiting distribution being a centred bi-Boolean Gaussian distribution. Similarly, it is easy to see that a Kac/Loeve type theorem also holds in the bi-Boolean setting (see [22,Theorem 3.2] for the bi-free version) and a general bi-Boolean limit theorem can be deduced from [11, Theorem 6.5], which we record as follows without proof.…”

Section: Combinatorial Aspectsmentioning

confidence: 96%

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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Independences and partial $R$-transforms in bi-free probability

Cited by 25 publications

References 22 publications

Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

Free probability for pairs of faces III: 2-Variables bi-free partial S- and T-transforms

Bi-Boolean Independence for Pairs of Algebras

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