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 self-similar
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.
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