Spectral representation of semistable processes, and semistable laws on Banach spaces

We introduce the notion of semistable processes and semistable random measures; and give a characterization of semistable laws on Banach spaces. Using this charcterization, we discuss the existence of semistable random measures, define the stochastic integrals with respect to these measures, and obtain the spectral representations of arbitrary (not necessairly symmetric) semistable and stable processes. In addition, we give a criterion of independence for stochastic integrals.