A random integral calculus on generalized s-selfdecomposable probability measures

It is known that the class $\mathcal{U}_{\beta}$, of generalized s-selfdecom-posable probability distributions, can be viewed as an image via random integral mapping $\mathcal{J}^{\beta}$ of the class $ID$ of all infinitely divisible measures. We prove that a composition of the mappings $\mathcal{J}^{\beta_1}, \mathcal{J}^{\beta_2}, ..., \mathcal{J}^{\beta_n}$ is again random integral mapping but with a new inner time. In a proof some form of Lagrange interpolation formula is needed. Moreover, some elementary formulas concerning the distributions of products of powers of independent uniformly distributed random variables as established as well.