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

TL;DR

This paper characterizes the set-indexed fractional Brownian motion (sifBm) by its fractal properties, proving it is the unique Gaussian process with self-similarity and stationary increments, and explores limitations based on the Hurst parameter.

Contribution

It provides a complete characterization of sifBm through a strengthened stationarity definition and analyzes the parameter range for self-similarity.

Findings

sifBm satisfies a strengthened stationarity property

sifBm is uniquely characterized by self-similarity and stationary increments

limitations of the Hurst parameter range are identified

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<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.