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

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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⊂ℝN 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.

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Ayache, A., Linde, W. Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion. J Theor Probab 21, 69–96 (2008). https://doi.org/10.1007/s10959-007-0101-2

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  • DOI: https://doi.org/10.1007/s10959-007-0101-2

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