Further examples of GGC and HCM densities

Jedidi, Wissem; Simon, Thomas

doi:10.3150/12-bej431

Cited by 15 publications

(23 citation statements)

References 19 publications

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“…First, we analyze in more detail the Kanter random variable K α , which plays an important role in the proof of all four theorems. The range α < 1/5 is particularly investigated, and two conjectures made in [32] and [17] are answered in the negative. A curious Airy-type function is displayed in the case α = 1/5.…”

Section: Theorem 4 One Hasmentioning

confidence: 99%

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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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Section: Theorem 4 One Hasmentioning

confidence: 99%

“…On the one hand, for every t ∈ (0, 1), the function [49]. On the other hand, by the aforementioned Corollary 3.2 in [32], the function z → f Kα (b α + z) is CM. Hence, by e.g.…”

Section: 21mentioning

confidence: 99%

Some properties of the free stable distributions

Simon¹,

Wang²

2020

Ann. Inst. H. Poincaré Probab. Statist.

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“…To do so, we first observe that by Corollary 3 in [13], it is enough to show that − log(β a,b ) ∈ M. This latter property follows from Steutel's theorem as had already been noticed in Example VI.12.21 in [18], but we will sketch the argument for the sake of completeness. Using Malmstén's formula for the Gamma function -see e.g.…”

Section: Proofsmentioning

confidence: 75%

“…For instance, they are connected to real stable densities via the Kanter random variable -see (2.4) and (7.1) in [17]. It was conjectured in [13] that β −s a,b ∈ G for all s ≥ 1 -see Conjecture 2 therein. It does not seem possible to express the Laplace exponent of β −s a,b or β −s a,b − 1 in a sufficiently explicit way in order to show that it is a Bernstein function.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

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On the infinite divisibility of inverse Beta distributions

Bosch

Simon²

2015

Bernoulli

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We show that all negative powers β −s a,b of the Beta distribution are infinitely divisible. The case b ≤ 1 follows by complete monotonicity, the case b > 1, s ≥ 1 by hyperbolically complete monotonicity and the case b > 1, s < 1 by a Lévy perpetuity argument involving the hypergeometric series. We also observe that β −s a,b is self-decomposable if and only if 2a + b + s + bs ≥ 1, and that in this case it is not necessarily a generalized Gamma convolution. On the other hand, we prove that all negative powers of the Gamma distribution are generalized Gamma convolutions, answering to a recent question of L. Bondesson.

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Unimodality of Hitting Times for Stable Processes

Letemplier

Simon

2014

Lecture Notes in Mathematics

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Abstract. We show that the hitting times for points of real α−stable Lévy processes (1 < α ≤ 2) are unimodal random variables. The argument relies on strong unimodality and several recent multiplicative identities in law. In the symmetric case we use a factorization of Yano et al. [15], whereas in the completely asymmetric case we apply an identity of the second author [11]. The method extends to the general case thanks to a fractional moment evaluation due to Kuznetsov et al. [6]. Introduction and statement of the resultA real random variable X is said to be unimodal if there exists a ∈ R such that its distribution function P[X ≤ x] is convex on (−∞, a) and concave on (a, +∞). When X is absolutely continuous, this means that its density is non-decreasing on (−∞, a] and nonincreasing on [a, +∞). The number a is called a mode of X and might not be unique. A random variable with a single mode is called strictly unimodal. The problem of unimodality has been intensively studied for infinitely divisible random variables and the reader can consult to Chapter 10 in [10] for details and references. This problem has also been settled in the framework of hitting times of processes and Rösler -see Theorem 1.2 in [9] -showed that hitting times for points of real-valued diffusions are always unimodal. However, much less is known when the underlying process has jumps, for example when it is a Lévy process.In this paper we consider a real strictly α−stable process (1 < α ≤ 2), which is a Lévy process {X t , t ≥ 0} starting from zero and having characteristic exponentwhere ρ ∈ [1 − 1/α, 1/α] is the positivity parameter of {X t , t ≥ 0} that is ρ = P[X 1 ≥ 0]. We refer to [16] and to Chapter 3 in [10] for an account on stable laws and processes. In particular, comparing the parametrisations (B) and (C) in the introduction of [16] shows that the characteristic exponent of {X t , t ≥ 0} takes the more familiar formwith ρ = 1/2 + (1/πα) tan −1 (θ tan(πα/2)) and c = cos(πα(ρ − 1/2)). The constant c is a scaling parameter which could take any arbitrary positive value without changing our purposes below. We are interested in the hitting times for points of {X t , t ≥ 0}: τ x = inf{t > 0, X t = x}, x ∈ R.2000 Mathematics Subject Classification. 60E05, 60G18, 60G51, 60G52.

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Further examples of GGC and HCM densities

Cited by 15 publications

References 19 publications

Some properties of the free stable distributions

Some properties of the free stable distributions

On the infinite divisibility of inverse Beta distributions

Unimodality of Hitting Times for Stable Processes

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