Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion

Antoine Ayache () and Werner Linde ()
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Antoine Ayache: Université Lille 1
Werner Linde: FSU Jena, Institut für Stochastik

Journal of Theoretical Probability, 2008, vol. 21, issue 1, 69-96

Abstract: Abstract We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so–called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self–adjoint integral operator in a suitable L 2(μ)–space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion X H with Hurst parameter H∈(0,1), indexed by a self–similar set T⊂ℝ N of Hausdorff dimension D. This rate turns out to be of order n −H/D (log n)1/2. In the case T=[0,1] N we present a concrete wavelet representation of X H leading to an approximation of X H with the optimal rate n −H/N (log n)1/2.

Keywords: Random Gaussian fields; Approximation of operators and processes; Fractional Brownian motion; Self–similar sets; Metric entropy; Hölder operators; Wavelet representation (search for similar items in EconPapers)
Date: 2008
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-007-0101-2

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