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A random integral calculus on generalized \(s\)-selfdecomposable probability measures. (English) Zbl 1322.60141

Summary: The class \(\mathcal{U}_{\beta}\) of generalized \(s\)-selfdecomposable probability distributions can be viewed as an image, via the random integral mapping \(\mathcal{J}^{\beta}\), of the class \(\mathrm{ID}\) of all infinitely divisible measures. We prove that a composition of the mappings \(\mathcal{J}^{\beta_1}, \mathcal{J}^{\beta_2},\dots, \mathcal{J}^{\beta_n}\), \(\beta_1>0, \dots, \beta_n>0\), is again a random integral but with a new deterministic inner time. Moreover, some elementary formulas concerning the distributions of products of powers of independent uniformly distributed random variables are established.

MSC:

60H99 Stochastic analysis
60H05 Stochastic integrals
60E07 Infinitely divisible distributions; stable distributions
60G51 Processes with independent increments; Lévy processes
60F05 Central limit and other weak theorems
60B11 Probability theory on linear topological spaces
60B10 Convergence of probability measures
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