On non-standard limits of Brownian semi-stationary processes

In this paper we present some new asymptotic results for high frequency statistics of Brownian semi-stationary (BSSBSS) processes. More precisely, we will show that singularities in the weight function, which is one of the ingredients of a BSSBSS process, may lead to non-standard limits of the realised quadratic variation. In this case the limiting process is a convex combination of shifted integrals of the intermittency function. Furthermore, we will demonstrate the corresponding stable central limit theorem. Finally, we apply the probabilistic theory to study the asymptotic properties of the realised ratio statistics, which estimates the smoothness parameter of a BSSBSS process.