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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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##### References:
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