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 Electronic Journal of Probability > Vol. 10 (2005) > Paper 41 open journal systems 


A Rate-Optimal Trigonometric Series Expansion of the Fractional Brownian Motion

Endre Iglói,


Abstract
Let $B^{(H)}(t),tinlbrack -1,1]$, be the fractional Brownian motion with Hurst parameter $Hin (1/2,1)$. In this paper we present the series representation $B^{(H)}(t)=a_{0}txi_{0}+sum_{j =1}^{infty }a_{j}( (1-cos (jpi t))xi_{j}+sin (jpi t)widetilde{xi }_{j}), tin lbrack -1,1]$, where $a_{j},jin mathbb{N}cup {0}$, are constants given explicitly, and $xi _{j},jin mathbb{N}cup {0}$, $widetilde{xi }_{j},jin mathbb{N}$, are independent standard Gaussian random variables. We show that the series converges almost surely in $C[-1,1]$, and in mean-square (in $L^{2}(Omega )$), uniformly in $tin lbrack -1,1]$. Moreover we prove that the series expansion has an optimal rate of convergence.


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Pages: 1381-1397

Published on: November 19, 2005
Bibliography
References
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Electronic Journal of Probability. ISSN: 1083-6489