The random integral representation hypothesis revisited : new classes of s-selfdecomposable laws

For $\,0<\alpha\le \infty$, new subclasses $\,\mathcal{U}^{<\alpha>}$ of the class $\,\mathcal{U}$, of s-selfdecomposable probability measures, are studied. They are described by random integrals, by their characteristic functions and their L\'evy spectral measures. Also their relations with the classical L\'evy class $L$ of selfdecomposable distributions are investigated.