Reproducing Kernel Hilbert Space Methods for wide-sense self-similar Processes

It has recently been observed that wide-sense self-similar processes have a rich linear structure analogous to that of wide-sense stationary processes. In this paper, a reproducing kernel Hilbert space (RKHS) approach is used to characterize this structure. The RKHS associated with a selfsimilar process on a variety of simple index sets has a straightforward description, provided that the scale-spectrum of the process can be factored. This RKHS description makes use of the Mellin transform and linear self-similar systems in much the same way that Laplace transforms and linear time-invariant systems are used to study stationary processes. The RKHS results are applied to solve linear problems including projection, polynomial signal detection and polynomial amplitude estimation, for general wide-sense self-similar processes. These solutions are applied specifically to fractional Brownian motion (fBm). Minimum variance unbiased estimators are given for the amplitudes of polynomial trends in fBm, and two new innovations representations for fBm are presented. 1. Introduction. This paper is concerned with the linear problems associated with wide-sense self-similar processes, that is, with processes whose first and second moments are essentially scale-invariant. In the linear problems we are referring to, the solution is constrained to be a linear functional of an observed random process, and the criterion of optimality is such that the solution is determined by the first and second moments of the observed process. Examples include linear projection with the mean-square error metric, maximum signal-to-noise ratio signal detection and minimum-variance unbiased linear estimation. For Gaussian processes, these linear solutions are still optimal when the linearity constraint is removed. The main finding is that wide-sense self-similar processes have essentially the same structure as wide-sense stationary processes, and that concepts such as autocorrelation and spectral density, used to study stationary processes, have simple analogs useful in the study of self-similar processes. The connection between these two classes was first made by Lamperti (1962), who pointed out a simple invertible transformation which connects any self-similar process with a stationary counterpart, and which is a central idea in this paper. Other recent workmak ing use of Lamperti’s transformation includes Albin (1998),