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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