A weak law of large numbers for estimating the correlation in bivariate Brownian semistationary processes

TL;DR

This paper establishes weak laws of large numbers for the realized covariation of certain bivariate stochastic processes, showing convergence to the integrated stochastic correlation in non-semimartingale settings.

Contribution

It introduces new weak laws of large numbers for realized covariation in bivariate processes that are not semimartingales, including moving average and Brownian semistationary processes.

Findings

Realized covariation converges to stochastic correlation.

Results apply to non-semimartingale processes.

Provides theoretical foundation for estimating stochastic correlation.

Abstract

This article presents various weak laws of large numbers for the so-called realised covariation of a bivariate stationary stochastic process which is not a semimartingale. More precisely, we consider two cases: Bivariate moving average processes with stochastic correlation and bivariate Brownian semistationary processes with stochastic correlation. In both cases, we can show that the (possibly scaled) realised covariation converges to the integrated (possibly volatility modulated) stochastic correlation process.