Abstract
Consider the fractional Brownian motion process $B_H(t), t\in [0,T]$, with parameter $H\in (0,1)$. Meyer, Sellan and Taqqu have developed several random wavelet representations for $B_H(t)$, of the form $\sum_{k=0}^\infty U_k(t)\epsilon_k$ where $\epsilon_k$ are Gaussian random variables and where the functions $U_k$ are not random. Based on the results of Kühn and Linde, we say that the approximation $\sum_{k=0}^n U_k(t)\epsilon_k$ of $B_H(t)$ is optimal if $$ \displaystyle \left( E \sup_{t\in [0,T]} \left| \sum_{k=n}^\infty U_k(t) \epsilon_k\right|^2 \right)^{1/2} =O \left( n^{-H} (1+\log n)^{1/2} \right), $$ as $n\rightarrow\infty$. We show that the random wavelet representations given in Meyer, Sellan and Taqqu are optimal.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yves Meyer.
Rights and permissions
About this article
Cite this article
Ayache, A., Taqqu, M. Rate Optimality of Wavelet Series Approximations of Fractional Brownian Motion. J. Fourier Anal. Appl. 9, 451–471 (2003). https://doi.org/10.1007/s00041-003-0022-0
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00041-003-0022-0