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

For $0<\alpha \le \mathrm{\infty }$, 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.