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

Author

Listed:
  • Antoine Ayache

    (Université Lille 1
    UMR CNRS 8020, CLAREE, IAE de Lille)

  • Werner Linde

    (FSU Jena, Institut für Stochastik)

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.

Suggested Citation

  • Antoine Ayache & Werner Linde, 2008. "Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion," Journal of Theoretical Probability, Springer, vol. 21(1), pages 69-96, March.
    • Handle: RePEc:spr:jotpro:v:21:y:2008:i:1:d:10.1007_s10959-007-0101-2
    DOI: 10.1007/s10959-007-0101-2
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    References listed on IDEAS

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    1. Dzhaparidze, Kacha & Zanten, Harry van, 2005. "Optimality of an explicit series expansion of the fractional Brownian sheet," Statistics & Probability Letters, Elsevier, vol. 71(4), pages 295-301, March.
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    Cited by:

    1. Helga Schack, 2009. "An Optimal Wavelet Series Expansion of the Riemann–Liouville Process," Journal of Theoretical Probability, Springer, vol. 22(4), pages 1030-1057, December.
    2. Palle Jorgensen & Feng Tian, 2019. "Realizations and Factorizations of Positive Definite Kernels," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1925-1942, December.

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