A Set-indexed Fractional Brownian Motion |
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Authors: | Erick Herbin Ely Merzbach |
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Institution: | (1) Dassault Aviation, 78 quai Marcel Dassault, 92552 Saint-Cloud Cedex, France;(2) Dept. of Mathematics, Bar Ilan University, 52900 Ramat-Gan, Israel |
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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 Lévy 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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Keywords: | Fractional Brownian motion Gaussian processes stationarity self-similarity set-indexed processes |
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