Limit Theorems for Functionals of Higher Order Differences of Brownian Semi-Stationary Processes
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 w
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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. Keywords Brownian semi-stationary processes • Central limit theorem • Gaussian processes • High frequency observations • Higher order differences • Multipower variation • Stable convergence
O.E. Barndorff-Nielsen () Department of Mathematics, University of Aarhus, Ny Munkegade, DK–8000 Aarhus C, Denmark e-mail: [email protected] J.M. Corcuera Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain e-mail: [email protected] M. Podolskij Department of Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 4, © Springer-Verlag Berlin Heidelberg 2013
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Mathematics Subject Classification (2010): Primary 60F05, 60G15, 62M09; Secondary 60G22, 60H07
1 Introduction Brownian semi-stationary processes (BS S ) has been originally introduced in [2] for modeling turbulent flows in physics. This class consists of processes .Xt /t 2R of the form Z t Z t g.t s/s W .ds/ C q.t s/as ds; (1) Xt D C 1
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where is a constant, g; q W R>0 ! R are memory functions, .s /s2R is a c`adl`ag intermittency process, .as /s2R a c`adl`ag drift process and W is the Wiener measure. When .s /s2R and .as /s2R are stationary then the process .Xt /t 2R is also stationary, which explains the name Brownian semi-stationary processes. In the following we concentrate on BS S models without the drift part (i.e. a 0), but we come back to the original process (1) in Example 1. The path properties of the process .Xt /t 2R crucially depend on the behaviour of the weight function g near 0. When g.x/ ' x ˇ (here g.x/ ' h.x/ means that g.x/= h.x/ is slowly varying at 0) with ˇ 2 . 12 ; 0/ [ .0; 12 /, X has r-H¨older continuous paths for any r < ˇ C 12 and, more importantly, X is not a semimartingale, because g 0 is not square integrable in the neighborhood of 0 (see e.g. [1
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