Reproducing kernel Hilbert space methods for wide-sense self-similar processes

Carl J. Nuzman, H. Vincent Poor

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1199-1219
Number of pages21
JournalAnnals of Applied Probability
Volume11
Issue number4
DOIs
StatePublished - Nov 2001

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Detection
  • Estimation
  • Fractional Brownian motion
  • Innovations
  • Lamperti's transformation
  • Mellin transform
  • Reproducing kernel Hilbert space
  • Self-similar

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