Electronic Journal of Probability

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

Endre Iglói

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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.

Article information

Electron. J. Probab., Volume 10 (2005), paper no. 41, 1381-1397.

Accepted: 19 November 2005
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60G18: Self-similar processes

fractional Brownian motion function series expansion rate of convergence Gamma-mixed Ornstein--Uhlenbeck process

This work is licensed under aCreative Commons Attribution 3.0 License.


Iglói, Endre. A Rate-Optimal Trigonometric Series Expansion of the Fractional Brownian Motion. Electron. J. Probab. 10 (2005), paper no. 41, 1381--1397. doi:10.1214/EJP.v10-287. https://projecteuclid.org/euclid.ejp/1464816842

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  • Dzhaparidze, Kacha; van Zanten, Harry. A series expansion of fractional Brownian motion. Probab. Theory Related Fields 130 (2004), no. 1, 39–55.
  • Dzhaparidze, Kacha; van Zanten, Harry. Optimality of an explicit series expansion of the fractional Brownian sheet. Statist. Probab. Lett. 71 (2005), no. 4, 295–301. http://www.math.vu.nl/stochastics/publications.php http://www.math.vu.nl/stochastics/publications.php
  • Iglói, E.; Terdik, G. Long-range dependence through gamma-mixed Ornstein-Uhlenbeck process. Electron. J. Probab. 4 (1999), no. 16, 33 pp. (electronic).
  • Kühn, Thomas; Linde, Werner. Optimal series representation of fractional Brownian sheets. Bernoulli 8 (2002), no. 5, 669–696.
  • van der Vaart, Aad W.; Wellner, Jon A. Weak convergence and empirical processes. With applications to statistics. Springer Series in Statistics. Springer-Verlag, New York, 1996. xvi+508 pp. ISBN: 0-387-94640-3