This thesis analysis the use of Brownian semi-stationary (BSS) processes to model the main statistical features present in turbulent time series, and some asymptotic properties of certain classes of smooth processes.
Turbulence is a complex phenomena governed by the Navier-Stokes equations. These equations do not represent a fully functional model and, consequently, it has been necessary to develop phenomenological models capturing main aspects of turbulent dynamics. The BSS processes were proposed as an option to model turbulent time series. In this thesis we proved, through a simulation-based approach, the potential of BSS processes to model turbulent velocity time series. It turns out that this family of processes reproduces accurately some of the main features present in turbulent time series, such as the distribution of the velocity increments and the statistics of the Kolmogorov variable. We also studied the distributional properties of the increments of BSS processes with the intent to better understand why the BSS processes seem to accurately reproduce the temporal turbulent dynamics.
BSS processes in general are not semimartingales. However, there are conditions which make a BSS process a bounded variation process with differentiable paths. It is natural to inquire if it is possible to obtain an asymptotic theory for this class of BSS processes. This problem is investigated and some partial results are presented. The asymptotic theory for BSS processes naturally leads to the study of the same problem for multiple Lebesgue integrals of Brownian motion. This thesis also presents some research about the asymptotic problem in the context of integrals of Brownian motion.