A class of multivariate infinitely divisible distributions related to arcsine density

Abstract

Two transformations ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ of Lévy measures on ${\mathbb{R}}^d$ based on the arcsine density are studied and their relation to general Upsilon transformations is considered. The domains of definition of ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ are determined and it is shown that they have the same range. The class of infinitely divisible distributions on ${\mathbb{R}}^d$ with Lévy measures being in the common range is called the class $A$ and any distribution in the class $A$ is expressed as the law of a stochastic integral $\int_0^1\cos(2^{-1}\uppi t)\,\mathrm{d}X_t$ with respect to a Lévy process $\{X_t\}$. This new class includes as a proper subclass the Jurek class of distributions. It is shown that generalized type $G$ distributions are the image of distributions in the class $A$ under a mapping defined by an appropriate stochastic integral. ${\mathcal{A}}_2$ is identified as an Upsilon transformation, while ${\mathcal{A}}_1$ is shown not to be.

arXiv commentPublished in at http://dx.doi.org/10.3150/10-BEJ348 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin note: substantial text overlap with arXiv:1007.0687

References in corpus (2)

• Transformations of infinitely divisible distributions via improper stochastic integrals
• Classes of infinitely divisible distributions on R^d related to the class of selfdecomposable distributions

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