## Electronic Journal of Probability

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

Endre Iglói

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.ejp/1464816842

Digital Object Identifier
doi:10.1214/EJP.v10-287

Mathematical Reviews number (MathSciNet)
MR2183006

Zentralblatt MATH identifier
1109.60032

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

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