Asymptotic behaviours for the trajectory fitting estimator in Ornstein–Uhlenbeck process with linear drift

We study the asymptotic behaviours for the trajectory fitting estimator in the Ornstein–Uhlenbeck process with linear drift. Deviation inequality, moderate deviations, Berry–Esseen bound and the law of iterated logarithm (LIL) of this estimator can be obtained. Moreover, as an application of the Berry–Esseen bound, we can get the precise rate in LIL. The main method of this paper is the deviation inequality for multiple Wiener-Itô integrals P. Major, On a multivariate version of Bernsteins inequality. Electron. J. Prob. 12 (2007), pp. 966–988; P. Major, Tail behavior of multiple integrals and U-statistics. Probab. Surv. 2 (2005), pp. 448–505].