Classical scale mixtures of Boolean stable laws

Arizmendi, Octavio

doi:10.1090/tran/6792

Cited by 20 publications

(26 citation statements)

References 47 publications

(96 reference statements)

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“…Proof By the definition we have f α,β,λ (α) = (λα) 2 , substituting this in the expression (2.13) we obtain that α|λ| = 2(α −δ) √ β(η − α). Taking the Taylor expansion around z = α for λ = 0 we obtain…”

Section: Proposition 26 the R-transform Of The Measure μ(α β λ) Camentioning

confidence: 94%

“…Examples of free regular distributions include positive free stable distributions, free Poisson distributions and powers of free Poisson distributions [16]. A general criterion in [3,Theorem 4.6] shows that some boolean stable distributions [1] and many probability distributions [2,3,15] and so we have the reduced formula (3.10). The drift is given by ξ α,β,λ = lim u→−∞ r α,β,λ (u) = 0.…”

Section: Free Regularitymentioning

confidence: 99%

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On Free Generalized Inverse Gaussian Distributions

Szpojankowski

2018

Complex Anal. Oper. Theory

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We study here properties of free Generalized Inverse Gaussian distributions (fGIG) in free probability. We show that in many cases the fGIG shares similar properties with the classical GIG distribution. In particular we prove that fGIG is freely infinitely divisible, free regular and unimodal, and moreover we determine which distributions in this class are freely selfdecomposable. In the second part of the paper we prove that for free random variables X, Y where Y has a free Poisson distribution oneX +Y if and only if X has fGIG distribution for special choice of parameters. We also point out that the free GIG distribution maximizes the same free entropy functional as the classical GIG does for the classical entropy.

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Section: Proposition 26 the R-transform Of The Measure μ(α β λ) Camentioning

confidence: 94%

Section: Free Regularitymentioning

confidence: 99%

On Free Generalized Inverse Gaussian Distributions

Szpojankowski

2018

Complex Anal. Oper. Theory

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“…for some ρ ∈ [0, 1]. By (2), this means that a free 1-stable distribution is up to translation the law of the free independent sum C a,b d = aX 1,1/2 + bT, for a ≥ 0, b ∈ R, where T has Voiculescu transform − log z and will be called henceforth the exceptional free 1-stable random variable. For example, φ 1/2 is the Voiculescu transform of X 1,1/2 , whereas φ 0 is that of 2 π (T + log(π/2)) and φ 1 that of − 2 π (T + log(π/2)).…”

Section: Introductionmentioning

confidence: 99%

Some properties of the free stable distributions

Simon¹,

Wang²

2020

Ann. Inst. H. Poincaré Probab. Statist.

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We investigate certain analytical properties of the free α−stable densities on the line. We prove that they are all classically infinitely divisible when α ≤ 1, and that they belong to the extended Thorin class when α ≤ 3/4. The Lévy measure is explicitly computed for α = 1, showing that the free 1-stable random variables are not Thorin except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when α > 1. In the one-sided case we prove, refining unimodality, that the densities are whale-shaped that is their successive derivatives vanish exactly once. Finally, we derive a collection of results connected to the fine structure of the one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, a new identity for the Beta-Gamma algebra, and several intrinsic properties of whale-shaped densities.where X 1 , X 2 are free independent copies of a random variable X with density f , a, b are arbitrary positive real numbers, c is a positive real number depending on a, b, and d is a real number. As in the classical framework, it turns out that there exist solutions to (1) only if c = (a α + b α ) 1/α for some fixed α ∈ (0, 2]. We will be mostly concerned with free strictly stable densities, which correspond to the case d = 0. In this framework the Voiculescu transform of f writes, up to multiplicative normalization,We refer e.g. to [43] for some background on the free additive convolution, to [11] for the original solution to the equation (1), and to the introduction of [29] for the above parametrization (α, ρ), which mimics that of the strict classical framework. Let us also recall that free stable laws appear as limit distributions of spectra of large random matrices with possibly unbounded variance -see [10,18], and that their domains of attraction have been fully characterized in [12,13]. In the following, we will denote by X α,ρ the random variable whose Voiculescu transform is given by (2), and set f α,ρ for its density. The analogy with the classical case extends to the fact, observed in Corollary 1.3 of [29], that with our parametrization one hasThe explicit form of the Voiculescu transform also shows that X α,ρ d = −X α,1−ρ . In this paper, some focus will put on the one-sided case and we will use the shorter notations X α,1 = X α and f α,1 = f α .Throughout, the random variable X α,ρ will be mostly handled as a classical random variable via its 2010 Mathematics Subject Classification. 60E07; 46L54; 62E15; 33E20.

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“…It seems that as of now there is no description of that free class U (s-selfdecomposable distributions) as a collection of limits in free analog of the scheme p˚q. It is worth mentioning that in [1] (already quoted above) the free class U was defined by using the unimodality property of Lévy spectral measures of distributions in U.…”

Section: Rzt0u´pmentioning

confidence: 99%

“…On the other hand, in [1], Section 5, free analog of measures from the class U, have defined them via the unimodality property of their Lévy spectral measures; comp. [13].…”

mentioning

confidence: 99%