arXiv subjects
Publications and source records attributed to David Berger.
We derive some estimates for the integral modulus of continuity of probability densities of infinitely divisible distributions. The paper is splitted into two parts. The first part deals with general infinitely divisible distributions. The second part is mainly concerned with densities of random integrals with respect to a Lévy process. We will see major differences between compact and non-compact supports.
An infinitely divisible distribution on $\mathbb{R}$ is a probability measure $μ$ such that the characteristic function $\hatμ$ has a Lévy-Khintchine representation with characteristic triplet $(a,γ, ν)$, where $ν$ is a Lévy measure, $γ\in\mathbb{R}$ and $a\ge 0$. A natural extension of such distributions are quasi-infinitely distributions. Instead of a Lévy measure, we assume that $ν$ is a "signed Lévy measure", for further information on the definition see [\ref{Lindner}]. We show that a distribution $μ=pδ_{x_0}+(1-p)μ_{ac}$ with $p>0$ and $x_0 \in \mathbb{R}$, where $μ_{ac}$ is the absolutely continuous part, is quasi-infinitely divisible if and only if $\hatμ(z)\neq0$ for every $z\in\mathbb{R}$. We apply this to show that certain variance mixtures of mean zero normal distributions are quasi-infinitely divisible distributions, and we give an example of a quasi-infinitely divisible distribution that is not continuous but has infinite quasi-Lévy measure. Furthermore, it is shown that replacing the signed Lévy measure by a seemingly more general complex Lévy measure does not lead to new distributions. Last but not least it is proven that the class of quasi-infinitely divisible distributions is not open, but path-connected in the space of probability measures with the Prokhorov metric.