Corcuera, José Manuel4; Hedevang, Emil5; Pakkanen, Mikko S.6; Podolskij, Mark5
1 Department of Economics and Business Economics - Center for Research in Econometric Analysis of Time Series (CREATES), Department of Economics and Business Economics, Aarhus BSS, Aarhus University2 Department of Mathematics, Science and Technology, Aarhus University3 Department of Economics and Business Economics, Aarhus BSS, Aarhus University4 Universiat de Barcelona5 Department of Mathematics, Science and Technology, Aarhus University6 Department of Economics and Business Economics, Aarhus BSS, Aarhus University
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.
Stochastic Processes and Their Applications, 2013, Vol 123, Issue 7, p. 2552-2574
Brownian semi-stationary processes; High frequency data; Limit theorems; Stable convergence; Turbulence