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
The performance of a wind turbine in terms of power production (the power curve) is important to the wind energy industry. The current International Electrotechnical Commission 61400-12-1 standard for power curve evaluation recognizes only the mean wind speed at hub height and the air density as relevant to the power production. However, numerous studies have shown that the power production depends on several variables, in particular turbulence intensity. This paper presents a model and a method that are computationally tractable and able to account for some of the influence of turbulence intensity on the power production. The model and method are parsimonious in the sense that only a single function (the zero-turbulence power curve) and a single auxiliary parameter (the equivalent turbulence factor) are needed to predict the mean power at any desired turbulence intensity. The method requires only 10 min statistics but can be applied to data of a higher temporal resolution as well.
We discuss continuous cascade models and their potential for modelling the energy dissipation in a turbulent flow. Continuous cascade processes, expressed in terms of stochastic integrals with respect to Lévy bases, are examples of ambit processes. These models are known to reproduce experimentally observed properties of turbulence: The scaling and self-scaling of the correlators of the energy dissipation and of the moments of the coarse-grained energy dissipation. We compare three models: a normal model, a normal inverse Gaussian model and a stable model. We show that the normal inverse Gaussian model is superior to both, the normal and the stable model, in terms of reproducing the distribution of the energy dissipation; and that the normal inverse Gaussian model is superior to the normal model and competitive with the stable model in terms of reproducing the self-scaling exponents. Furthermore, we show that the presented analysis is parsimonious in the sense that the self-scaling exponents are predicted from the one-point distribution of the energy dissipation, and that the shape of these distributions is independent of the Reynolds number. arXiv:1305.0690v2 [cond-mat.stat-mech]
Some of the recent developments in the rapidly expanding field of Ambit Stochastics are here reviewed. After a brief recall of the framework of Ambit Stochastics, two topics are considered: (i) Methods of modelling and inference for volatility/intermittency processes and fields; (ii) Universal laws in turbulence and finance in relation to temporal processes. This review complements two other recent expositions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.