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 $ℝ^d$ 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 $ℝ^d$ 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 $∫_0^1\cos(2^{−1}πt) \mathrm{d}X_t$ 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.