The Lévy-Itô decomposition in free probability

Barndorff–Nielsen, Ole E.; Thorbjørnsen, Steen

doi:10.1007/s00440-003-0322-y

Cited by 13 publications

(30 citation statements)

References 21 publications

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“…In the classical case, after a suitable scaling, the point process N n converges to a Poisson random measure. It turns out that in the non-commutative case the free point process converges to a free Poisson random measure, which was recently defined in [15] and [2]. The following three theorems are the main results of our paper.…”

Section: Introductionmentioning

confidence: 75%

“…

We continue here the study of free extreme values begun in [3]. We study the convergence of the free point processes associated with free extreme values to a free Poisson random measure ([15], [2]). We relate this convergence to the free extremal laws introduced in [3] and give the limit laws for free order statistics.

1The basic element in both classical and free theory of extremes is a probability measure µ.

…”

mentioning

confidence: 99%

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Free point processes and free extreme values

Arous

Kargin

2009

Probab. Theory Relat. Fields

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Section: Introductionmentioning

confidence: 75%

“…

1The basic element in both classical and free theory of extremes is a probability measure µ.

…”

mentioning

confidence: 99%

Free point processes and free extreme values

Arous

Kargin

2009

Probab. Theory Relat. Fields

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“…We also extend the N/V -limit to the case of a free Lévy white noise as defined in [6]. In particular, such a result holds for a free Lévy process without free Gaussian part [1].…”

Section: Introductionmentioning

confidence: 78%

Approximation of a free Poisson process by systems of freely independent particles

Bożejko

Silva

Kuna

et al. 2018

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Let σ be a non-atomic, infinite Radon measure on R d , for example, dσ(x) = z dx where z > 0. We consider a system of freely independent particles x 1 , . . . , x N in a bounded set Λ ⊂ R d , where each particle x i has distribution 1 σ(Λ) σ on Λ and the number of particles, N , is random and has Poisson distribution with parameter σ(Λ). If the particles were classically independent rather than freely independent, this particle system would be the restriction to Λ of the Poisson point process on R d with intensity measure σ. In the case of free independence, this particle system is not the restriction of the free Poisson process on R d with intensity measure σ. Nevertheless, we prove that this is true in an approximative sense: if bounded sets Λ (n) (n ∈ N) are such that Λ (1) ⊂ Λ (2) ⊂ Λ (3) ⊂ · · · and ∞ n=1 Λ (n) = R d , then the corresponding particle system in Λ (n) converges (as n → ∞) to the free Poisson process on R d with intensity measure σ. We also prove the following N/V -limit: Let N (n) be a determinstic sequence of natural numbers such that lim n→∞ N (n) /σ(Λ (n) ) = 1. Then the system of N (n) freely independent particles in Λ (n) converges (as n → ∞) to the free Poisson process. We finally extend these results to the case of a free Lévy white noise (in particular, a free Lévy process) without free Gaussian part.

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“…Noncommutative Lévy processes have most actively been studied in the framework of free probability, see e.g. [4] and the references therein. Using q-deformed cumulants, Anshelevich [2] constructed and studied noncommutative Lévy processes for q-commutation relations (1.1).…”

Section: 3)mentioning

confidence: 99%

Noncommutative Lévy Processes for Generalized (Particularly Anyon) Statistics

Bożejko

Lytvynov

Wysoczański

2012

Commun. Math. Phys.

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Let T = R d . Let a function Q : T 2 → C satisfy Q(s, t) = Q(t, s) and |Q(s, t)| = 1. A generalized statistics is described by creation operators ∂ † t and annihilation operators ∂ t , t ∈ T , which satisfy the Q-commutation relations:From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which Q(s, t) is equal to q if s < t, and toq if s > t. Here q ∈ C, |q| = 1. We start the paper with a detailed discussion of a Q-Fock space and operators (∂ † t , ∂ t ) t∈T in it, which satisfy the Q-commutation relations. Next, we consider a noncommutative stochastic process (white noise)Here λ ∈ R is a fixed parameter. The case λ = 0 corresponds to a Q-analog of Brownian motion, while λ = 0 corresponds to a (centered) Q-Poisson process. We study Q-Hermite (QCharlier respectively) polynomials of infinitely many noncommutatative variables (ω(t)) t∈T . The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding Lévy processes. To this end, we recursively define Q-cumulants of a field (ξ(t)) t∈T . This allows us to define a Q-Lévy process as a field (ξ(t)) t∈T whose values at different points of T are Q-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a Q-Lévy process, and derive a Nualart-Schoutens-type chaotic decomposition for such a process.

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The Lévy-Itô decomposition in free probability

Cited by 13 publications

References 21 publications

Free point processes and free extreme values

Free point processes and free extreme values

Approximation of a free Poisson process by systems of freely independent particles

Noncommutative Lévy Processes for Generalized (Particularly Anyon) Statistics

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