Moment inequalities for mixing long-span high-frequency data and strongly consistent estimation of OU integrated diffusion process

Mixing is not much used in the high-frequency literature so far. However, mixing is a common weakly dependent property of continuous and discrete stochastic processes, such as Gaussian, Ornstein-Uhlenberck (OU), Vasicek, CIR, CKLS, logistic diffusion, generalized logistic diffusion, and double-well diffusion processes. So, long-span high-frequency data typically have weak dependence, and using mixing to study them is also an alternative approach. In this paper, we give some moment inequalities for long-span high-frequency data with phi-mixing, rho-mixing, and alpha-mixing. These inequalities are effective tools for studying asymptotic properties. Applying these inequalities, we investigate the strong consistency of parameter estimation for the OU-integrated diffusion process. We also derive the mean square error of the estimation of the OU process and the optimal interval for the drift parameter estimator.