On the non-classical infinite divisibility of power semicircle distributions

Arizmendi, Octavio; Pérez‐Abreu, Víctor

doi:10.31390/cosa.4.2.03

Cited by 17 publications

(19 citation statements)

References 40 publications

(53 reference statements)

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“…Therefore, φ(µ) ≤ γ 1 /γ 0 from Theorem 3.3. The last proposition extends Theorem 3.7 in [5] which gives an upper bound for the divisibility indicator in terms of the Boolean kurtosis: Kurt ⊎ (µ) ≥ φ(µ). Indeed, for a probability measure with mean zero and finite fourth moment, we have Kurt ⊎ (µ) = (γ 1 + β 2 1 )/γ 0 ≥ γ 1 /γ 0 .…”

Section: On Jacobi Parameters and Free Meixner Lawssupporting

confidence: 66%

Semigroups related to additive and multiplicative, free and Boolean convolutions

Arizmendi¹

2013

Studia Math.

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Belinschi and Nica introduced a composition semigroup of maps on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know quantitatively if a measure is freely infinitely divisible or not.In the first half of the paper, we further investigate this indicator: we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to infinity; we derive an upper bound of the indicator in terms of Jacobi parameters. This upper bound is achieved only by free Meixner distributions. We also prove Bożejko's conjecture which says the Boolean powers µ ⊎ , t ∈ [0, 1], of a probability measure µ are freely infinitely divisible if the measure µ is freely infinitely divisible.In the other half of this paper, we introduce an analogous composition semigroup for multiplicative convolutions and define free divisibility indicators for these convolutions. Moreover, we prove that a probability measure on the unit circle is freely infinitely divisible relative to the free multiplicative convolution if and only if the indicator is not less than one. We also prove how the multiplicative divisibility indicator changes under free and Boolean powers and then the multiplicative analogue of Bożejko's conjecture. We include an appendix, where the Cauchy distributions and point measures are shown to be the only fixed points of the Boolean-to-free Bercovici-Pata bijection.

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Section: On Jacobi Parameters and Free Meixner Lawssupporting

confidence: 66%

Semigroups related to additive and multiplicative, free and Boolean convolutions

Arizmendi¹

2013

Studia Math.

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“…(a) The first identity in Proposition 2 is actually the consequence of an integral transformation due to Zolotarev -see (2.2.19) with β = 1 in [58]. We have offered a separate proof which is perhaps clearer, and which enhances the similarities between the free and the classical case echoing those between (3) and (4). Observe in particular the identity…”

Section: 3mentioning

confidence: 85%

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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“…The law β a = β 1−a,1+a is FID if (and only if) 1 2 ≤ |a| < 1 [AH13a]. The free infinite divisibility for ultraspherical distributions u p was conjectured for p ≥ 1 in [AP10,Remark 4.4], and Arizmendi and Belinschi [AB13] showed that the ultraspherical distribution u n (and also the beta distribution β 1 2 ,n+ 1 2 ) is FID for n = 1, 2, 3, · · · . For beta prime distributions, β ′ 2/3,1/2 is a free stable law and so is FID [BP99, p. 1054].…”

Section: Resultsmentioning

confidence: 99%