## Bernoulli

- Bernoulli
- Volume 25, Number 3 (2019), 2245-2278.

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

Andrea Granelli and Almut E.D. Veraart

#### 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.

#### Article information

**Source**

Bernoulli, Volume 25, Number 3 (2019), 2245-2278.

**Dates**

Received: August 2017

Revised: June 2018

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1560326444

**Digital Object Identifier**

doi:10.3150/18-BEJ1052

**Mathematical Reviews number (MathSciNet)**

MR3961247

**Zentralblatt MATH identifier**

07066256

**Keywords**

bivariate Brownian semistationary process central limit theorem fourth moment theorem high frequency data moving average process multivariate setting stable convergence

#### Citation

Granelli, Andrea; Veraart, Almut E.D. A central limit theorem for the realised covariation of a bivariate Brownian semistationary process. Bernoulli 25 (2019), no. 3, 2245--2278. doi:10.3150/18-BEJ1052. https://projecteuclid.org/euclid.bj/1560326444

#### Supplemental materials

- Supplement to “A central limit theorem for the realised covariation of a bivariate Brownian semistationary process”. We collect technical details and proofs in the supplementary article, which should be read in conjunction with the present paper.Digital Object Identifier: doi:10.3150/18-BEJ1052SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.