An Optimal Wavelet Series Expansion of the Riemann–Liouville Process

Helga Schack ()
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Helga Schack: FSU Jena

Journal of Theoretical Probability, 2009, vol. 22, issue 4, 1030-1057

Abstract: 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.

Keywords: Riemann–Liouville process; Approximation of Gaussian processes; Wavelet series representation; Rate optimality; Daubechies wavelets; 60G15; 41A58; 60G18; 41A25 (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10959-008-0187-1

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