Abstract
We present some new asymptotic results for functionals of higher order differences of Brownian semi-stationary processes. In an earlier work [8] 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 semi-stationary 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.
Mathematics Subject Classification (2010): Primary 60F05, 60G15, 62M09; Secondary 60G22, 60H07
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Acknowledgements
Ole Barndorff-Nielsen and Mark Podolskij gratefully acknowledge financial support from CREATES funded by the Danish National Research Foundation.
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Barndorff-Nielsen, O.E., Corcuera, J.M., Podolskij, M. (2013). Limit Theorems for Functionals of Higher Order Differences of Brownian Semi-Stationary Processes. In: Shiryaev, A., Varadhan, S., Presman, E. (eds) Prokhorov and Contemporary Probability Theory. Springer Proceedings in Mathematics & Statistics, vol 33. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33549-5_4
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DOI: https://doi.org/10.1007/978-3-642-33549-5_4
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