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Minimal integral representations of stable processes. (English) Zbl 1121.60032
The notion of minimal representations was introduced by C. D. Hardin jun. [J. Multivariate Anal. 12, 385–401 (1982; Zbl 0493.60046)] for symmetric $$\alpha$$-stable processes. The present paper defines minimal integral representations for general stochastic processes. In the case of strictly stable or symmetric stable processes, a characterization is given in terms of rigid subsets of $$L^p$$-spaces, a notion that is introduced and investigated in this paper. This characterization is used to establish the minimality of the representation for a large class of examples of stable processes.

##### MSC:
 60G07 General theory of stochastic processes 60G57 Random measures 60E07 Infinitely divisible distributions; stable distributions 60G25 Prediction theory (aspects of stochastic processes)