On the linear structure of self-similar processes

Self-similar processes have a rich linear structure, based on scale invariance, which is analogous to the shift-invariant structure of stationary processes. The analogy is made explicit via Lamperti's (1962) transformation. This transformation is used here to characterize the reproducing kernel Hilbert space (RKHS) associated with self-similar processes and hence to solve problems of prediction, whitening, and Gaussian signal detection. Some specific results for the fractional Brownian motion illustrate the general concepts.