Extending self-similarity for fractional Brownian motion

Abstract-The fractional Brownian motion (fBm) model has proven to be valuable in modeling many natural processes because of its persis-tence for large time lags. However, the model is characterized by one single parameter that cannot distinguish between short- and long-term correlation effects. This work investigates the idea of extending self-similarity to create a correlation model that generalizes discrete fBm referred to as asymptotic fBm (afBm). Namely, afBm is parameterized by variables controlling short- and long-term correlation effects. We propose a fast parameter estimation algorithm for afSm based on the Haar transform, and we demonstrate the performance of this parameter estimation algorithm with numerical simulations. I