  In this paper we present some new asymptotic results for high frequency
statistics of Brownian semi-stationary processes. More precisely, we will show
that singularities in the weight function, which is one of the ingredients of a
BSS 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 BSS process.
