Theory of Probability and Mathematical Statistics
Fractional Brownian motion: Optimality of random series representation via Haar basis
Antoine Ayache and Florent Bouly
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Abstract: A long time ago, it was shown that random series representations of fractional Brownian motion (fBm) generated by smooth orthonormal wavelet bases of L2(ℝ) are optimal, in the sense that they converge at the best possible rate provided by the sequence of the l-approximation numbers of this process. The goal of our article is to extend this optimality result to the random series representation of fBm generated by the Haar basis of L2(ℝ), which offers the advantage to provide explicit results and simple computations. One of the main challenges in this context is that, in contrast to smooth wavelet functions, fractional primitives of the discontinuous Haar functions are badly localized. In order to overcome it, we will use a new strategy relying on Abel transform.
Keywords: Approximation of processes, approximation numbers, wavelet bases, Haar basis, Abel transform
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