A class of multivariate infinitely divisible distributions related to arcsine density

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.

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