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


 
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1. Title Title of document A Rate-Optimal Trigonometric Series Expansion of the Fractional Brownian Motion
 
2. Creator Author's name, affiliation, country Endre Iglói; University of Debrecen
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) fractional Brownian motion; function series expansion; rate of convergence; Gamma-mixed Ornstein--Uhlenbeck process
 
3. Subject Subject classification 60G15; 60G18
 
4. Description Abstract Let $B^{(H)}(t),t\in\lbrack -1,1]$, be the fractional Brownian motion with Hurst parameter $H\in (1/2,1)$. In this paper we present the series representation $B^{(H)}(t)=a_{0}t\xi_{0}+\sum_{j =1}^{\infty }a_{j}( (1-\cos (j\pi t))\xi_{j}+\sin (j\pi t)\widetilde{\xi }_{j}), t\in \lbrack -1,1]$, where $a_{j},j\in \mathbb{N}\cup {0}$, are constants given explicitly, and $\xi _{j},j\in \mathbb{N}\cup {0}$, $\widetilde{\xi }_{j},j\in \mathbb{N}$, 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 $t\in \lbrack -1,1]$. Moreover we prove that the series expansion has an optimal rate of convergence.
 
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7. Date (YYYY-MM-DD) 2005-11-19
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/287
 
10. Identifier Digital Object Identifier 10.1214/EJP.v10-287
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 10
 
12. Language English=en
 
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