Equivalence of Infinitely Divisible Distributions

Abstract

If $F$ is an infinitely divisible distribution function without a Gaussian component whose Levy spectral measure $M$ is absolutely continuous and $M(\mathbb{R}^1\backslash\{0\}) = \infty$, then $F$ is shown to have an a.e. positive density over its support; this support of $F$ is always an interval of the form $(-\infty, \infty), (-\infty, a\rbrack$ or $\lbrack a, \infty)$. In addition, sufficient conditions are obtained for two infinitely divisible distribution functions without Gaussian components to be absolutely continuous with respect to each other, i.e., equivalent.