A random integral calculus on generalized s-selfdecomposable probability measures

It is known that the class $\mathcal{U}_β$, of generalized s-selfdecom-posable probability distributions, can be viewed as an image via random integral mapping $\mathcal{J}^β$ of the class $ID$ of all infinitely divisible measures. We prove that a composition of the mappings $\mathcal{J}^{β_1}, \mathcal{J}^{β_2}, ..., \mathcal{J}^{β_n}$ 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.

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