Random integral representations for free-infinitely divisible and tempered stable distributions

There are given sufficient conditions under which mixtures of dilations of Lévy spectral measures, on a Hilbert space, are Lévy measures again. We introduce some random integrals with respect to infinite-dimensional Lévy processes, which in turn give some integral mappings. New classes (convolution semigroups) are introduced. One of them gives an unexpected relation between the free (Voiculescu) and the classical Lévy–Khintchine formulae while the second one coincides with tempered stable measures (Mantegna and Stanley) arisen in statistical physics.