Pre-averaging fractional processes contaminated by noise, with an application to turbulence

In this article, we consider the problem of estimating fractional processes based on noisy high-frequency data. Generalizing the idea of pre-averaging to a fractional setting, we exhibit a sequence of consistent estimators for the unknown parameters of interest by proving a law of large numbers for associated variation functionals. In contrast to the semimartingale setting, the optimal window size for pre-averaging depends on the unknown roughness parameter of the underlying process. We evaluate the performance of our estimators in a simulation study and use them to empirically verify Kolmogorov's 2/3-law in turbulence data contaminated by instrument noise.