Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions

Halgreen, C.

doi:10.1007/bf00533246

Cited by 122 publications

(77 citation statements)

References 4 publications

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“…Classical GIG distribution is selfdecomposable [14,31] (more strongly, hyperbolically completely monotone [10, p. 74]), and hence it is natural to ask whether free GIG distribution is freely selfdecomposable. A distribution μ is said to be freely selfdecomposable (FSD) [5] if for any c ∈ (0, 1) there exists a probability measure μ c such that μ = (D c μ) …”

Section: Free Selfdecomposabilitymentioning

confidence: 99%

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: Free Selfdecomposabilitymentioning

confidence: 99%

On Free Generalized Inverse Gaussian Distributions

Szpojankowski

2018

Complex Anal. Oper. Theory

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“…The expectation exists provided that ν > 2, and the nth moment exists provided that ν > 2n. However, it is possible to obtain the Lévy-Khinchin representation of the characteristic function of the symmetric t-distribution T (ν, δ, µ) directly from the results of Halgreen (1979), by choosing α = |β| = 0, δ > 0, and λ = − 1 2 ν < 0. We obtain…”

Section: Density and Characteristic Functionsmentioning

confidence: 99%

Student processes

Heyde

Leonenko

2005

Adv. Appl. Probab.

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“…Self-decomposable distributions can be consider as candidate for the unit period distribution of asset returns. Halgreen (1977) is shown that, the hyperbolic distributions are selfdecomposable.…”

Section: Laws Of Class Lmentioning

confidence: 97%

Pure -Jump Levy Processes and Self-decomposability in Financial Modeling

Önalan¹

2011

JMR

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In this study, we review the connections between Lévy processes with jumps and self-decomposable laws. Self-decomposable laws constitute a subclass of infinitely divisible laws. Lévy processes additive processes and independent increments can be related using self-similarity property. Sato (1991) defined additive processes as a generalization of Lévy processes. In this way, additive processes are those processes with inhomogeneous (in general) and independent increments and Lévy processes correspond with the particular case in which the increments are time homogeneous. Hence Lévy processes are considerable as a particular type. Self-decomposable distributions occur as limit law an Ornstein-Uhlenbeck type process associated with a background driving Lévy process. Finally as an application, asset returns are representing by a normal inverse Gaussian process. Then to test applicability of this representation, we use the nonparametric threshold estimator of the quadratic variation, proposed by Cont and Mancini (2007).

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Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions

Cited by 122 publications

References 4 publications

On Free Generalized Inverse Gaussian Distributions

On Free Generalized Inverse Gaussian Distributions

Student processes

Pure -Jump Levy Processes and Self-decomposability in Financial Modeling

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