High-frequency financial data are not only discretely sampled in time but the time separating successive observations is often random. We review the paper of Aït-Sahalia and Mykland (2003), that measure the effects of discreteness sampling and ignoring the randomness of the sampling for estimating the m.l.e of a continuous-time diffusion model. In that article, three different assumptions and restrict in one made on the sampling intervals, and the corresponding likelihood function, asymptotic normality, and covariance matrix are obtained. It is concluded that the effects due to discretely sampling are smaller than the effect of simply ignoring the sampling randomness. This study focuses on rechecking the results in the paper of A¡Lıt-Sahalia and Mykland (2003) including theory, simulation and application. We derive a different likelihood function expression from A¡Lıt-Sahalia and Mykland (2003)¡¦s result. However, the asymptotic covariance are consistent for both approaching in the O-U process. Furthermore, we conduct an empirical study on the high frequency transaction time data by using non-homogeneous Poisson Processes.