In this paper we present the asymptotic analysis of the realised quadratic variation for multivariate symmetric $\beta$-stable Lévy 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 Lévy process when the original process is symmetric $\beta$-stable, while the limit is conditionally $\beta$-stable in case of integrals with respect to locally $\beta$-stable motions. These asymptotic results are mostly related to the work (Diop et al., 2013), 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 Lévy setting to obtain confidence regions.

在本文中，我们提出了多元对称的已实现二次方差的渐近分析 $\beta$稳定的Lévy流程， $\beta \in （0，\mathrm{2个}）$，以及某些纯跳半mart。主要重点是针对已实现的二次方差及其谱的函数极限定理的推导。我们将证明限制过程是一个矩阵值$\beta$原始过程对称时的Lévy稳定过程 $\beta$-稳定，而限制是有条件的 $\beta$在局部积分的情况下是稳定的 $\beta$稳定的动作。这些渐近结果主要与工作有关（Diop等，2013），该工作调查了问题的单变量版本。此外，我们将展示二次方差矩阵的特征值和特征向量估计的意义，这对于主成分分析是有用的。最后，我们建议在Lévy设置中采用一致的二次采样程序来获得置信区域。