On quasi‐infinitely divisible distributions with a point mass

An infinitely divisible distribution on R is a probability measure μ such that the characteristic function μ̂ has a Lévy–Khintchine representation with characteristic triplet (a,γ,ν), where ν is a Lévy measure, γ∈R and a≥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 [10]. We show that a distribution μ=pδx0+(1−p)μac with p>0 and x0∈R, where μac is the absolutely continuous part, is quasi‐infinitely divisible if and only if μ̂(z)≠0 for every z∈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.