Electronic Journal of Probability

Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems

Werner Linde and Antoine Ayache

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The aim of the present paper is to investigate series representations of the Riemann-Liouville process $R^\alpha$, $\alpha >1/2$, generated by classical orthonormal bases in $L_2[0,1]$. Those bases are, for example, the trigonometric or the Haar system. We prove that the representation of $R^\alpha$ via the trigonometric system possesses the optimal convergence rate if and only if $1/2 < \alpha\leq 2$. For the Haar system we have an optimal approximation rate if $1/2 < \alpha <3/2$ while for $\alpha > 3/2$ a representation via the Haar system is not optimal. Estimates for the rate of convergence of the Haar series are given in the cases $\alpha > 3/2$ and $\alpha = 3/2$. However, in this latter case the question whether or not the series representation is optimal remains open. Recently M. A. Lifshits answered this question (cf. [13]). Using a different approach he could show that in the case $\alpha = 3/2$ a representation of the Riemann-Liouville process via the Haar system is also not optimal.

Article information

Electron. J. Probab., Volume 14 (2009), paper no. 94, 2691-2719.

Accepted: 21 December 2009
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 26A33: Fractional derivatives and integrals 47B06: Riesz operators; eigenvalue distributions; approximation numbers, s- numbers, Kolmogorov numbers, entropy numbers, etc. of operators 41A30: Approximation by other special function classes

Approximation of operators and processes Rie-mann--Liouville operator Riemann--Liouville process Haar system trigonometric system

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Linde, Werner; Ayache, Antoine. Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems. Electron. J. Probab. 14 (2009), paper no. 94, 2691--2719. doi:10.1214/EJP.v14-727. https://projecteuclid.org/euclid.ejp/1464819555

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