nth-order fractional Brownian motion and fractional Gaussian noises

Abstract

A generalization of fractional Brownian motion (fBm) of parameter H in ]0, 1[ is proposed. More precisely, this work leads to nth-order fBm (n-fBm) of H parameter in ]n - 1, n[, where n is any strictly positive integer. They include fBm for the special case n = 1. Properties of these new processes are investigated. Their covariance function are given, and it is shown that they are self similar. In addition, their spectral shape is assessed as 1/fα belonging to ]1; +∞[, providing a larger framework than classical fBm. Special interest is given to their nth-order stationary increments, which extend fractional Gaussian noises. Covariance function and power spectral densities are calculated. Properties and signal processing tasks such as a Cholesky-type synthesis technique and a maximum likelihood estimation method of the H parameter are presented. Results show that the estimator is efficient (unbiased and reaches the Cramér-Rao lower bound) for a large majority of tested values.