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An Optimal Wavelet Series Expansion of the Riemann–Liouville Process

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  • Helga Schack

    (FSU Jena)

Abstract

Consider the Riemann–Liouville process R α ={R α (t)} t∈[0,1] with parameter α>1/2. Depending on α, wavelet series representations for R α (t) of the form ∑ k=1 ∞ u k (t)ε k are given, where the u k are deterministic functions, and {ε k } k≥1 is a sequence of i.i.d. standard normal random variables. The expansion is based on a modified Daubechies wavelet family, which was originally introduced in Meyer (Rev. Mat. Iberoam. 7:115–133, 1991). It is shown that these wavelet series representations are optimal in the sense of Kühn–Linde (Bernoulli 8:669–696, 2002) for all values of α>1/2.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:4:d:10.1007_s10959-008-0187-1
    DOI: 10.1007/s10959-008-0187-1
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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.
    2. Eduard Belinsky & Werner Linde, 2002. "Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators," Journal of Theoretical Probability, Springer, vol. 15(3), pages 589-612, July.
    3. 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.
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