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A Rate-Optimal Trigonometric Series Expansion of the Fractional Brownian Motion
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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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Full text: PDF
Pages: 1381-1397
Published on: November 19, 2005
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Bibliography
References
[1] K. Dzhaparidze and H. van Zanten. A series expansion of fractional Brownian motion. Probab. Theory Relat. Fields 130 (2004), 3955. Math Review:(2005i:60065).
[2] K. Dzhaparidze and H. van Zanten. Optimality of an explicit series expansion of fractional Brownian sheet. Statistics and Probability Letters 71 (2005), 295301. Math Review .
[3] E. Iglói and G. Terdik. Long-range dependence through Gamma-mixed Ornstein-Uhlenbeck process. Electronic Journal of Probability (EJP) 4 (1999), 133. Math Review
[5] A. W. van der Vaart and J. A. Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Verlag, New York, 1996. Math. Review:MR1385671 (97g:60035).
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