ABSTRACT
Self-similar processes are invariant in distribution under judicious scaling of time and space. They are important in probability because of their connection to limit theorems and they are of great interest in modeling. Self-similar processes are also considered in physics, particularly in connection to the so-called "renormalization group theory" and "critical phenomena." Although the increments of Brownian motion are independent, those of other H-sssi processes can display long-range dependence or long memory. If the process is Gaussian, long-range dependence manifests itself by the presence of cycles of any order and, ultimately, by a spectral density that diverges at the origin like a power function. This chapter provides two equivalent representations for that process, a "moving average" representation and a "harmonizable" one. The increments of fractional Brownian motion are called fractional Gaussian noise.
