On the singularity of least squares estimator for mean-reverting α-stable motions

We study the problem of parameter estimation for mean-reverting a-stable motion, dXt = (a0 - ϑ0Xt)dt + dZt observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discussed in the singular case (a0, ϑ0) = (0, 0). If a0 = 0, then the mean-reverting a-stable motion becomes Ornstein-Uhlenbeck process and is studied in [7] in the ergodic case ϑ0 > 0. For the Ornstein-Uhlenbeck process, asymptotics of the least squares estimators for the singular case (ϑ0 = 0) and for ergodic case (ϑ0 > 0) are completely different.