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Integral-geometric construction of self-similar stable processes. (English) Zbl 0757.60035
A generalization of fractional Brownian motions is considered. A real- valued stochastic process \(\{X(t); t\in R^ d\}\) is called an \((\alpha,H)\)-process if the following conditions are fulfilled: (i) \(\{X(t)\}\) is a symmetric stable family of index \(\alpha\); (ii) \(\{X(t)\}\) has stationary increments with respect to the action of Euclidean solid motion; (iii) \(\{X(t)\}\) is a self-similar process of exponent \(H\). Existence conditions for \((\alpha,H)\)-processes are shown. A series of examples of \((\alpha,H)\)-processes is constructed by using integral-geometric method. Some properties of the conjugate class (i.e. the set of processes which are constructed by a common integral-geometric structure) are discussed.

MSC:
60G18 Self-similar stochastic processes
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