Invariant measures under random integral mappings and marginal distributions of fractional Lévy processes (Q1933721)

This is another work of this author concerning random integral mappings which map infinitely divisible laws \(\nu\) into the laws of integrals \(\int_{(a,\,b]}h(t)dY_\nu(r(t))\), where \(0\leq a<b<\infty\); \(h:\mathbb{R}^+\to \mathbb{R}\); \(Y_\nu(\cdot)\) is an \(\mathbb{R}^d\)-valued Lévy process such that the law of \(Y_\nu(1)\) is \(\nu\); \(r:\mathbb{R}^+\to \mathbb{R}^+\) is a monotone time change. Proposition 1 investigates classes of probability measures which are invariant under the mapping. The classes covered by the proposition include, among others, selfdecomposable and generalized \(s\)-selfdecomposable distributions. It is already known that all selfdecomposable distributions are generalized \(s\)-selfdecomposable ones. Proposition 2 gives criteria for the converse inclusion. Finally, the assertions obtained are specialized to so called moving average fractional Lévy processes, thereby strengthening some results given by \textit{S. Cohen} and \textit{M. Maejima} [Stat. Probab. Lett. 81, No. 11, 1664--1669 (2011; Zbl 1227.60050)].