Makoto Maejima

In the paper by Fan\cite{F06}, he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this paper, we firstly introduce a new definition, stronger than Fan's one in general, of selfsimilarity via linear combinations of non-commutative stochastic processes, although their two definitions are equivalent for non-commutative stochastic processes with freely independent increments. We secondly prove the converse of Fan's result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for representing the background driving free L{\'e}vy processes of freely selfdecomposable distributions. A relationship between freely selfdecomposable distributions and their background driving free L{\'e}vy processes in terms of their free cumulant transforms is also given, and several examples are discussed.

In this paper we find and develop a stochastic integral representation for the class of strictly stable distributions. We establish an explicit relationship between stochastic integral and shot-noise series representations of strictly stable distributions, which shows that the class of distributions representable by stochastic integral is larger than the class representable by a shot-noise series. This inclusion is proper when the stability index is greater than 1. We also give an explicit description of distributions possessing both representations.