In this paper we study the asymptotic behaviour of power and multipower variations of processes Y :where g : (0, ∞) → R is deterministic, σ > 0 is a random process, W is the stochastic Wiener measure and Z is a stochastic process in the nature of a drift term. Processes of this type serve, in particular, to model data of velocity increments of a fluid in a turbulence regime with spot intermittency σ . The purpose of this paper is to determine the probabilistic limit behaviour of the (multi)power variations of Y as a basis for studying properties of the intermittency process σ . Notably the processes Y are in general not of the semimartingale kind and the established theory of multipower variation for semimartingales does not suffice for deriving the limit properties. As a key tool for the results, a general central limit theorem for triangular Gaussian schemes is formulated and proved. Examples and an application to the realised variance ratio are given.

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

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