Let \$B\$ be a fractional Brownian motion with Hurst index \$H ge 1/2\$. Denote by \$x_1 < x_2 < cdots\$ the positive, real zeros of the Bessel function \$J_{-H\$ of the first kind of order \$-H\$, and by \$y_1 < y_2 < cdots\$ the positive zeros of \$J_{1-H\$. We prove the series representation begin{equation* B_t = sum_{n=1^infty frac{sin x_n t{x_n, X_n + sum_{n=1^infty frac{1-cos y_n t{y_n, Y_n, end{equation* where \$X_1, X_2, ldots\$ and \$Y_1, Y_2, ldots\$ are independent, Gaussian random variables with mean zero and \$Var X_n = 2c_H^2x_n^{-2HJ^{-2_{1-H(x_n)\$, \$Var Y_n = 2c_H^2y_n^{-2HJ^{-2_{-H(y_n)\$, where the constant \$c_H^2\$ is defined by \$c_H^2 = pi^{-12HGamma(2H)sin pi H\$. With probability \$1\$, the random series converges absolutely and uniformly in \$t in [0,1]\$. To keep the exposition transparant, we deliberately exclude the case \$H < 1/2\$. The expansion is still valid in this case, but the proof requires additional technicalities.

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CWI
CWI. Probability, Networks and Algorithms [PNA]
Stochastics

Dzhaparidze, K.O, & van Zanten, J.H. (2002). A series expansion of fractional Brownian motion with Hurst index exceeding 1/2. CWI. Probability, Networks and Algorithms [PNA]. CWI.