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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.

60G18 Self-similar stochastic processes
Full Text: DOI
[1] DOI: 10.2307/3315040 · Zbl 0589.60044 · doi:10.2307/3315040
[2] DOI: 10.1007/BF00532163 · Zbl 0488.60066 · doi:10.1007/BF00532163
[3] Processus stochastiques et mouvement brownien (1965)
[4] (1990)
[5] Appl 30 pp 329– (1988)
[6] DOI: 10.1007/BF00718037 · Zbl 0627.60039 · doi:10.1007/BF00718037
[7] DOI: 10.1137/1010093 · Zbl 0179.47801 · doi:10.1137/1010093
[8] DOI: 10.1016/0304-4149(89)90082-3 · Zbl 0713.60050 · doi:10.1016/0304-4149(89)90082-3
[9] DOI: 10.1214/aop/1176993063 · Zbl 0555.60025 · doi:10.1214/aop/1176993063
[10] Harmonic Analysis and the Theory of Probability (1955)
[11] DOI: 10.1007/BF00532163 · Zbl 0488.60066 · doi:10.1007/BF00532163
[12] Nagoya Math. J 105 pp 19– (1987) · Zbl 0594.60054 · doi:10.1017/S0027763000000714
[13] Joint distributions of some self-similar stable processes (1989)
[14] DOI: 10.1137/1102019 · doi:10.1137/1102019
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