TY - JOUR

T1 - Stationarity and self-similarity characterization of the set-indexed fractional Brownian motion

AU - Herbin, Erick

AU - Merzbach, Ely

PY - 2009/10

Y1 - 2009/10

N2 - The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (J. Theor. Probab. 19(2):337-364, 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 < H < 1/2 is studied, as illustrated by some examples. When the indexing collection is totally ordered, the sifBm can be defined for 0 < H < 1.

AB - The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (J. Theor. Probab. 19(2):337-364, 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 < H < 1/2 is studied, as illustrated by some examples. When the indexing collection is totally ordered, the sifBm can be defined for 0 < H < 1.

KW - Fractional Brownian motion

KW - Gaussian processes

KW - Self-similarity

KW - Set-indexed processes

KW - Stationarity

UR - http://www.scopus.com/inward/record.url?scp=70350719885&partnerID=8YFLogxK

U2 - 10.1007/s10959-008-0180-8

DO - 10.1007/s10959-008-0180-8

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AN - SCOPUS:70350719885

SN - 0894-9840

VL - 22

SP - 1010

EP - 1029

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 4

ER -