The Annals of Applied Probability

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

Carl J. Nuzman and H. Vincent Poor

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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.

Article information

Ann. Appl. Probab., Volume 11, Number 4 (2001), 1199-1219.

First available in Project Euclid: 5 March 2002

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Zentralblatt MATH identifier

Primary: 60G18: Self-similar processes
Secondary: 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32] 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

Self-similar reproducing kernel Hilbert space Lamperti’s transformation Mellin transform fractional Brownian motion detection estimation innovations


Nuzman, Carl J.; Poor, H. Vincent. Reproducing Kernel Hilbert Space Methods for wide-sense self-similar Processes. Ann. Appl. Probab. 11 (2001), no. 4, 1199--1219. doi:10.1214/aoap/1015345400.

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