Stationarity and Self-similarity Characterization of the Set-indexed Fractional Brownian Motion - CentraleSupélec Access content directly
Journal Articles Journal of Theoretical Probability Year : 2009

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

Dates and versions

hal-00652063 , version 1 (14-12-2011)

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