Laws of large numbers for Hayashi-Yoshida-type functionals

TL;DR

This paper extends the laws of large numbers to Hayashi-Yoshida-type functionals for asynchronously observed stochastic processes, revealing that asymptotic behavior depends on both the functional and observation scheme.

Contribution

It introduces new asymptotic results for sums of functionals of asynchronously observed processes, generalizing the Hayashi-Yoshida estimator.

Findings

Asymptotic behavior differs from equidistant synchronous cases.

Dependence on observation scheme asymptotics is crucial.

Results apply to general functionals beyond quadratic covariation.

Abstract

In high-frequency statistics and econometrics sums of functionals of increments of stochastic processes are commonly used and statistical inference is based on the asymptotic behaviour of these sums as the mesh of the observation times tends to zero. Inspired by the famous Hayashi-Yoshida estimator for the quadratic covariation process based on two asynchronously observed stochastic processes we investigate similar sums based on increments of two asynchronously observed stochastic processes for general functionals. We find that our results differ from corresponding results in the setting of equidistant and synchronous observations which has been well studied in the literature. Further we observe that in the setting of asynchronous observations the asymptotic behaviour is not only determined by the nature of the functional but also depends crucially on the asymptotics of the observation scheme.