A Class of Probability Distributions that is Closed with Respect to Addition as Well as Multiplication of Independent Random Variables

Bondesson, Lennart

doi:10.1007/s10959-013-0523-y

Cited by 32 publications

(46 citation statements)

References 20 publications

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“…Hence, the ID character of C a,b follows from that of T, which is a consequence of Theorem 1 and the convergence in law (15) (…”

Section: 3mentioning

confidence: 80%

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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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“…Hence, the ID character of C a,b follows from that of T, which is a consequence of Theorem 1 and the convergence in law (15) (…”

Section: 3mentioning

confidence: 80%

“…to be compared with that of (15). This shows that distributions of the free Gumbel random variable − log U and that of the drifted exceptional 1-free stable random variable T − 1 can be viewed as "log free stable" distributions.…”

Section: 5mentioning

confidence: 98%

Some properties of the free stable distributions

Simon¹,

Wang²

2020

Ann. Inst. H. Poincaré Probab. Statist.

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“…This result is only a limit case of [6, Theorem 6.2.4]. The following recent result from [8] needs to be mentioned. It can be proved by the help of Proposition 4.…”

Section: Generalized Gamma Convolutionsmentioning

confidence: 95%

A class of scale mixtures of $\operatorname{Gamma}(k)$-distributions that are generalized gamma convolutions

Behme¹,

Bondesson²

2017

Bernoulli

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Let k > 0 be an integer and Y a standard Gamma(k) distributed random variable. Let X be an independent positive random variable with a density that is hyperbolically monotone (HM) of order k. Then Y ·X and Y /X both have distributions that are generalized gamma convolutions (GGCs). This result extends a result of Roynette et al. from 2009 who treated the case k = 1 but without use of the HM-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of Lévy processes are mentioned.2010 Mathematics subject classification. 60E05, 60E10 (primary), 60G17, 60G51, 60J60 (secondary)

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“…In the case α ≤ 1/2, this representation entails that Z −1 α factorizes into the product of a Gamma random variable Γ α and another infinite Beta product. Using a recent key-result by Bondesson [5] on the stability of the GGC property by independent multiplication of random variables, we can then deduce, without much effort, that this latter factorization has indeed an HCM density.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 93%

“…. n. By the main result of [5], the law of B −1 a 1 −c,b 1 × · · · × B −1 an−c,bn − 1 is hence a GGC as well. A combination of Theorem 5.4.1 and Property (ii) p.68 in [2] shows finally that g is HCM.…”

Section: Proof Of the Theoremmentioning

confidence: 90%