Limit theorems for multivariate Brownian semistationary processes and feasible results

TL;DR

This paper introduces multivariate Brownian semistationary processes, analyzes their asymptotic behavior, and provides limit theorems and feasible results, including an example with gamma kernels.

Contribution

It extends the theory of BSS processes to the multivariate case and establishes new limit theorems for their realized covariation.

Findings

Central limit theorem for multivariate Gaussian processes

Weak laws of large numbers for BSS processes

Feasible results with explicit gamma kernel example

Abstract

In this paper we introduce the \textit{multivariate} Brownian semistationary (BSS) processes and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general stationary multivariate Gaussian processes, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.