Type A Distributions: Infinitely Divisible Distributions Related to Arcsine Density

Makoto Maejima, Victor Perez-Abreu, Ken-iti Sato

Published 2010-07-05Version 1

Two transformations $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ of L\'{e}vy 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 L\'{e}vy measures being in the common range are called type $A$ distributions and expressed as the law of a stochastic integral $\int_0^1\cos (2^{-1}\pi t)dX_t$ with respect to L\'{e}vy 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.