Laws of large numbers for Hayashi–Yoshida-type functionals

The main object in the statistical analysis of high-frequency financial data are sums of functionals of increments of stochastic processes, 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 based on two asynchronously observed stochastic processes, we investigate similar sums for general functionals. We find that our results differ from corresponding results for synchronous observations, a case which has been well studied in the literature, and we observe that the asymptotic behaviour in the setting of asynchronous observations is not only determined by the nature of the functional, but also depends crucially on the asymptotics of the observation scheme. Several examples are discussed, including the case of f ( x 1 , x 2 ) = | x 1 | p 1 | x 2 | p 2 $f(x_{1},x_{2}) = |x_{1}|^{p_{1}} |x_{2}|^{p_{2}}$ which has various applications in empirical finance.