  We will focus on estimating the integrated covariance of two diffusion
processes observed in a nonsynchronous manner. The observation data is
contaminated by some noise, which is possibly correlated with the returns of
the diffusion processes, while the sampling times also possibly depend on the
observed processes. In a high-frequency setting, we consider a modified version
of the pre-averaged Hayashi-Yoshida estimator, and we show that such a kind of
estimators has the consistency and the asymptotic mixed normality, and attains
the optimal rate of convergence.
