  In this paper we present the asymptotic analysis of the realised quadratic
variation for multivariate symmetric $\beta$-stable L\'evy processes, $\beta
\in (0,2)$, and certain pure jump semimartingales. The main focus is on
derivation of functional limit theorems for the realised quadratic variation
and its spectrum. We will show that the limiting process is a matrix-valued
$\beta$-stable L\'evy process when the original process is symmetric
$\beta$-stable, while the limit is conditionally $\beta$-stable in case of
integrals with respect to symmetric $\beta$-stable motions. These asymptotic
results are mostly related to the work [5], which investigates the univariate
version of the problem. Furthermore, we will show the implications for
estimation of eigenvalues and eigenvectors of the quadratic variation matrix,
which is a useful result for the principle component analysis. Finally, we
propose a consistent subsampling procedure in the L\'evy setting to obtain
confidence regions.
