On estimation of quadratic variation for multivariate pure jump semimartingales

TL;DR

This paper analyzes the asymptotic behavior of realized quadratic variation for multivariate pure jump semimartingales, deriving limit theorems and proposing methods for eigenvalue estimation and confidence region construction.

Contribution

It extends univariate results to multivariate processes, deriving functional limit theorems and introducing a subsampling method for inference in the Lévy process setting.

Findings

Limit process is a matrix-valued β-stable Lévy process for symmetric β-stable processes.

Derived asymptotic distributions for eigenvalues and eigenvectors of the quadratic variation matrix.

Proposed a consistent subsampling procedure for confidence regions in Lévy processes.

Abstract

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.