Realization of free white noises

Glöckner, P.G.; Schürmann, Michael; Speicher, Roland

doi:10.1007/bf01189934

Cited by 19 publications

(17 citation statements)

References 17 publications

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“…One knows (see, e.g., [6,15,17]) that these operators p(d) give a realization of compound Poisson elements, i.e., their moments are given by Eq. (5).…”

Section: Compound Poisson Casementioning

confidence: 99%

Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

Mingo

Speicher

2006

Journal of Functional Analysis

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105

112

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We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of "second order freeness" and interpret the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order. (J.A. Mingo), speicher@mast.queensu.ca (R. Speicher).

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“…One knows (see, e.g., [6,15,17]) that these operators p(d) give a realization of compound Poisson elements, i.e., their moments are given by Eq. (5).…”

Section: Compound Poisson Casementioning

confidence: 99%

Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

Mingo

Speicher

2006

Journal of Functional Analysis

Self Cite

105

112

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“…Since X i is symmetric, λ i ∈ R and ξ i , Ω = 0 so ξ i ∈ H 0 . Clearly ξ i ∈ D, and since X i , a Note that there are bosonic and free versions of the operator decomposition (22) (see [Sch91] and [GSS92]) but they only hold for operators with (freely) infinitely divisible joint distributions. The preceding proposition reflects the fact that all states are infinitely divisible in the Boolean sense [BN06, Proposition 4.8].…”

Section: (S)mentioning

confidence: 99%

Appell polynomials and their relatives II. Boolean theory

Anshelevich¹

2009

Indiana Univ. Math. J.

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ABSTRACT. The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolvent-type form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, in the Boolean and free theory, coincide. This is true even in the multivariate case. A number of applications of this fact are described, to the Belinschi-Nica and Bercovici-Pata maps, conditional freeness, and the Laha-Lukacs type characterization.A number of properties which hold for the Meixner class in the free and classical cases turn out to hold in general in the Boolean theory. Examples include the behavior of the Jacobi coefficients under convolution, the relationship between the Jacobi coefficients and cumulants, and an operator model for cumulants. Along the way, we obtain a multivariate version of the Stieltjes continued fraction expansion for the moment generating function of an arbitrary state with monic orthogonal polynomials.

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“…It turns out that they can always be represented as a linear combination of the corresponding creators, annihilators, conservation operators and time (which contains the projection Γ (0 0T ) to the vacuum in the boolean case), cf. [GSS92,BG01].…”

Section: Markov Structure Of Boolean Lévy Processesmentioning

confidence: 99%

“…[Sch95,BGS99,Fra01]. It is natural to conjecture that realizations of these processes can be constructed using the corresponding quantum stochastic calculi, like in the case of tensor independence, but so far this has not been proven rigorously for the general case (but see [GSS92,BG01], where it has been shown for quantum stochastic processes with free or boolean additive increments).…”

Section: Introductionmentioning

confidence: 99%

Unification of boolean, monotone, anti-monotone, and tensor independence and L�vy processes

Franz

2003

Mathematische Zeitschrift

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The unification of the boolean and the tensor product of quantum probability spaces given by Lenczewski [Len98] is modified to include the monotone product and its mirror image, the anti-monotone product, and used to reduce the theories of boolean, monotone, and anti-monotone Lévy processes on dual semi-groups to the theory of Lévy processes on involutive bialgebras. This leads to a representation theorem for these processes. Further applications are a proof of the Schoenberg correspondence for boolean, monotone, and anti-monotone convolution semi-groups, a realization of boolean, monotone, and anti-monotone creation, annihilation, and conservation operators on boson Fock spaces, and a discussion of the Markov structure of these Lévy processes.

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Realization of free white noises

Cited by 19 publications

References 17 publications

Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

Appell polynomials and their relatives II. Boolean theory

Unification of boolean, monotone, anti-monotone, and tensor independence and L�vy processes

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