Stationarity and Self-similarity Characterization of the Set-indexed Fractional Brownian Motion

Abstract : The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to statisfy a strenghtened definition of increment stationarity. This new definition for stationarity property allows to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dimensional fractional Brownian motion, the limitation of its definition for a self-similarity parameter 0
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Submitted on : Wednesday, December 14, 2011 - 5:45:17 PM
Last modification on : Friday, July 26, 2019 - 2:14:28 PM

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Erick Herbin, Ely Merzbach. Stationarity and Self-similarity Characterization of the Set-indexed Fractional Brownian Motion. Journal of Theoretical Probability, Springer, 2009, 22 (4), pp.1010-1029. ⟨10.1007/s10959-008-0180-8⟩. ⟨hal-00652063⟩

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