Original article at: http://www.math.washington.edu/~ejpecp/viewarticle.php?id=2041

Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems

Werner Linde, FSU Jena
Antoine Ayache, UMR CNRS 8524 , Laboratoire Paul Painleve

Abstract

The aim of the present paper is to investigate series representations of the Riemann--Liouville process Rα, α>1/2, generated by classical orthonormal bases in L2 [0,1]. Those bases are, for example, the trigonometric or the Haar system. We prove that the representation of Rα via the trigonometric system possesses the optimal convergence rate if and only if 1/2<α≤ 2. For the Haar system we have an optimal approximation rate if 1/2<α<3/2 while for α>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 α>3/2 and α=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 α= 3/2 a representation of the Riemann­Liouville process via the Haar system is also not optimal.

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Original article at: http://www.math.washington.edu/~ejpecp/viewarticle.php?id=2041