@article{EJP287,
author = {Endre Iglói},
title = {A Rate-Optimal Trigonometric Series Expansion of the Fractional Brownian Motion},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {10},
year = {2005},
keywords = {fractional Brownian motion; function series expansion; rate of convergence; Gamma-mixed Ornstein--Uhlenbeck process},
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.},
pages = {no. 41, 1381-1397},
issn = {1083-6489},
doi = {10.1214/EJP.v10-287},
url = {http://ejp.ejpecp.org/article/view/287}}