  In this article we will introduce the realised semicovariance for Brownian
semistationary (BSS) processes, which is obtained from the decomposition of the
realised covariance matrix into components based on the signs of the returns,
and study its in-fill asymptotic properties. More precisely, a weak convergence
in the space of c\`adl\`ag functions endowed with the Skorohod topology for the
realised semicovariance of a general Gaussian process with stationary
increments is proved first. The methods are based on Breuer-Major theorems and
on a moment bound for sums of products of Gaussian vector's functions.
Furthermore, we establish a corresponding stable convergence. Finally, a weak
law of large numbers and a central limit theorem for the realised
semicovariance of multivariate BSS processes are established. These results
extend the limit theorems for the realised covariation to a result for
non-linear functionals.
