Tyrone E. Duncan
University of Kansas
Fractional Gaussian noise is a stochastic process that is the formal derivative of a fractional Brownian motion. Fractional Brownian motions are a family of Gaussian processes that are indexed by the Hurst parameter, H, in the interval (0, 1). These processes have a property of self similarity of the probability laws under a scale change in time. For H in (1/2, 1) these processes have a property of long range dependence that is characterized by the relatively slow decay of the correlation (or covariance) function. These properties of self similarity and long range dependence have been observed empirically, initially in hydrology by Hurst, subsequently in economic data, and most recently in telecommunications. This presentation will focus on the fractional Brownian motions with a long range dependence, that is, H in (1/2, 1) and will contrast the results with those for Brownian motion, that is, H = _. To analyze systems with fractional Gaussian noise, it is important to have an effective stochastic calculus for fractional Brownian motion by analogy to the use of white Gaussian noise in systems via a stochastic calculus for Brownian motion. Various properties of linear systems excited by fractional Gaussian noise are considered. Explicit solutions for the linear systems are given, conditions for stability and instability are described and limiting distributions are characterized. Some Radon-Nikodym derivatives, that are important for statistical applications, are described. Some applications to linear filtering and the calculation of mutual information are given. Some relations to the spectral description of a fractional Gaussian noise are given.