Limit Theorems for Functionals of Higher Order Differences of Brownian Semi-Stationary Processes

CREATES Research Paper No. 2009-60

28 Pages Posted: 29 Dec 2009

See all articles by Ole E. Barndorff-Nielsen

Ole E. Barndorff-Nielsen

University of Aarhus - Thiele Centre, Department of Mathematical Sciences

José Manuel Corcuera

University of Barcelona - Faculty of Mathematics

Mark Podolskij

Aarhus University - School of Business and Social Sciences

Date Written: December 28, 2009

Abstract

We present some new asymptotic results for functionals of higher order differences of Brownian semi-stationary processes. In an earlier work we have derived a similar asymptotic theory for first order differences. However, the central limit theorems were valid only for certain values of the smoothness parameter of a Brownian semistationary process, and the parameter values which appear in typical applications, e.g. in modeling turbulent flows in physics, were excluded.

The main goal of the current paper is the derivation of the asymptotic theory for the whole range of the smoothness parameter by means of using second order differences. We present the law of large numbers for the multipower variation of the second order differences of Brownian semi-stationary processes and show the associated central limit theorem. Finally, we demonstrate some estimation methods for the smoothness parameter of a Brownian semi-stationary process as an application of our probabilistic results.

Keywords: Brownian semi-stationary processes, central limit theorem, Gaussian processes, high frequency observations, higher order differences, multipower variation, stable convergence

JEL Classification: C10, C13, C14

Suggested Citation

Barndorff-Nielsen, Ole E. and Corcuera Valverde, José Manuel and Podolskij, Mark, Limit Theorems for Functionals of Higher Order Differences of Brownian Semi-Stationary Processes (December 28, 2009). CREATES Research Paper No. 2009-60, Available at SSRN: https://ssrn.com/abstract=1528887 or http://dx.doi.org/10.2139/ssrn.1528887

Ole E. Barndorff-Nielsen (Contact Author)

University of Aarhus - Thiele Centre, Department of Mathematical Sciences ( email )

Ny Munkegade
Aarhus, DK 8000
Denmark

José Manuel Corcuera Valverde

University of Barcelona - Faculty of Mathematics ( email )

Gran Via de les Corts
Catalanes 585
Barcelona 08007
Spain

Mark Podolskij

Aarhus University - School of Business and Social Sciences ( email )

Building 350
DK-8000 Aarhus C
Denmark

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