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2009 Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems
Werner Linde, Antoine Ayache
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Electron. J. Probab. 14: 2691-2719 (2009). DOI: 10.1214/EJP.v14-727


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.


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Werner Linde. Antoine Ayache. "Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems." Electron. J. Probab. 14 2691 - 2719, 2009.


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

zbMATH: 1193.60051
MathSciNet: MR2576756
Digital Object Identifier: 10.1214/EJP.v14-727

Primary: 60G15
Secondary: 26A33 , 41A30 , 47B06

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

Vol.14 • 2009
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