Asymptotic theory for Brownian semi-stationary processes with application to turbulence

TL;DR

This paper develops asymptotic results for power variations of Brownian semi-stationary processes, introduces new estimators for their smoothness, and applies these methods to turbulence data analysis.

Contribution

It provides new connections to fractional diffusion models and develops estimators with gaps, enabling valid central limit theorems for turbulence data analysis.

Findings

Asymptotic limit theorems for power variations of BSS processes.

New estimators for the smoothness parameter with proven CLTs.

Application of the theory to turbulence data.

Abstract

This paper presents some asymptotic results for statistics of Brownian semi-stationary (BSS) processes. More precisely, we consider power variations of BSS processes, which are based on high frequency (possibly higher order) differences of the BSS model. We review the limit theory discussed in [Barndorff-Nielsen, O.E., J.M. Corcuera and M. Podolskij (2011): Multipower variation for Brownian semistationary processes. Bernoulli 17(4), 1159-1194; Barndorff-Nielsen, O.E., J.M. Corcuera and M. Podolskij (2012): Limit theorems for functionals of higher order differences of Brownian semi-stationary processes. In "Prokhorov and Contemporary Probability Theory", Springer.] and present some new connections to fractional diffusion models. We apply our probabilistic results to construct a family of estimators for the smoothness parameter of the BSS process. In this context we develop estimates with gaps, which allow to obtain a valid central limit theorem for the critical region. Finally, we apply our statistical theory to turbulence data.