Karhunen–Loève expansion of a set indexed fractional Brownian motion

This study presents the Karhunen–Loève expansion of a set indexed fractional Brownian motion (sifBM) XH={XAH}A∈A, based on “characterization of set indexed fractional Brownian motion by flows”. The characterization was proven by Herbin and Merzbach (2006), and it says that a set indexed process is a set indexed fractional Brownian motion if and only if its projections on all the increasing paths are one-parameter time changed fractional Brownian motions. The Karhunen–Loève expansion of a sifBM is: XAH=[μ(A)]H∑n=1∞eBnNnJν(γn(μ(A))H+12), for all A∈A.Where {eBn} is an orthonormal sequence of set indexed centered Gaussian variables, Jν is a Bessel function, Nn,γn are constants and {Bn}∈A. In addition, several cases of an expansion of a sifBM on [0,1]d are presented.