A set-indexed fractional Brownian motion

Abstract : We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Levy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no "really nice" set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.
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https://hal-ecp.archives-ouvertes.fr/hal-00652069
Contributor : Erick Herbin <>
Submitted on : Wednesday, December 14, 2011 - 5:56:38 PM
Last modification on : Friday, July 26, 2019 - 2:28:53 PM

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E. Merzbach, Erick Herbin. A set-indexed fractional Brownian motion. Journal of Theoretical Probability, Springer, 2006, 19 (2), pp.337-364. ⟨hal-00652069⟩

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