An Optimal Wavelet Series Expansion of the Riemann–Liouville Process

Abstract

Consider the Riemann–Liouville process R α ={R α (t)} t∈[0,1] with parameter α>1/2. Depending on α, wavelet series representations for R α (t) of the form ∑ k=1 ∞ u k (t)e k are given, where the u k are deterministic functions, and {e k } k≥1 is a sequence of i.i.d. standard normal random variables. The expansion is based on a modified Daubechies wavelet family, which was originally introduced in Meyer (Rev. Mat. Iberoam. 7:115–133, 1991). It is shown that these wavelet series representations are optimal in the sense of Kuhn–Linde (Bernoulli 8:669–696, 2002) for all values of α>1/2.