Stationarity and Self-Similarity Characterization of the Set-Indexed Fractional Brownian Motion

Erick Herbin () and Ely Merzbach ()
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Erick Herbin: Ecole Centrale Paris
Ely Merzbach: Bar Ilan University

Journal of Theoretical Probability, 2009, vol. 22, issue 4, 1010-1029

Abstract: Abstract The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin–Merzbach (J. Theor. Probab. 19(2):337–364, 2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to satisfy a strengthened definition of increment stationarity. This new definition for stationarity property allows us 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

Keywords: Fractional Brownian motion; Gaussian processes; Stationarity; Self-similarity; Set-indexed processes; 62G05; 60G15; 60G17; 60G18 (search for similar items in EconPapers)
Date: 2009
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-008-0180-8

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