Spectral representation of semistable processes, and semistable laws on Banach spaces (Q1089984)

The spectral representation of symmetric stable processes are useful in studying sample path properties as well as prediction and estimation problems; cf. \textit{C. D. Hardin} jun., ibid. 12, 385-401 (1982; Zbl 0493.60046). In this paper the authors give a definition of stochastic integrals for a suitable class of functions relative to semistable and stable random measures. Moreover they obtain spectral representations for arbitrary (not necessarily symmetric) semistable and stable processes of index \(0<\alpha <2\), \(\alpha\neq 1.\) A condition for independence of infinitely divisible Banach-valued random variables, in terms of their Lévy measure, is given. As a consequence a criterion for independence of stochastic integrals is deduced. Reviewer's remark: The spectral representation theorem for general (i.e., possibly skewed or asymmetric) stable processes is also due to \textit{C. D. Hardin} jun., Univ. North Carolina, Center for Stochastic Processes, Technical Report {\#}79 (September 1984).