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

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

2007-06-23

arxiv.org/abs/0706.3472v3

Mathematics

Probability

18 pages

Scientific paper

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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