Infinite Divisibility of Probability Distributions on the Real Line

Steutel, F.W.; Harn, Klaas Van

doi:10.1201/9780203014127

Cited by 311 publications

(435 citation statements)

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“…Some references for tempered stable distributions and tempered stable Lévy processes are Rosiński (2002), Cont and Tankov (2003), Schoutens (2003) and Steutel and van Harn (2004). Note, that several names and origins, and many different parameterisations are used for the tempered stable distributions and processes.…”

Section: The Tempered Stable Ladder Processmentioning

confidence: 99%

Old and New Examples of Scale Functions for Spectrally Negative Lévy Processes

Hubalek

Kyprianou

2011

Seminar on Stochastic Analysis, Random Fields and Applications VI

102

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We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations. Spectrally negative Lévy processes and scale functionsLet X = {X t : t ≥ 0} be a Lévy process defined on a filtered probability space (Ω, F, F, P), where {F t : t ≥ 0} is the filtration generated by X satisfying the usual conditions. For x ∈ R denote by P x the law of X when it is started at x and write simply P 0 = P. Accordingly we shall write E x and E for the associated expectation operators. In this paper we shall assume throughout that X is spectrally negative meaning here that it has no positive jumps and that it is not the negative of a subordinator. It is well known that the latter allows us to talk about the Laplace exponent ψ(θ) := log E[e θX1 ] for (θ) ≥ 0 where in particular we have the Lévy-Khintchine representationwhere a ∈ R, σ ≥ 0 is the Gaussian coefficient and Π is a measure concentrated on (−∞, 0) satisfying (−∞,0) (1∧x 2 )Π(dx) < ∞. The, so-called, Lévy triple (a, σ, Π) completely characterises the process X. For later reference we also introduce the function Φ : [0, ∞) → [0, ∞) as the right inverse of ψ on (0, ∞) so that for all q ≥ 0 Φ(q) = sup{θ ≥ 0 : ψ(θ) = q}.(2)

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Section: The Tempered Stable Ladder Processmentioning

confidence: 99%

Old and New Examples of Scale Functions for Spectrally Negative Lévy Processes

Hubalek

Kyprianou

2011

Seminar on Stochastic Analysis, Random Fields and Applications VI

102

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“…Since X(t) is not degenerate, this would imply that X(t) is Gaussian (see [35,p. 200,Corollary 9.9]).…”

Section: Fractional Laplace Motionmentioning

confidence: 99%

Fractional Laplace motion

2006

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Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic selfsimilarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

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“…First recall a well known result on a characterization of the Gaussian distribution in terms of the tail behavior; see [30,Corollary 4.9.9]: a non-degenerate infinitely divisible random variable X has a normal distribution if, and only if, it satisfies lim sup…”

Section: Non Infinite Divisibility Of Tracy-widom Distributionsmentioning

confidence: 99%

The Tracy–Widom distribution is not infinitely divisible

Domínguez-Molina

2017

Statistics & Probability Letters

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The classical infinite divisibility of distributions related to eigenvalues of some random matrix ensembles is investigated. It is proved that the β-Tracy-Widom distribution, which is the limiting distribution of the largest eigenvalue of a β-Hermite ensemble, is not infinitely divisible. Furthermore, for each fixed N ≥ 2 it is proved that the largest eigenvalue of a GOE/GUE random matrix is not infinitely divisible.

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Infinite Divisibility of Probability Distributions on the Real Line

Cited by 311 publications

References 0 publications

Old and New Examples of Scale Functions for Spectrally Negative Lévy Processes

Old and New Examples of Scale Functions for Spectrally Negative Lévy Processes

Fractional Laplace motion

The Tracy–Widom distribution is not infinitely divisible

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