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

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 [4,5] 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.Keywords: Brownian semi-stationary processes, high frequency data, limit theorems, stable convergence, turbulence.AMS 2010 subject classifications. Primary 60F05, 60F15, 60F17; Secondary 60G48, 60H05. t −∞ g(t − s)σ s W (ds) + t −∞ q(t − s)a s ds

Hybrid scheme for Brownian semistationary processes

“…. , m, of the BSS process X given by (2.1), for some m ∈ N. Barndorff-Nielsen et al [6] and Corcuera et al [16] discuss how the roughness index α can be estimated consistently as m → ∞. The method is based on the change-of-frequency (COF) statistics…”

“…By now these processes have been applied in various contexts, most notably in the study of turbulence in physics [7,16] and in finance as models of energy prices [4,11]. A BSS process X is defined via the integral representation…”

A Weak Limit Theorem for Numerical Approximation of Brownian Semi-stationary Processes

“…In the past years stochastic analysis, probabilistic properties and statistical inference for Lévy semi-stationary processes have been studied in numerous papers. We refer to [2,3,6,7,11,12,15,17,20,25] for the mathematical theory as well as to [5,26] for a recent survey on theory of ambit fields and their applications. For practical applications in sciences numerical approximation of Lévy (Brownian) semi-stationary processes, or, more generally, of ambit fields, is an important issue.…”

“…When introducing the approximation at (16), we obviously obtain two types of error: The Riemann sum approximation error and tail approximation error. Condition (17) guarantees that the Riemann sum approximation error will dominate.…”

“…where κ x (y) = h λ (y) cos(yx) and the approximation follows by the inequality |κ x (y)| ≤ | h λ (y)| and condition (17). Let us denote by κ x (y) the derivative of κ x (y) with respect to y.…”