Random self-decomposability and autoregressive processes (Q1957153)

The authors introduce the random self-decomposability. We say that a distribution with the characteristic function \(\psi\) is random self-decomposable if for each \(p\in[0,1]\) and \(c\in[0,1]\) there exists a probability distribution with the characteristic function \(\psi_{c,p}\) such that \(\psi(t)=\psi_{c,p}(t)(p+(1-p)\psi(ct))\). The authors show that \({\mathcal C}_{RSD}={\mathcal C}_{SD}\cap {\mathcal C}_{GID}\), where \({\mathcal C}_{RSD}\), \({\mathcal C}_{SD}\) and \({\mathcal C}_{GID}\) are the classes of random self-decomposable, self-decomposable and geometrically infinitely divisible distributions, respectively. Also, they provide the connections between the random self-decomposability and AR structures.