Type A Distributions: 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. Infinitely divisible distributions on ${\mathbb{R}}^d$ with Lévy measures being in the common range are called type $A$ distributions and expressed as the law of a stochastic integral ${\int }_0^1\cos (2^{-1}\pi t)d{X}_t$ with respect to 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 type $A$ distributions under a mapping defined by an appropriate stochastic integral. ${\mathcal{A}}_2$ is identified as an Upsilon transformation, while ${\mathcal{A}}_1$ is shown to be not.

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

Cited by in corpus (1)

• Description of limits of ranges of iterations of stochastic integral mappings of infinitely divisible distributions