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

Limit theorems for the realised semicovariances of multivariate Brownian semistationary processes