A central limit theorem for the realised covariation of a bivariate Brownian semistationary process

Abstract

This article presents a weak law of large numbers and a central limit theorem for the scaled realised covariation of a bivariate Brownian semistationary process. The novelty of our results lies in the fact that we derive the suitable asymptotic theory both in a multivariate setting and outside the classical semimartingale framework. The proofs rely heavily on recent developments in Malliavin calculus.

Limit theorems for multivariate Brownian semistationary processes and feasible results

“…However, this is true under Assumption 3.1 by using the same computations as carried out in the proof of Theorem 3.2 of [17]. Let us sketch them.…”

“…Moreover, we will adopt the following three assumptions. These are the analogues of Assumption (CLT) and conditions in Section 4.1 of [7], and Assumptions 2.1, 2.2, 4.1, and 4.2 of [17]. The only difference is that they also focus on the convergence of the autocorrelations r (n) , which for the sake of brevity and clarity of exposition we decided not to focus on in this work.…”

“…In Section 6 we will show that these assumptions are satisfied for an important and practical case which is found in many real-world applications: the gamma kernel. Finally, since the assumptions of this work and of [7] and [17] are similar, we refer the reader to Section 4.3 of [7] and Section 2.1 of [17] for further discussions on the assumptions.…”

“…New results based on different mathematical tools, mainly those presented in the works of Peccati, Nourdin, and coauthors (see [20] and the references therein), have been obtained. Barndorff-Nielsen et al [7] presented the multipower variation for BSS processes, while Granelli and Veraart [17] obtained the realised covariation for the bivariate BSS without drift. It is important to mention that in the general multivariate setting we have the work [3] for the semimartingale case, but no work for the case of BSS processes outside the semimartingale framework.…”

“…We remark that, despite the more general theory developed here, no additional assumptions will be added other than those already introduced in [7] and [17] (but used in a multivariate setting).…”