Time-frequency and dual-frequency representation of fractional brownian motion

Abstract

Fractional Brownian motion (fBm) is a useful non-stationary model for certain fractal and long-range dependent processes of interest in telecommunications, physics, biology, and finance. Conventionally, the power spectrum of fBm is claimed to be a fractional power-law. However, fBm is not a wide-sense stationary process, so the precise meaning of this spectrum is unclear. In this paper, we model and analyze fBm in the context of harmonizable random processes. We derive and interpret exact expressions for novel useful complex valued second-order moment functions for fBm. These moment functions are time-frequency and dual-frequency correlation functions, connecting the random process to its infinitesimal random Fourier generator. In particular, we derive and discuss the time-frequency Rihaczek spectrum, and the dual-frequency Loeve spectrum. Our main finding is that the dual-frequency spectrum of fBm has its spectral support confined to three discrete lines. This leads to the surprising conclusion that for fBm, the DC component of the infinitesimal Fourier generator is correlated with ail other frequencies of the Fourier generator. We propose and apply multitaper based estimators for the moment functions, and numerical estimates based on synthetic fBm data and real world earthquake data confirm our theoretical results